the ricci flow an introduction标准

MathematicalSurveys

andMonographs

Volume I10

the Ricci Flow:An Introduction

Bennett ChowDan Knopf

A; I American mathematical Society

EDITORIAL COMMITTEEJerry L. Bona Michael P. LossPeter S. Landweber, Chair Tudor Stefan Ratiu

J. T. Stafford

2000 Mathematics Subject Classification. Primary 53C44, 53C21, 58J35, 53C25, 35K55,57M50, 35K57.

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Library of Congress Cataloging-in-Publication DataChow, Bennett.The Ricci flow: an introduction/Bennett Chow, Dan Knopf.

p.cm.- (Mathematical surveys and monographs, ISSN 0076-5376; v. 110)Includes bibliographical references and index.ISBN 0-8218-3515-7 (alk. paper)1. Global differential geometry. 2. Ricci flow. 3. Riemannian manifolds. I. Knopf, Dan,

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Contents

Preface viiA guide for the reader viiiA guide for the hurried reader xAcknowledgments xi

Chapter 1. The Ricci flow of special geometries 1

1. Geometrization of three-manifolds 2

2. Model geometries 43. Classifying three-dimensional maximal model geometries 6

4. Analyzing the Ricci flow of homogeneous geometries 8

5. The Ricci flow of a geometry with maximal isotropy SO (3) 11

6. The Ricci flow of a geometry with isotropy SO (2) 15

7. The Ricci flow of a geometry with trivial isotropy 17

Notes and commentary 19

Chapter 2. Special and limit solutions 211. Generalized fixed points 21

2. Eternal solutions 243. Ancient solutions 284. Immortal solutions 345. The neckpinch 386. The degenerate neckpinch 62


Chapter 3. Short time existence 671. Variation formulas 672. The linearization of the Ricci tensor and its principal symbol 71

3. The Ricci-DeTurck flow and its parabolicity 784. The Ricci-DeTurck flow in relation to the harmonic map flow 845. The Ricci flow regarded as a heat equation 90Notes and commentary 92

Chapter 4. Maximum principles 931. Weak maximum principles for scalar equations 932. Weak maximum principles for tensor equations 973. Advanced weak maximum principles for systems 1004. Strong maximum principles 102

Hi

iv CONTENTS


Chapter 5. The Ricci flow on surfaces 1051. The effect of a conformal change of metric 1062. Evolution of the curvature 1093. How Ricci solitons help us estimate the curvature from above 1114. Uniqueness of Ricci solitons 1165. Convergence when X (M2) < 0 120

6. Convergence when X (M2) = 0 1237. Strategy for the case that X (M2 > 0) 1288. Surface entropy 1339. Uniform upper bounds for R and IVRI 13710. Differential Harnack estimates of LYH type 14311. Convergence when R (., 0) > 0 14812. A lower bound for the injectivity radius 14913. The case that R (-, 0) changes sign 15314. Monotonicity of the isoperimetric constant 15615. An alternative strategy for the case X (M2 > 0) 165Notes and commentary 171

Chapter 6. Three-manifolds of positive Ricci curvature 1731. The evolution of curvature under the Ricci flow 1742. Uhlenbeck's trick 1803. The structure of the curvature evolution equation 1834. Reduction to the associated ODE system 1875. Local pinching estimates 1896. The gradient estimate for the scalar curvature 1947. Higher derivative estimates and long-time existence 2008. Finite-time blowup 2099. Properties of the normalized Ricci flow 21210. Exponential convergence 218Notes and commentary 221

Chapter 7. Derivative estimates 2231. Global estimates and their consequences 2232. Proving the global estimates 2263. The Compactness Theorem 231Notes and commentary 232

Chapter 8. Singularities and the limits of their dilations 2331. Classifying maximal solutions 2332. Singularity models 2353. Parabolic dilations 2374. Dilations of finite-time singularities 2405. Dilations of infinite-time singularities 2466. Taking limits backwards in time 250

CONTENTS v


Chapter 9. Type I singularities 2531. Intuition 2532. Positive curvature is preserved 2553. Positive sectional curvature dominates 2564. Necklike points in finite-time singularities 2625. Necklike points in ancient solutions 2716. Type I ancient solutions on surfaces 274Notes and commentary 277

Appendix A. The Ricci calculus 2791. Component representations of tensor fields 2792. First-order differential operators on tensors 2803. First-order differential operators on forms 2834. Second-order differential operators 2845. Notation for higher derivatives 2856. Commuting covariant derivatives 286Notes and commentary 286

Appendix B. Some results in comparison geometry 2871. Some results in local geometry 2872. Distinguishing between local geometry and global geometry 2953. Busemann functions 3034. Estimating injectivity radius in positive curvature 312Notes and commentary 315

Bibliography 317

Index 323

Preface

This book and its planned sequel(s) are intended to compose an introduc-tion to the Ricci flow in general and in particular to the program originatedby Hamilton to apply the Ricci flow to approach Thurston's GeometrizationConjecture. The Ricci flow is the geometric evolution equation in which onestarts with a smooth Riemannian manifold (M'2, go) and evolves its metricby the equation

at g =-2 Rc,

where Rc denotes the Ricci tensor of the metric g. The Ricci flow wasintroduced in Hamilton's seminal 1982 paper, "Three-manifolds with pos-itive Ricci curvature". In this paper, closed 3-manifolds of positive Riccicurvature are topologically classified as spherical space forms. Until thattime, most results relating the curvature of a 3-manifold to its topology in-volved the influence of curvature on the fundamental group. For example,Gromov-Lawson and Schoen-Yau classified 3-manifolds with positive scalarcurvature essentially up to homotopy. Among the many results relating cur-vature and topology that hold in arbitrary dimensions are Myers' Theoremand the Cartan-Hadamard Theorem.

A large number of innovations that originated in Hamilton's 1982 andsubsequent papers have had a profound influence on modern geometric anal-ysis. Here we mention just a few. Hamilton's introduction of a nonlinearheat-type equation for metrics, the Ricci flow, was motivated by the 1964harmonic heat flow introduced by Eells and Sampson. This led to the re-newed study by Huisken, Ecker, and many others of the mean curvatureflow originally studied by Brakke in 1977. One of the techniques that hasdominated Hamilton's work is the use of the maximum principle, both forfunctions and for tensors. This technique has been applied to control vari-ous geometric quantities associated to the metric under the Ricci flow. Forexample, the so-called pinching estimate for 3-manifolds with positive cur-vature shows that the eigenvalues of the Ricci tensor approach each otheras the curvature becomes large. (This result is useful because the Ricciflow shrinks manifolds with positive Ricci curvature, which tends to makethe curvature larger.) Another curvature estimate, due independently toIvey and Hamilton, shows that the singularity models that form in dimen-sion three necessarily have nonnegative sectional curvature. (A singularity

vii

viii PREFACE

model is a solution of the flow that arises as the limit of a sequence of dila-tions of an original solution approaching a singularity.) Since the underlyingmanifolds of such limit solutions are topologically simple, a detailed analy-sis of the singularities which arise in dimension three is therefore possible.For example, showing that certain singularity models are shrinking cylin-ders is one of the cornerstones for enabling geometric-topological surgeriesto be performed on singular solutions to the Ricci flow on 3-manifolds. TheLi-Yau-Hamilton-type differential Harnack quantity is another innovation;this yields an a priori estimate for a certain expression involving the cur-vature and its first and second derivatives in space. This estimate allowsone to compare a solution at different points and times. In particular, itshows that in the presence of a nonnegative curvature operator, the scalarcurvature does not decrease too fast. Another consequence of Hamilton'sdifferential Harnack estimate is that slowly forming singularities (those inwhich the curvature of the original solution grows more quickly than theparabolically natural rate) lead to singularity models that are stationarysolutions.

The convergence theory of Cheeger and Gromov has had fundamentalconsequences in Riemannian geometry. In the setting of the Ricci flow,Gromov's compactness theorem may be improved to obtain C°° convergenceof a sequence of solutions to a smooth limit solution. For singular solutions,Perelman's recent No Local Collapsing Theorem allows one to dilate aboutsequences of points and times approaching the singularity time in such a waythat one can obtain a limit solution that exists infinitely far back in time.The analysis of such limit solutions is important in Hamilton's singularitytheory.

A guide for the reader

The reader of this book is assumed to have a basic knowledge of Rie-mannian geometry. Familiarity with algebraic topology and with nonlinearsecond-order partial differential equations would also be helpful but is notstrictly necessary.

In Chapters 1 and 2, we begin the study of the Ricci flow by consideringspecial solutions which exhibit typical properties of the Ricci flow and whichguide our intuition. In the case of an initial homogeneous metric, the solutionremains homogenous so that its analysis reduces to the study of a system ofODE. We study examples of such solutions in Chapter 1. Solutions whichexist on long time intervals such as those which exist since time -oo or untiltime +oo are very special and can appear as singularity models. In Chapter2, we both present such solutions explicitly and provide some intuition forhow they arise. An important example of a stationary solution is the so-called "cigar soliton" on R2; intuitive solutions include the neckpinch anddegenerate neckpinch. In Chapter 2, we discuss degenerate neckpinches

A GUIDE FOR THE READER ix

heuristically. We also rigorously construct neckpinch solutions under certainsymmetry assurriptions.

The short-time existence theorem for the Ricci flow with an arbitrarysmooth initial metric is proved in Chapter 3. This basic result allows one touse the Ricci flow as a practical tool. In particular, a number of smoothingresults in Riemannian geometry can be proved using the short-time existenceof the flow combined with the derivative estimates of Chapter 7. Since theRicci flow system of equations is only weakly parabolic, the short-time ex-istence of the flow does not follow directly from standard parabolic theory.Hamilton's original proof relied on the Nash-Moser inverse function theo-rem. Shortly thereafter, DeTurck gave a simplified proof by showing thatthe Ricci flow is equivalent to a strictly parabolic system.

In Chapter 4, the maximum principle for both functions and tensors ispresented. This provides the technical foundations for the curvature, gradi-ent of scalar curvature, and Li-Yau-Hamilton differential Harnack estimatesproved later in the book.

In Chapter 5, we give a comprehensive treatment of the Ricci flow onsurfaces. In this lowest nontrivial dimension, many of the techniques usedin three and higher dimensions are exhibited. The concept of stationarysolutions is introduced here and used to motivate the construction of var-ious quantities, including Li-Yau-Hamilton (LYH) differential Harnack es-timates. The entropy estimate is applied to solutions on the 2-sphere. Inaddition, derivative and injectivity radius estimates are first proved here. Wealso discuss the isoperimetric estimate and combine it with the Gromov-typecompactness theorem for solutions of the Ricci flow to rule out the cigar soli-ton as a singularity model for solutions on surfaces. In higher dimensions,the cigar soliton is ruled out by Perelman's No Local Collapsing Theorem.

The original topological classification by Hamilton of closed 3-manifoldswith positive Ricci curvature is proved in Chapter 6. Via the maximumprinciple for systems, the qualitative behavior of the curvature tensor maybe reduced in this case to the study of a system of three ODE in three un-knowns. Using this method, we prove the pinching estimate for the curvaturementioned above, which shows that the eigenvalues of the Ricci tensor areapproaching each other at points where the scalar curvature is becominglarge. This pinching estimate compares curvatures at the same point. Thenwe estimate the gradient of the scalar curvature in order to compare the cur-vatures at different points. The combination of these estimates shows thatthe Ricci curvatures are tending to constant. Using this fact together withestimates for the higher derivatives of curvature, we prove the convergenceof the volume-normalized Ricci flow to a spherical space form.

The derivative estimates of Chapter 7 show that assuming an initialcurvature bound allows one to bound all derivatives of the curvature for ashort time, with the estimate deteriorating as the time tends to zero. (Thisdeterioration of the estimate is necessary, because an initial bound on thecurvature alone does not imply simultaneous bounds on its derivatives.) The

x PREFACE

derivative estimates established in this chapter enable one to prove the long-time existence theorem for the flow, which states that a unique solution tothe Ricci flow exists as long as its curvature remains bounded.

In Chapter 8, we begin the analysis of singularity models by discussingthe general procedures for obtaining limits of dilations about a singular-ity. This involves taking a sequence of points and times approaching thesingularity, dilating in space and time, and translating in time to obtain asequence of solutions. Depending on the type of singularity, slightly differentmethods must be employed to find suitable sequences of points and timesso that one obtains a singularity model which can yield information aboutthe geometry of the original solution near the singularity just prior to itsformation.

In Chapter 9, we consider Type I singularities, those where the curvatureis bounded proportionally to the inverse of the time remaining until the sin-gularity time. This is the parabolically natural rate of singularity formationfor the flow. In this case, one sees the Ii8 X S2 cylinder (or its quotients) assingularity models. The results in this chapter provide additional rigor tothe intuition behind neckpinches.

In the two appendices, we provide elementary background for tensorcalculus and some comparison geometry.

REMARK. At present (December 2003) there has been sustained ex-citement in the mathematical community over Perelman's recent ground-breaking progress on Hamilton's Ricci flow program intended to resolveThurston's Geometrization Conjecture. Perelman's results allow one to viewsome of the material in Chapters 8 and 9 in a new light. We have retainedthat material in this volume for its independent interest, and hope to presentPerelman's work in a subsequent volume.

A guide for the hurried reader

The reader wishing to develop a nontechnical appreciation of the Ricciflow program for 3-manifolds as efficiently as possible is advised to followthe fast track outlined below.

In Chapter 1, read Section 1 for a brief introduction to the Geometriza-tion Conjecture.

In Chapter 2, the most important examples are the cigar, the neckpinchand the degenerate neckpinch. Read the discussion of the cigar soliton inSection 2. Read the statements of the main results on neckpinches derivedin Section 5. And read the heuristic discussion of degenerate neckpinches inSection 6.

In Chapter 3, review the variation formulas derived in Section 1.In Chapter 4, read the proof of the first scalar maximum principle (The-

orem 4.2) and at least the statements of the maximum principles that follow.In Chapter 5, review the entropy estimates derived in Section 8 and the

differential Harnack estimates derived in Section 10. The entropy estimates

ACKNOWLEDGMENTS xi

will be used in Section 6 of Chapter 9. The Harnack estimates for surfacesare prototypes of those that apply in higher dimensions.

We suggest that Chapter 6 be read in its entirety.In Chapter 7, read the statements of the main results. These give pre-

cise insight into the smoothing properties of the Ricci flow and its long-timebehavior. The Compactness Theorem stated in Section 3 is an essential in-gredient in important technique of analyzing singularity formation by takinglimits of parabolic dilations.

In Chapter 8, read Sections 1-3.1. These present the classification ofmaximal-time solutions to the flow, and introduce the method of parabolicdilation.

In Chapter 9, read Section 1 for a heuristic description of singularity for-mation in 3-manifolds. Then read Sections 2 and 3 to gain an understandingof why positive curvature dominates near singularities in dimension three.Finally, review the results of Sections 4 and 6 to gain insight into the tech-nique of dimension reduction.

Acknowledgments

During the preparation of this volume, Bennett Chow received supportfrom NSF grant DMS-9971891, while Dan Knopf was partially supportedby DMS-0328233.

B.C. would like to thank the following mathematicians from whom hehas benefitted both mathematically and personally. He thanks ProfessorShiing-Shen Chern whom he met in 1979, and who provided him with theinspiration to study differential geometry. From his graduate school days atPrinceton and San Diego: Huai-Dong Cao, Craig Hodgson, Nikos Kapouleas,Igor Rivin, Fadi Twainy, and Cumrun Vafa. From his postdoc years atMIT, Rice, and NYU: Zhiyong Gao, Victor Guillemin, Richard Melrose,Louis Nirenberg, Dan Stroock, and Deane Yang. From the years at Min-nesota: his colleagues Scot Adams, Bob Gulliver, Conan Leung, Wei-MingNi, and Jiaping Wang, as well as his former students Sun-Chin (Michael)Chu, Alexei Krioukov, Lii-Perng Liou, Dan O'Loughlin, Dong-Ho Tsai, andYu Yuan. From San Diego: his colleagues Li-Tien Cheng, Zheng-Xu (John)He, Lei Ni, and Nolan Wallach, as well as his former student David Glick-enstein. Special thanks to B.C.'s advisor, Professor Shing-Tung Yau, andteacher, Professor Richard Hamilton, for their invaluable teachings, support,and encouragement for more than 20 years, without which this book wouldnot have been written. Very special thanks to Classic Dimension for encour-agement, support and inspiration. B.C. is indebted to his parents for thelove, support, and encouragement they have given him all of these years, andto his father for teaching him mathematics beginning in his youth. B.C. ded-icates this book to his parents Yutze and Wanlin and to the memory of hissister Eleanor.

xii PREFACE

The authors would also like to thank Sigurd Angenent, Albert Chau,Hsiao-Bing Cheng, Christine Guenther, Joel Hass, Gerhard Huisken, TomIlmanen, Jim Isenberg, Tom Ivey, Peter Li, Chang-Shou Lin, Fang-HuaLin, Peng Lu, Feng Luo, Rick Schoen, Miles Simon, Gang Tian, Mu-TaoWang, Brian White, and Xi-Ping Zhu. They thank Helena Noronha for herencouragement, Ed Dunne and Sergei Gelfand at AMS for their help, andAndre Neves for proofreading and offering many useful suggestions.

In addition to the other mathematicians mentioned above, D.K. extendsspecial thanks to his advisor Kevin McLeod. D.K. thanks his parents for allthat they are, and especially his father for his loving patience and support.Thanks to Erik, Susie, Yoyi, Hilarie, and everyone at ERC. D.K. dedicatesthis book to his wife, Stephanie.

Bennett Chow and Dan Knopf

CHAPTER 1

The Ricci flow of special geometries

The Ricci flow

aatg=-2Rc

9 (0) = go

and its cousin the normalized Ricci flow

_ fMn R dµatg

-2 Rc --n9

fMn

dµ

g(0) =go

are methods of evolving the metric of a Riemannian manifold (.Ma, go) thatwere introduced by Hamilton in [58]. They differ only by a resealing of spaceand time. Hamilton has crafted a well-developed program to use these flowsto resolve Thurston's Geometrization Conjecture for closed 3-manifolds. Theintent of this volume is to provide a comprehensive introduction to the foun-dations of Hamilton's program. Perelman's recent ground-breaking work[105, 106, 107] is aimed at completing that program.

Roughly speaking, Thurston's Geometrization Conjecture says that anyclosed 3-manifold can be canonically decomposed into pieces in such a waythat each admits a unique homogeneous geometry. (See Section 1 below.)As we will learn in the chapters that follow, one cannot in general expecta solution (,M3,g (t)) of the Ricci flow starting on an arbitrary closed 3-manifold to converge to a complete locally homogeneous metric. Instead, onemust deduce topological and geometric properties of M3 from the behaviorof g (t). Hamilton's program outlines a highly promising strategy to do so.

By way of an intuitive introduction to this strategy, this chapter ad-dresses the following natural question:

If go is a complete locally homogeneous metric, how will g (t) evolve?

The observations we collect in examining this question are intended tohelp the reader develop a sense and intuition for the properties of the flowin these special geometries. While knowledge of the Ricci flow's behaviorin homogeneous geometries does not appear necessary for understandingits topological consequences, such knowledge is valuable for understandinganalytic aspects of the flow, particularly those related to collapse.

1

2 1. THE RICCI FLOW OF SPECIAL GEOMETRIES

REMARK 1.1. In dimension n = 3, the categories TOP, PL, and DIFF allcoincide. In this volume, any manifold under consideration - regardless ofits dimension - is assumed to be smooth (C°O). Moreover, unless explicitlystated otherwise, every manifold is assumed to be without boundary, so thatit is closed if and only if it is compact.

1. Geometrization of three-manifolds

To motivate our interest in locally homogeneous metrics on 3-manifolds,we begin with a heuristic discussion of the Geometrization Conjecture, whichwill be reviewed in somewhat more detail in the successor to this volume.(Good references are [115] and [123].) We begin that discussion with a briefreview of some basic 3-manifold topology. One says an orientable closedmanifold M3 is prime if M3 is not diffeomorphic to the 3-sphere S3 andif a connected sum decomposition M3 = Mi#M2 is possible only if M3or M2 is itself diffeomorphic to S3. One says an orientable closed manifoldM3 is irreducible if every separating embedded 2-sphere bounds a 3-ball.It is well known that the only orientable 3-manifold that is prime but notirreducible is S2 X S1.

A consequence of the Prime Decomposition Theorem [85, 97] isthat an orientable closed manifold M3 can be decomposed into a finiteconnected sum of prime factors

1M3 ^ (#j X3) # (#kYk) # (#P (s2 X `Sl))

Each X? is irreducible with finite fundamental group and universal cover ahomotopy 3-sphere. Each yk is irreducible with infinite fundamental groupand a contractible universal cover. The prime decomposition is unique up tore-ordering and orientation-preserving diffeomorphisms of the factors. Fromthe standpoint of topology, the Prime Decomposition Theorem reduces thestudy of closed 3-manifolds to the study of irreducible 3-manifolds.

There is a further decomposition of irreducible manifolds, but we mustrecall more nomenclature before we can state it precisely. Let E2 be a two-sided compact properly embedded surface in a manifold-with-boundary N3.Assume that E2 has no components diffeomorphic to the 2-disc D2, and thatE2 either lies in 9N3 or intersects aN3 only in 8E2. Under these conditions,one says E2 is incompressible if for each D2 C N3 with D2 n E2 = 8D,there exists a disc D. C E2 with 8D* = 8D.

Let M3 be irreducible. The Torus Decomposition Theorem [79, 80]says that there exists a finite (possibly empty) collection of disjoint incom-pressible 2-tori T2 such that each component N3 of M 3\ U T2 is either geo-metrically atoroidal or a Seifert fiber space, and that a minimal such collec-tion {Ti } is unique up to homotopy. One says an irreducible manifold-with-boundary N3 is geometrically atoroidal if every incompressible torusT2 C N3 is isotopic to a component of aN. One says a compact manifoldN3 is a Seifert fiber space if it admits a foliation by S' fibers.

1. GEOMETRIZATION OF THREE-MANIFOLDS 3

REMARK 1.2. Seifert's original definition [116] of a fiber space requiredthe existence of a fiber-preserving diffeomorphism of a tubular neighborhoodof each fiber to a neighborhood of a fiber in some quotient of Sl x D2 bya cyclic group action. But Epstein showed later [39] that if a compact 3-manifold is foliated by S' fibers, then each fiber must possess a tubularneighborhood with the prescribed property. Hence the simpler definition wehave given above is equivalent to the original.

One says M3 is Haken if it is prime and contains an incompressiblesurface other than S2. In [122], Thurston proved the important result that ifM3 is Haken (in particular, if M3 admits a nontrivial torus decomposition)then M3 admits a canonical decomposition into finitely many pieces N3 suchthat each possesses a unique geometric structure. For now, the reader shouldregard a geometric structure as a complete locally homogeneous Riemannianmetric gi on N3. A more thorough discussion of geometric structures willbe found in Sections 2 and 3 below.

Thurston's Geometrization Conjecture asserts that a geometric de-composition holds for non-Haken manifolds as well, namely that every closed3-manifold can be canonically decomposed into pieces such that each admitsa unique geometric structure. In light of the Torus Decomposition Theo-rem and more recent results from topology, one can summarize what iscurrently known about the Geometrization Conjecture as follows. Let N3be an irreducible orientable closed 3-manifold. If its fundamental group7rl (N3) contains a subgroup isomorphic to the fundamental group Z ® Z ofthe torus, then either 7r, (A() has a nontrivial center or else N3 contains anincompressible torus. (See [114] and also [124].) If 7r1(N3) has a nontrivialcenter, then results of Casson-Jungreis [22] and Gabai [42] imply that N3is a Seifert fiber space, all of which are known to be geometrizable. On theother hand, if N3 contains an incompressible torus, then it is Haken, hencegeometrizable by Thurston's result. Thus only the following two cases ofThurston's Conjecture are open today:

CONJECTURE 1.3 (Elliptization). Let N3 be an irreducible orientableclosed 3-manifold of finite fundamental group 7rl(N3). Then N3 is diffeo-morphic to a quotient S3 /I, of the 3-sphere by a finite subgroup r of 0 (4). Inparticular, N3 admits a Riemannian metric of constant positive curvature.

CONJECTURE 1.4 (Hyperbolization). Let N3 be an irreducible orientableclosed 3-manifold of infinite fundamental group 9r1 (N3) such that 7r1 (N3)contains no subgroup isomorphic to Z ® Z. Then N3 admits a completehyperbolic metric of finite volume.

REMARK 1.5. A complete proof of the Elliptization Conjecture wouldimply the Poincare Conjecture, which asserts that any homotopy 3-sphereis actually a topological 3-sphere. (See Chapter 6.)

REMARK 1.6. Noteworthy progress toward the Hyperbolization Conjec-ture has come from topology. For example, Gabai has proved [43] that if


N3 is closed, irreducible, and homotopic to a hyperbolic 3-manifold, thenN3 is homeomorphic to a hyperbolic 3-manifold.

2. Model geometries

A geometric structure on an n-dimensional manifold M' may beregarded as a complete locally homogeneous Riemannian metric g. A metricis locally homogeneous if it looks the same at every point. More precisely,one says that g is locally homogeneous if for all x, y E Mn there existneighborhoods Ux C Mn of x and l fy C_ Mn of y and a g-isometry 'yx,Lfx --; U. such that 'yyy (x) = y. In this section, we shall develop a moreabstract way to think about geometric structures.

One says that a Riemannian metric g on Mn is homogeneous (towit, globally homogeneous) if for every x, y E Mn there exists a g-isometryyxy : M' --+ Mn such that yyy (x) = y. These concepts are equivalent when.M'1 is simply connected.

PROPOSITION 1.7 (Singer [120]). Any locally homogeneous metric g ona simply-connected manifold is globally homogeneous.

We also have the following well-known fact.

PROPOSITION 1.8. If (Mn, g) is homogeneous, then it is complete.

Thus if (,M', g) is locally homogeneous, we say that its geometry is givenby the homogeneous model (Mn, g), where Mn is the universal cover ofM' and g is the complete metric obtained by lifting the metric g. Thus tostudy geometric structures, it suffices to study homogeneous models.

A model geometry in the sense of Klein is a tuple (Mn, C, G.), whereM' is a simply-connected smooth manifold and CC is a group of diffeo-morphisms that acts smoothly and transitively on Mn such that for eachx E MTh, the isotropy group (point stabilizer)

C. {yEC:y(x)=x}is isomorphic to 9*. We say a model geometry (M", C, C*) is a maximalmodel geometry if CC is maximal among all subgroups of the diffeomor-phism group Z (Mn) that have compact isotropy groups. (An example ofparticular relevance occurs when Mn = C is a 3-dimensional unimodularLie group. These maximal models were classified by Milnor in [98] and arebriefly reviewed in Section 4 of this chapter.)

The concepts of model geometry and homogeneous model are essentiallyequivalent (up to a subtle ambiguity that will be addressed in Remark 1.13).One direction is easy to establish.

LEMMA 1.9. Every model geometry (.Mn, C, G*) may be regarded as acomplete homogeneous space (Mn,g).

PROOF. One may obtain a complete homogeneous c-invariant Riemann-ian metric on Mn as follows. Choose any x E .Mn. If gX is any scalar product

2. MODEL GEOMETRIES 5

on T1.M', one obtains a scalar product gx by averaging gx un-der the action of Cx = g*. Then since c is transitive, there exists for anyy E M' an element 'yyx E G such that -yyx (y) = x. So one may define ascalar product gy for all V, W E TyMn by

gy (V, W) - (,'yxgx) (V, W) = gx ((-Yyx)* V, ('Yyx)* W) .

To see that gy is well defined, suppose that `yyx is another element of G suchthat -y'yx (y) = x. Then ryyx o ryy E !9, so by Gx-invariance of gx, we have

yxgx)(V,W)=9x((Yyx°7yx)*(1yx)*V, (Yyx0 'Yyx)*(Yyx)*W)

=gx (-Yyx)*V, (-Yyx)*W) = ((Yyx)*gx) (V, W).

Because the action of 9 is smooth, one obtains in this way a smooth Rie-mannian metric g on .A41. Because the action of g is transitive, g is completeby Proposition 1.8.

To complete the connection between homogeneous models and modelgeometries, we want to establish the converse of the construction above.Namely, we want to show that every homogeneous model (M',§) may beregarded as a model geometry (Mn, c, 9*). We begin by recalling someclassical facts about the set Isom (Mn, g) of isometries of a Riemannianmanifold. Isom (Mn, g) is clearly a subgroup of the diffeomorphism group0 (Ma). Moreover, it is clear that (Mn,g) is homogeneous if and only ifIsom (Mn, g) acts transitively on Mn. The following facts are classical. (See[100] and [87].)

PROPOSITION 1.10. Let (Mn, g) be a smooth Riemannian manifold.

(1) Isom (A4', g) is a Lie group and acts smoothly on Mn.(2) For each x E Mn, the isotropy group

Ix (Mn, g) {-Y E Isom W" g) : 7 (x) = x}

is a compact subgroup of Isom (.M', g).(3) For each x E Mn, the linear isotropy representation

Ax : Ix (Mn, g) --a 0 (T M', g (x))

defined by

Ax ('Y) = y* : TZMn -p TXMn

is an injective group homomorphism onto a closed subgroup of theorthogonal group 0 (TxMn,g (x)). In particular,

Ix (Mn, g) Ax (Ix (M', g)) C 0 (TxMn, g (x))

is compact.


COROLLARY 1.11.

(1) Given x, y e M' and a linear isometry

f : (TTM" g (x)) - (TTM", g (y)) ,

there exists at most one isometry -y E Isom (M', g) with y (x) = yand y* = f.

(2) If M" is compact, then Isom (.M", g) is compact.

When (Mn, g) is homogeneous, all isotropy groups look the same.

LEMMA 1.12. If Isom (Nt", g) acts transitively on M", then all isotropygroups Ix (M', g) are isomorphic.

PROOF. Given any x, y E M", there exists an isometry y : M" --+ .M"such that y (x) = y. Define

y# : Ix (Mn' g) _ Iy (M", g)by

-1'#(13) ='Y°/3oy-1Clearly, y# is a group isomorphism with inverse (Y-1)#'

Thus one may regard a complete homogeneous space (M', g) as a modelgeometry (M", Isom (Ma, g) , Ix (Mn' g)).

REMARK 1.13. There is an ambiguity in the equivalence of model geome-tries and homogeneous models that we now address. The variety of possibleRiemannian metrics g compatible with a given model geometry (Ma, Q, Q*)depends strongly on the size of the isotropy group Q*. If the isotropy groupQ* is large enough, for example if Q,, 0 (n), then one may show easily thata unique C,,-invariant metric g, hence a unique homogenous model (M", g)is determined up to a scalar multiple. For smaller isotropy groups Q*, theremay be more choices for g, and these may endow M" with somewhat dif-ferent geometric properties. On the other hand, there may be closed propersubgroups of Isom (.A4', g) acting transitively on (M", g), so that a homo-geneous model (M", g) may be regarded as a model geometry (M',!91 Q*)under different groups (M", Q', Q*). For the purpose of giving a standarddescription of the geometric structures that occur in the GeometrizationConjecture, this ambiguity is addressed by choosing the isotropy groups tobe as large as possible, in other words by considering only maximal modelgeometries. In fact, Thurston proved that there are exactly eight maximalmodel geometries that have compact 3-manifold representatives.

3. Classifying three-dimensional maximal model geometries

In dimension three, a surprisingly rich collection of model geometriesis obtained by taking M3 = Q3 to be a Lie group acting transitively onitself by multiplication. Since we are interested only in closed manifoldsmodeled on Q3, we can make a further restriction. One says a Lie group

3. MAXIMAL MODEL GEOMETRIES

Signature Lie group DescriptionSU (2) compact, simple

(_J,_1,0) Isom (1R2) solvable

(-1, -1, +1) SL (2, R) noncompact, simple(-1,0,0) nil nilpotent(-1,0,+1) sol solvable(0,0,0) R e R e R commutative

TABLE 1. 3-dimensional unimodular simply-connected Lie groups

7

9n is unimodular if its volume form is bi-invariant. This is equivalent tothe statement that the 1-parameter family of diffeomorphisms generated byany left-invariant vector field preserves volume. Only unimodular Lie groupsadmit compact quotients, because only those groups admit quotients of finitevolume. Thus one can make the following observation.

LEMMA 1.14. If (Mn, g) is a homogeneous model such that Isom (Ma, g)is diffeomorphic to M', then M' is a Lie group. If moreover there exists adiscrete subgroup r of Isom (Mn, g) such that Mn/F is compact, then MThis unimodular.

In [98], Milnor classified all 3-dimensional unimodular Lie groups. (Hismethod is sketched in Section 4 below.) Any simply-connected 3-dimensionalunimodular Lie group M3 must be isomorphic to one of the six groups listedin Table 1. Remarkably, the groups in this list make up five of the eightmaximal models that appear in the Geometrization Conjecture. (Thurston

discards Isom (R2) because it is not maximal; see Remark 1.13.)The key to classifying all maximal model geometries (M3, G,71) with

compact representatives is to look more closely at the isotropy group G*. Let(M3, c, 71) be a maximal model geometry, and let C' denote the connectedcomponent of the identity in G. Then g' acts transitively on M3, so byLemma 1.12, all isotropy groups C of the 9' action are isomorphic. If Gxdenotes the connected component of the identity in 9, then the collection

forms a covering space of M'. Since M" is simply connected, thiscovering must be trivial. It follows that all g are connected, hence isomor-phic to a connected closed subgroup !9* of SO (3). Since the only propernontrivial Lie subalgebra of so (3) is so (2), one concludes that c*.is eitherSO (3) itself or SO (2) or else the trivial group.

Following Thurston's argument in Chapter 3.8 of [123], one is thus ableto classify all 3-dimensional maximal model geometries which admit compactrepresentatives:

PROPOSITION 1.15. Let (.M3, g, 9*) be a maximal model geometry repre-sented by at least one compact 3-manifold. Then exactly one of the followingis true:

(1) 9* ^_' SO (3), in which case either


(a) .M3 is a Lie group isomorphic to SU (2);(b) M3 is a Lie group isomorphic to R3; or(c) M3 is diffeomorphic to hyperbolic 3-space 7-13.

(2) 9* = SO (2), in which case either(a) M3 is a trivial bundle over a 2-dimensional maximal model,

so that(i) M3 is diffeomorphic to S2 x III; or

(ii) M3 is diffeomorphic to 7-12 x R;(b) M3 constitutes a nontrivial bundle over a 2-dimensional max-

imal model, so that(i) M3 is a Lie group isomorphic to nil; or

(ii) A43 is a Lie group isomorphic to SL (2, R),(3) 9* is trivial, in which case M3 is a Lie group isomorphic to sol.

4. Analyzing the Ricci flow of homogeneous geometries

In a homogeneous geometry, every point looks the same as any other. Asa consequence, the Ricci flow is reduced from a system of partial differentialequations to a system of ordinary differential equations. In this section, weshall outline how this reduction may be done for the five model geometriesthat can be realized as a pair (C, C), where 9 is a simply-connected unimod-ular Lie group. The theory in these cases is most elegant, but the remainingthree model geometries can be treated in a similar fashion.

Let gn be any Lie group, and let g be the Lie algebra of all left-invariantvector fields on C. Since a left-invariant metric on c is equivalent to a scalarproduct on g, the set of all such metrics can be identified with the set Sof symmetric positive-definite n x n matrices. S, is an open convex subsetof I18n(n+1)/2. For each left-invariant metric g on 9, the Ricci flow may thusbe regarded as a path t 1-4 g (t) E SS . One would thus expect the Ricci flowto reduce to a system of n (n + 1) /2 coupled ordinary differential equations.In fact, we can do better.

Let us equip G with a left-invariant moving frame {Fi}. Then the struc-ture constants c for the Lie algebra g of left-invariant vector fields onare defined by

[Fi, Fj] = c Fk,and the .adjoint representation of g is the map ad : g -p gl (g) gl (n, IR)given by

(ad V) W - [V,W] =13

If g is a left-invariant metric on C, then the adjoint ad* : g --> gl (g) withrespect to g of the map ad : g --b gl (g) is given by

((ad X) * Y, Z) = (Y, (ad X) Z) = (Y, [X, Z]) .

Now suppose C is a 3-dimensional Lie group. Let {Fi} be a left-invariantmoving frame chosen to be orthonormal with respect to a given left-invariant

4. THE RICCI FLOW OF HOMOGENEOUS GEOMETRIES

metric g.. Then one may define a vector space isomorphisin g ---> A2g by

Fi H-4 Fi+1mod3 A Fi+2mod3

9

Composing this with the commutator, regarded as the linear map A29__+9sending V A W F--> [V, W], yields a vector space endomorphism r : g --> gwhose matrix with respect to the ordered basis 0 = (F1, F2, F3) is

1 1

0213 C31

F = C22

3 C31

1

C12

C212

C312

Observe that the matrix IFp displays all the structure constants of g.Suppose further that c is unimodular. In this case, one has tr ad V = 0

for every V E g, which implies in particular that F is self adjoint with respectto g. So by an orthogonal change of basis 3 a = (F1,-P2, F3), the matrixr,3 of r can be transformed into

(1.1) Fa =

3 3C23 C31

2C31

2x

211

2v

where the c are the Lie algebra structure constants for {Pi}. We can23

arrange that A, µ, v E {-1, 0, 1} without changing the metric g if we arewilling to let the frame {Fi} be merely orthogonal. For instance, if A, p, vare all nonzero, we get A, A, v = ±1 by putting

1

F1 =vI

F1, F2 = 71 F2F3 aµ

F3-:X

The general case is just as easy. We can also arrange that A < a < v.For instance, if F1 = -F2, F2 = F1i F3 = F3, the orientation of themanifold is unchanged, but _ (µ, A, v). Finally if we are willingto reverse the orientation of G, we can arrange that there are at least asmany negative structure constants as positive, thereby displaying exactly sixdistinct signatures. This construction and classification was first describedin [98], so we call an orthogonal frame field for which A < A< v E {-1, 0, 1}a Milnor frame for the left-invariant metric g. Table 1 (above) displaysthe possible simply-connected unimodular Lie groups.

Let {F} be a Milnor frame for some left-invariant metric g on a 3-dimensional unimodular Lie group 9. Then there are positive constants A,B, and C so that g may be written with respect to the set {wi} of 1-formsdual to {Fi} as

C12

1

(1.2) g=Awl®wl+Bw2®w2+Cw3®w3.


If we want to evolve g by the Ricci flow, we must study its curvature. Forbrevity, we display the map ad* in the form

(ad Fl)* F1 (ad F2)* F1 (ad F3)* F1 1

(ad F1)* F2 (ad F2)* F2 (ad F3)* F2

(ad Fi)* F3 (ad F2)* F3 (ad F3)* F3 )

If {F,} is a Milnor frame, it is easy to compute that ad* is determined by

r 0 2A F3 -2)4

F2 1

-2µCF3 0 2µAF1

2vBF2 -2uAF1 0

The Levi-Civita connection of g is determined by ad and ad* via the formula

(1.4) VXY = 2 {[X, Y] - (adX)* Y - (adY)* X} .

Then once the connection is known, it is straightforward to compute thatthe curvature tensor of g is given by

(R(X,Y)Y,X) = 4 (adX)'Y+(adY)'XI2 - (adX)'X,(adY)'Y)

4 1[X, Y]12 - 2 ([[X,Y] ,Y] ,X) - I ([[Y, X], X], Y)

LEMMA 1.16. Suppose that {Fz} is a Milnor frame for a left-invariantmetric on a 3-dimensional unimodular Lie group 9. Then

(R (Fk, Fi) Fj, Fk) = 0

for all k and any i j.PROOF. Without loss of generality, we may assume that i = 1, j = 2,

and k = 3. Using (1.1), (1.3), and (1.4), and noting in particular thatOF,Fz = - Fp = 0 for any t, we compute that

(R (F3, F1) F2, F3) (VF3 (VF1F2) - VF, (VF3F2) - V [F3,Fi]F2, F3)

(V F3 F2, V F1 F3) - (O F1 F2, V F3 F3 )

- (V[F3,Fl)F2, F3)

_ (VF3F2, VF1F3)/ \

2((-A-µA+VA)F1, l-µ-vB+A IF2)

=0.//

5. A GEOMETRY WITH ISOTROPY SO (3) 11

By the lemma, any choice of Milnor frame for a left-invariant metricg on 9 lets us globally identify both g and Rc (g) with diagonal matrices.Hence, we may regard the Ricci flow as a coupled system of three (ratherthan six) ordinary differential equations for the positive quantities A, B, andC appearing in formula (1.2), now regarded as functions of time. (Comparewith Section 4 of Chapter 6.) We shall see in particular that the behaviorof solutions to this system depends on the structure constants of the Liealgebra g. Heuristically, the evolution of the metric will in some sense tryto expand the non-commutative directions. Put another way, one expectsthe Ricci flow to try to increase the isotropy of the initial metric.

5. The Ricci flow of a geometry with maximal isotropy SO (3)

In the remaining sections of this chapter, we will consider one exampleof the behavior of the Ricci flow on each isotropy class of model geometryin dimension three. An excellent study of the behavior of the normalizedRicci flow on all the homogeneous models is [76]. (The behavior of theunnormalized Ricci flow on model geometries was also studied from anotherperspective in [86].)

The fully isotropic geometries are S3, 1R3, and N3. We shall take S3 asour example, identifying it with the Lie group SU (2). Algebraically

SU (2) = S (z zJ

: W, Z E wl2 + Izl2 = 1} .

Clearly, SU (2) may be identified topologically with the standard 3-sphereof radius 1 embedded in R4. The signature of a Milnor frame on SU (2) isgiven byA=p=v=-1.

Some of the most significant differences in the behavior of the Ricci flowon the various 3-dimensional model geometries involve the issue of collapse.Recall that a Riemannian manifold (M', g) is said to be --collapsed ifits injectivity radius satisfies inj (x) < e for all x E M1 . Intuitively, sucha manifold appears to be of lower dimension when viewed at length scalesmuch larger than E. A manifold M' is said to collapse with boundedcurvature if it admits a family {g, : E > 0} of Riemannian metrics such thatSUP-EM" IRm [ge] (x)Ig. is bounded independently of E, but

sup inj gE (x)0.to CEMn

To introduce this issue, we shall consider the behavior of the Ricci flowon SU (2) with respect to a 1-parameter family of initial data exhibitingcollapse to a lower-dimensional manifold. Recall that the Hopf fibration

S1 +S3 --f S2

is induced by the projection 7r : S3 SU (2) -* C?1 S2 defined by7r (w, z) = [w, z]. In [14], Berger showed how to collapse S3 with bounded

curvature by shrinking the circles of the Hopf fibration. The GromovHausdorif limit of the collapse is (S2, h), where h is a metric of constantsectional curvature 4. Motivated by his example, we will consider a family ofleft-invariant initial metrics {g6 : 0 < e < 11 written with respect to a fixedMilnor frame {Fz} on SU (2) in the form

g, = EAW®®W1 + Bw2 ®W2 + CW3 ®w3.

The sectional curvatures of g, are then2

K(F2AF3) =1

ABC(B_C)2 - 3 + +E C

K (F3 A F1) = (EA - C)2 -3 B 2 2

EABC EA-C + EA + C

K(F1AF2)=EABC(EA-B)2-3EACB+ A+B.

Notice that the special case A = B = C = 1 recovers the classic E-collapsedBerger collapsed sphere metric whose sectional curvatures are

K(F2AF3)=4-3E, K(F3AF1)=e, K(F1AF2)=E.Note too that all geometric quantities are bounded as long as B - C = 0 (E)as E -f 0. In particular, the Ricci tensor of gf is determined by the following:

Re (FI, F1) = BC[(EA)2 - (B - C)2]

2 _ 2

Re (F2, F2) =EAC

[B2 _ (EA - C)2] = 4 - 2C, +2 BEAC2 _ 2

B 4 - 2Re

Our first observation is that regardless of how collapsed an initial metricgE may be, the Ricci flow starting at g, - that is, the Ricci flow of any ho-mogeneous metric on SU (2) - shrinks to a point in finite time and becomesasymptotically round. This is equivalent to the convergence result for thenormalized flow that was obtained in [76].

PROPOSITION 1.17. For any E E (0, 1] and any choice of initial dataEAo, B0, Co > 0, the unique solution 9e (t) to (1.5) exists for a maximalfinite time interval 0 < t < T < oo. The metric gF (t) becomes asymptoticallyround as t / T.

PROOF. Define D = EA. Then B, C, D satisfy the system

dB = -8+4C2 +D2 B2

d

_t

CD

C -84B2 +

D2 _ C2dt=+

dD -8

4B2 +C

2 _ D2dt=+

From the symmetry in these equations, we may assume without loss ofgenerality that Do < Co < B0. The computation

d (B-D)=4(B-D) C2-(B+D)2dt BCD

shows that the inequalities D < C 0. The estimate

dtB<-8+4C <-4,implies that the solution exists only on a finite time interval; in particular,there exists T < oo such that D \ 0 as t / T. The estimate

d

dt(B-D)

8 (B D-D)

C CDB2

< 0D

\l

shows that BDD < 8, where

Bo - Do>0.Do

Thus we have 0 < B - D < SD for 0 < t < T, whence the result follows.

REMARK 1.18. When B = C = D at t = 0, this symmetry persists. Itfollows easily that SU (2) remains round and shrinks at the rate

D(t) =Do-4t=Do-2(n-1)t.This is the 3-dimensional version of the `shrinking round sphere' discussedin Subsection 3.1 of Chapter 2.

Now we consider the 2-parameter family of metrics gE (t) obtained byletting the an initial metric gE evolve by the Ricci flow on a maximal timeinterval 0 < t < T. We will see that the Ricci flow on SU (2) (in contrastto the homogeneous geometries we shall consider below) strongly avoidscollapse. Instead of studying A, B, and C directly, it will be more convenientto introduce quantities E, F defined by

E+B+C, F---B-C

Then the Ricci flow of g, (t) is equivalent to the following system:

(1.5a) dA = (F2A2)

t BC

(1.5b) dtE=-16+ABC(A2-F2)E

(1.5c)dtF ABC

(eA2 - E ) F.

As initial conditions, we posit A (0, e) = Ao > 0, E (0, e) = Bo + Co > 0,and without loss of generality F (0, E) _ (Bo - Co) /e > 0. A solution ofthis system exists as long as ABC E (0, oo). Our next observation is thatsolutions exist on a common time interval, regardless of E.

LEMMA 1.19. There exists a time To > 0 depending only on the initialdata (Ao, Bo, Co) such that the solution ge (t) of (1.5) exists for all 0 < t < Toandall0<E<1.

PROOF. Since A (AE) = -16A < 0, there is ko such that ABC < ko/Afor as long as a solution exists. So it will suffice to prove that ABC > 0.The calculation

z

dt (BC) AB2C2[2AE2 - 4ABC - EE (A2 + F2)] < 8

(BCE )

shows that there is k1 > 0 depending only on Bo and Co such thatBCEk1-8t

for as long as a solution exists. Then since

dt(BC) = 8 (EA - E) > -8E > -

k1 8 8t (BC) ,

there is k2 > 0 depending only on Bo and Co such that

(1.6) BC > k2 (k1 - 8t)

for as long as a solution exists. This implies thatd _ 2 > 4 2

dtA - BCA k2(k1-8t)`4

whence the conclusion follows readily.

Our final observation is that there may be a jump discontinuity in thecollapsing structure at t = 0. This reflects the fact that the Ricci flow onSU (2) assiduously avoids collapse.

LEMMA 1.20. If (Ao, Bo, Co), is a family of initial data parameterizedby e E (0, 1) such that

then for all t E (0, To),

l oF(0,s) > 0,

limF(t,e)=0.E-0

PROOF. It will suffice to prove the result fort E [0, T], where rr E (0, To)is arbitrary. Define

2

PBC

and Q -ABC'

recalling that

Since

dtF= (EP_)F.

wt (AF)16F < 0,

there is by (1.6) some k3 < oc such that

d F < PF = BCAF < kk32 (k1 - 8t)

Hence there is a positive function f j (t) bounded on [0, T] such that

for all t E [0, 7-]. Then since

d 4F2 4 (fl (t))2dtA BC - k2 (kl - 8t)'

there are positive functions f2 (t) and f3 (t) bounded on [0, -r] such that

A (t, ) < f2 (t)

and thus

P (t,) < 4f2 (t)< fs (t)k2 (k1 - 8t) -

for all t E [0,,r]. Finally, since E2/ (BC) > 4, we have

Q (t ') A16

>16

- f2 (t)for all t E [0, T]. Hence as E \ 0, we have

EP QE

uniformly on [0,,r], which proves the result.

6. The Ricci flow of a geometry with isotropy SO (2)

Under the Ricci flow, the S2 factor of S2 x R shrinks homothetically whilethe R factor remains unchanged. As a consequence, the solution becomessingular in finite time, and the manifold converges in the pointed Gromov-Hausdorff sense to R. In Section 5 of Chapter 2, we will discuss Ricciflow singularities on portions of a manifold that are geometrically close to

Sn-1 x (-Q, e).On the other hand, the product 7-12 x 1[8 and the Lie groups nil and

SL (2, lI8) all share the property that certain directions selected by the Liealgebra expand under the Ricci flow while the remaining directions convergeto a finite (possibly zero) value. As a result, solutions of the normalized Ricciflow on compact representatives of the latter three geometries all collapsewith bounded curvature, exhibiting Gromov-Hausdorff convergence to 1- or2-dimensional manifolds. (See [76].)

We shall use nil as an example of a geometry with isotropy SO (2),because its behavior can be computed explicitly. Let 93 denote the 3-dimensional Heisenberg group. Algebraically, Q3 is isomorphic to the set


of upper-triangular 3 x 3 matrices

1 x zg3=

0 1 y :x,y,zER0 0 1

endowed with the usual matrix multiplication. Topologically, g3 is diffeo-morphic to II83 under the map

g31 x z

ry= 0 1 y) F-+ (x, y, z) E R3.0 0 1

Under this identification, left multiplication by -y corresponds to the map

L.y(a,b,c) = (a + x, b+y, c+xb+z).The signature of a Milnor frame on g3 is A= -1 and µ = v = 0. Any

left-invariant metric g on g3 may be written in a Milnor frame {FZ} for g as

g=Awl ®w1+BW2®w2+CW3®W3,

where A, B, C are positive. Then the sectional curvatures of (G3, g) are

K(F2AF3)_-3BC, K(F3AFl)=BA,

K(F1AF2)BC,

and its Ricci tensor is

Rc=2BCwl ®w1 -2-ACw2®w2-2B w3®w3.

Hence we get a solution (g3, g (t)) of the Ricci flow for given initial dataA0, B0, Co > 0 if A, B, and C evolve by

a

(1.7a) d A = -4 BC

(1.7b)dtB 4 C

(1.7c) 4dtC

B.

System (1.7) can be solved explicitly. Observing that B/C is a conservedquantity, we compute that

d(A) - -12 (C(A

)

2)2

= -12CCo) \B /2

Solving this givesA_ Co /Bo

B2 12t + BOCo/Aoand shows that

d

dtA-4 () (B2) A 12t+BOC0/A0

A.

7. A GEOMETRY WITH TRIVIAL ISOTROPY 17

Observing that AB is another conserved quantity, we can thus obtain thefull solution

(1.8a) A = A02/3B1/3C1/3 (12t + BoCo/Ao)`1/3

(1.8b) B = 113(12t + BoCo/Ao)1/3

(1.8c) C = A10/3Bo 1/3Co13 (12t + BoCo/Ao)1/3 .

Thus we have proved the following.

PROPOSITION 1.21. For any choice of initial data Ao, Be, Co > 0, theunique solution of (1.7) is given by (1.8). There exist constants 0 < cl <C2 < oo depending only on the initial data such that each sectional curvatureK is bounded for all t > 0 by

cl < K < c2

t t

and such that the diameter of any compact quotient N3 of g3 is bounded forallt>0 by

c1t1/6 < diam.A/3 < c2t1/6.

In particular, N3 is almost flat.

Recall that a Riemannian manifold (Mn, g) is said to be e-flat if itscurvature is bounded in terms of its diameter by

e(1.9) 1Rml <

diam2 (Mn, g)

One says M' is almost flat if it is e-flat for all e > 0.

COROLLARY 1.22. On any compact nil-geometry manifold, the normal-ized Ricci flow undergoes collapse, exhibiting Gromov-Hausdorfconvergenceto (R2, gcan) where gcan is the standard flat metric.

7. The Ricci flow of a geometry with trivial isotropy

The only geometry with trivial isotropy is sol, which may also be re-

garded as the group Isom (IEi) of rigid motions of Minkowski 2-space.The signature of a Milnor frame on sol is ) = -1, µ = 0, and v = 1.

Any left-invariant metric g may be written in a Milnor frame {F2} for g as

g=Awl®w1+Bw2®w2+Cw3®w3.The sectional curvatures are

K F2 nFs_ (A-C)2-4A2() ABC2

K (F3 A F1) _(A +

ABC)(A - C)2 - 4C2K(F1AF2)=

ABC

so the Ricci flow is equivalent to the system

d C2 - A2(1. l Oa)

dt A = 4 BC

(1.10b) d B = 4 (A +C)2

d A2 - C2(1.lOc) dtC = 4

ABObserving that AC = AoCo and B (C - A) Bo (Co - A3) are conservedquantities, we define G _ A/C and consider the simplified system

(l.ila)dtB

= 8+41 G2

2

(l.ilb) dtG = 81 Gwhich has a solution for all positive time. Observing that AB > 16, weconclude in particular that B --f oc as t --+ oo. If A0 = Co, then G - 1and B is a linear function of time. If not, then it is straightforward to showthat G is strictly monotone and approaches a limit Gc.. satisfying eitherGo<G,, <lorl<G,, <Go. Since

ddG

log B = 2G 1 - G'the condition Gc 1 is incompatible with the observation that B -4 oo. Itfollows easily that A, C -+ AoCo and that B grows linearly.

PROPOSITION 1.23. For any choice of initial data A0, B0, Co > 0, theunique solution of (1.10) exists for all positive time. For any E > 0, thereexists TE > 0 such that

A - AoCo 5 E, C - AoCoI < E, anddt

B - 16 <E

for all t > T, Moreover, there exist constants 0 < cl < c2 < oo dependingonly on the initial data such that each sectional curvature K is bounded forallt>0by

cl <<C2t - - tCOROLLARY 1.24. On any compact sol-geometry manifold, the normal-

ized Ricci flow undergoes collapse, exhibiting Gromov-Hausdorff convergenceto i .

To gain insight into the behavior of sol metrics on compact manifolds, werecall a construction from [67]. One chooses A E SL (2, Z) with eigenvaluesA+ > 1 > A- . Then in coordinates 0, x, y on R3, chosen so that the x, y axescoincide with the eigenvectors of A, one defines an initial metric by

(1.12) g= e2adB®d0+ep+7dx®dx+e,3--y dy®dy,

NOTES AND COMMENTARY 19

where a and 0 are constant and -y is a linear function of 0 such thate-a (dry/dB) > 0. It is not hard to check that a Milnor frame for g is givenby

F1 = e--y/2 a - ey/2ax ay

F2 = 4 e-a aae

F3 = e--y/2 a + ey/2 a

ax ay*

Indeed, one has [Fl, F2] = 2F3, [F2, F3] = -2F1, [F3, Fl] = 0, and can writethe metric (1.12) in the form

g = 2eow1 ®w1 + (2 w2 ®w2 + 2eAw3 ®w3.

Clearly, g descends to a metric on the product of the line and the torus T2,and A acts on R x T2 by (0, x, y) F--, (0 + 2ir, A_x, A+y). If

W (0 + 2ir) = W (0) + 2 log A+,

then A is an isometry, whence g becomes a well defined metric on the com-pact mapping torus NA, regarded as a twisted T2 bundle over S1. By (1.12),it is easy to see that a governs the length of the base circle, while /3 and -ydescribe the scale and skew of the fibers, respectively. Under the Ricci flow,the metric (1.12) evolves as follows: ,8 and -y remain fixed, while one has

a (t) = a (0) + In 1 + (2t.

Intuitively, one may interpret this by saying that the Ricci flow attempts to`untwist' the bundle NA.

Notes and commentary

Some good sources for basic 3-manifold topology are [71], [78], and[70]. Homogeneous Riemannian spaces are reviewed in Chapter 3 of [27]and Chapter 7 of [20]. The curvatures of left-invariant metrics on Lie groupsare studied in [98].

Detailed analyses of the behavior of the normalized and unnormalizedRicci flows on the remaining homogeneous geometries may be found in [76]and [86], respectively. Qualitatively, one sees a variety of behaviors. Forexample, metrics in the Isom (J2) family converge. Metrics in the 9-13 familyconverge under the normalized Ricci flow, and give rise to immortal solu-tions of the unnormalized flow. (See Section 4 of Chapter 2.) Under thenormalized flow, compact manifolds in the SL (2, IR) and fl2 x R familiescollapse to 2-dimensional limits, while compact manifolds in the S2 x Ifs andsol families collapse to 1-dimensional limits.

CHAPTER 2

Special and limit solutions

In this chapter, we continue our study of special and intuitive solutionsto the Ricci flow. We introduce self-similar solutions, often called Ricci soli-tons, which may be regarded as generalized fixed points of the flow. Wethen give examples of solutions that exist for infinite time: eternal solutions(those existing for all times -oo < t < oo), ancient solutions (those whichexist for times -oo < t < w), and immortal solutions (those which existfor a < t < oo). Because these solutions have infinite time to diffuse, theyshould have very special properties. Our interest in those properties arisesfor the following reason. A singularity model for the Ricci flow is a com-plete nonfat solution obtained as a limit of dilations about a singularity. (InChapters 8 and 9, we shall study singularity models and their formation.)Every singularity model exists for infinite time, and the special propertiesone expects to find in a singularity model should yield valuable informationabout the geometry of the original solution near the singularity just priorto its formation. In this chapter, we also study two important singularitiesdirectly: we present a rigorous analysis of neckpinch singularities (undercertain symmetry hypotheses) and a heuristic analysis of a degenerate neck-pinch.

1. Generalized fixed points

There is only a small class of genuine fixed points of the Ricci flow. ARiemannian manifold (M', g) is a fixed point of the unnormalized Ricci flow

(2.1) tg = -2Rc

if and only if the metric g is Ricci flat. A compact manifold (M', g) is afixed point of the normalized Ricci flow

2g=-2Rc+n (fm,,d

if and only if its average scalar curvature is constant (fMf R dµ/ fm. dp)p and

Rc = P91n

hence if and only if g is an Einstein metric.There is however a larger class of solutions which may be regarded as

generalized fixed points. These are called self-similar solutions or Ricci

21

22 2. SPECIAL AND LIMIT SOLUTIONS

solitons. We will study Ricci solitons in detail in a chapter of the successorto this volume. But because these solutions often arise as limits of dilationsabout singularities, we shall encounter some examples in this chapter. Hencewe provide a brief introduction to them here.

Suppose that (M'',g (t)) is a solution of the unnormalized Ricci flowon a time interval (a, w) containing 0, and set go = g (0). One says g (t)is a self-similar solution of the Ricci flow if there exist scalars a (t) anddiffeomorphisms ibt of A4' such that

(2.2) g (t) = a (t) Vt (go)

for all t c (a, w). Thus a self-similar solution is a solution of the evolutionequation (2.1) having the special form (2.2). A metric having this formchanges only by diffeomorphism and rescaling. To see the significance ofthis fact, let us denote by S2M' the bundle of symmetric (2,0)-tensorson .M' and by S2M"' the sub-bundle of positive-definite tensors. Thenthe space of Riemannian metrics on Mn may be written as fit (.M')C°° (Mn, S2 .Mn). Let D (M') denote the diffeomorphism group of Mn.Because the Ricci flow is invariant under diffeomorphism, it may be regardedas a dynamical system on the moduli space

9N(Mn)

/1)(Mn)

of Riemannian metrics modulo diffeomorphisms. In this context, Ricci soli-tons correspond to generalized fixed points.

On the other hand, suppose that (Mn, go) is a fixed Riemannian mani-fold such that the identity

(2.3) -2 Re (9o) = Gxgo + 2A90

holds for some constant A and some complete vector field X on Mn. In thiscase, we say go is a Ricci soliton. If X vanishes identically, a Ricci solitonis simply an Einstein metric. Consequently, any solution of (2.3) (whichis actually a coupled elliptic system for go and X) may be regarded as ageneralization of an Einstein metric.

REMARK 2.1. By rescaling, one may assume that A E {-1, 0, 1} in (2.3).These three cases correspond to shrinking, steady, and expanding soli-tons, respectively. And as we shall see below, these yield examples of ancient,eternal, and immortal solutions, respectively.

REMARK 2.2. In case the vector field X appearing in (2.3) is the gradientfield of a potential function -f, one has

VVf =Rc+Ag

and says go is a gradient Ricci soliton.

A vector field X that makes a Riemannian metric into a Ricci solitonmay not be unique. The next example illustrates this.

1. GENERALIZED FIXED POINTS 23

EXAMPLE 2.3. Let (IIBn, gcan) denote Euclidean space with its standardmetric. Since the metric is flat, the trivial solution of (2.3) obtained bychoosing X - 0 lets us regard this as a steady Ricci soliton. But it may alsobe regarded as an expanding gradient Ricci soliton, called the Gaussiansoliton, by taking A = 1 and choosing the potential function

f (x) = 2 IxI2 .

In this section, we will prove the following observation.

LEMMA 2.4. If (Mn, g (t)) is a solution of the Ricci flow (2.1) havingthe special form (2.2), then there exists a vector field X on Mn such that(Mn, go, X) solves (2.3). Conversely, given any solution (Mn, go, X) of(2.3), there exist 1-parameter families of scalars o- (t) and diffeomorphisms'cbt of MTh such that (Mn, g (t)) becomes a solution of the Ricci flow (2.1)when g (t) is defined by (2.2).

In words, this result says that there is a bijection between the familiesof self-similar solutions and Ricci solitons which allows us to regard theconcepts as equivalent.

PROOF OF LEMMA 2.4. First suppose that (Mn, g (t)) is a solution ofthe Ricci flow having the form (2.2). We may assume without loss of gener-ality that a (0) = 1 and bo = id. Then we have

-2 Rc (go) = atg (t) _ (0) go +Gy(o)go,t=o

where Y (t) is the family of vector fields generating the diffeomorphisms ,t.This implies that go satisfies (2.3) with A = 2v' (0) and X = Y (0).

Conversely, suppose that go satisfies (2.3). Define

v (t) - 1 + 2At,

and define a 1-parameter family of vector fields Yt on M' by

Yt (x) . (t) X (x).

Let 'fit denote the diffeomorphisms generated by the family Yt, where b0 =idm.., and define a smooth 1-parameter family of metrics on Mn by

g (t) _ a (t) . 'fit (go) .Then g (t) has the special form (2.2). The computation

a du8t9 = dt ''t (go) + a (t) ' t (Gyt9o) _t (290 + Gxgo)

implies by (2.3) that

at-g = t (-2 Rc [go]) = -2 Rc [g),

hence that g (t) is a solution of the Ricci flow.


2. Eternal solutions

An eternal solution of the Ricci flow is one that exists for all time.Such solutions should have very special properties. Intuitively, one may ar-rive at this expectation as follows. We shall see in Chapters 5 and 6 that thecurvatures of a solution to the Ricci flow evolve by reaction-diffusion equa-tions. Equations of this type involve a competition between the diffusionterm (which seeks to disperse concentrations of curvature uniformly overthe manifold as time moves forward) and the reaction term (which tends tocreate concentrations of curvature as time moves forward). On an eternalsolution, one can look arbitrarily far backwards or forwards in time withoutencountering any concentration phenomena. This means that such solutionsmust be very stable, with no concentrations of curvature at any finite timein either the past or the future.

2.1. The cigar soliton. Hamilton's cigar soliton is the complete Rie-mannian surface (II82, gE) , where

(24) 9E l + x2 + y2This manifold is also known in the physics literature as Witten's blackhole. (See [125].) As we shall see below, it is a steady soliton, hencecorresponds to an eternal self-similar solution. Recalling that the Christoffelsymbols F of a metric g are given by

23

k 1ke(a a23 = 2 g axe

gee +ax- 9Ze - axe 9z, J ,

it is easy to compute that the Christoffel symbols of gE with respect to thecoordinate system (z' = x, z2 = y) on lR2are

1'i = -x/ (1 + r2) r12 = -y/ (1 + r2) I'22 = x/ (1 + r2)

ri1 = y/ (1 + r2) r12 = -x/ (1 + r2) r22 = -y/ (1 + r2)where r _ x2 + y2. Because gE is rotationally symmetric, it is natural towrite it in polar coordinates as

(2.5)dr2 + r2 d92

9E =1 + r2

The scalar curvature of gE is

(2.6)

and its area form is

R =4

E1+r2'

1dpE _1 + r2

dx n dy.

Since r2/ (1 + r2) --> 1 as r -f oo, equations (2.5) and (2.6) show that themetric is asymptotic at infinity to a cylinder of radius 1. In fact, we shall seebelow that this convergence happens exponentially fast in the distance scale

2. ETERNAL SOLUTIONS 25

induced by the metric. One can also see the cigar's asymptotic approach tothe cylinder in another way: the geodesics in the cigar metric are exactlythose curves -y (t) = (r (t) , 0 (t)) given in polar coordinates by solutions tothe ODE system

B"+ 2 OY=0r (1 + r2)

r" 1 + r2 [(r')2 + (0,)2] = 0.

From this, it is not hard to see that 0 (t) can be written in the form

0(t)=a+bt+bJ d(r

r2 (T)where a, b E R are arbitrary constants. In particular, when ro = r (0) islarge, a Euclidean circle of radius ro in R2 is close to being geodesic for ashort time.

REMARK 2.5. The cigar soliton is actually a Kahler metricdz dz

9E= 1+Izl2on C R2, and hence a Kahler-Ricci soliton. In fact, the cigar is the simplestrepresentative of an entire family of Kahler-Ricci solitons that exist on C2ra

for all m E N and on certain other complex manifolds. Various other metricsin this family were introduced in the papers [88], [23, 24], and [40]. Seealso [77].

REMARK 2.6. The cigar metric on the topological cylinder

C={(x,y):xERandyEllB/7L}is

dx2 + dy21 + e-2a

The details are left as an easy exercise for the reader.

To write the cigar metric in its natural coordinates on 1182, define

.s aresinh r = log (r + 1 + r2)

It is easy to see that s represents the metric distance from the origin. Indeed,since

dr = cosh s ds = 1 + r2 ds,formula (2.5) becomes

(2.7) gE = ds2 + tanh2 s d02.

Then formula (2.6) becomes4 4 16

cosh2 8 (es + e-s)2


In particular, RE = 0 (e-2s) as s -f oo, which illustrates the exponentialdecay claimed above.

We will now show that the cigar is essentially unique. The argumentwill also construct the potential function, hence the gradient vector fieldthat makes gE into a soliton.

LEMMA 2.7. Up to homothety, the cigar is the unique rotationally sym-metric gradient Ricci soliton of positive curvature on 1f82.

PROOF. Without loss of generality, one may write any rotationally sym-metric metric g on 1R2 in the form

(2.8) g=ds2+cp(s)2 d02,where cp (s) is a strictly positive function. (Compare with equation (2.7)above.) We shall calculate using moving frames, as will be reviewed inSection 1 of Chapter 5. It is natural to write

9=bijwz®Win terms of the orthonormal coframe field {wl,w2} given by

wl = ds, w2 = cp (s) dB.

The orthonormal frame field {el, e2} dual to {wl,w2} is then given by

8 __ 1 8el 8s' e2

cp (s) 80

It is straightforward to check that for any vector field

x=xlas+x2_50

on one has

Vxel = /(s)X2 a =c (s)dO(X).e2cp (s) 80

V xe2 = -gyp (s)X 2 as = - (s) dO (X) . el

It follows that the Levi-Civita connection 1-forms {w? } defined with respectto {el,e2} by

have only two nonzero components:

(2.9) wi = -w2 = cp' (s) dB.

In general, Cartan's first and second structural equations are

dwi = w A w and S2i = dwi - wk A wj.

In the case at hand, we have dwl = 0, dw2 = '(s) wl A w2, and

SZi=dw?=cp"(s) dsAdO=s

2. ETERNAL SOLUTIONS 27

So the Gauss curvature K is given by

(2.10) K = Q (e2, el) = _ ((ss)

Recalling equation (2.3), Remark 2.1, and Remark 2.2, we see that g con-stitutes a steady gradient Ricci soliton if and only if there exists a functionf such that

(2.11) Kg = VV f.

Our assumptions are that K > 0 and that f is a radial function f = f (s).This implies in particular that f is a convex function. Recalling (2.9), weobserve that

V,,el = w2 (el) e2 = 0and

Ve2e2 = W2 (e2) el = - ,(s) e1

Since g = 6 with respect to the frame {el, e2}, it follows that (2.11) holds ifand only if both

and

K = el (elf) - (Veiel) f = f" (s)

K = e2 (e2f) - (Ve2e2) f (s) f' (S)W (s)

hold, hence by (2.10) if and only if

(2.12) -p"

(s) _ f (s) = P, (s) f, (s)so (s)

-W (s)

Solving the separable ODE f"/ f' = cp'/cp implied by the second equality in(2.12) implies that

(2.13) f' (s) = cYCp (s)

for some constant a. We cannot have a = 0 or else g would be flat. Sincecp > 0, this forces f' to have a sign. Since f is convex, this sign mustbe positive; so we may set a = 2a2 for some a 0. Then substitutingf' = 2a2cp into the first equality in (2.12) yields the ODE

cp" + 2a2cpcp' = 0.

Integrating this givescp' + a2 cp2 = b

for some constant b. In order to g to extend to a smooth metric at s = 0,we must have

cp (0) = 0 and co' (0) = 1

by Lemma 2.10 (below). This is possible if and only if b = 1. So we solvethe linear ODE W'+ a2cp2 = 1 to obtain

cp (s) = a tanh (as).

Hence by (2.8), g must have the form

g = ds2 + 2 tanh2 (as) dB2a

for some a > 0. We claim that g is nothing other than a constant multipleof the cigar soliton. To see this, make the substitution o = as to get

g = a (d02 + tank' o d02) = a gE.

REMARK 2.8. By (2.13), we have f' = aV and hence

f (s) =

a

log (cosh (as)).

In particular

X = - grad a e tank as a

EXERCISE 2.9. Find the analog of the cigar metric with curvature K < 0.

3. Ancient solutions

An ancient solution of the Ricci flow is one which exists on a maximaltime interval -oo < t < w, where w < oc.

3.1. The round sphere. The shrinking round sphere is in a sense thecanonical ancient solution of the Ricci flow. Let gear, denote the standardround metric on Sn of radius 1, and consider the 1-parameter family ofconformally equivalent metrics

g (t) = r (t)2gcan,

where r (t) is to be determined. Observe that g (t) is a solution of the Ricciflow if and only if

2r dt ' Scan = 9 = -2 Rc [g] _ -2 Rc [gcan] _ 2 (n - 1) gcan,

hence if and only if r (t) is a solution of the ODE

dr n-1dt r

Evidently, setting

(2.14) r(t)Vr2 a-2(n-1)t= 2(n-1) T-tyields an ancient solution (Sn, g (t)) of the Ricci flow that exists for the timeinterval -oo < t < T, where T < oo is the singularity time defined by

20T 2(n-1)

3. ANCIENT SOLUTIONS 29

FIGURE 1. Shrinking round n-sphere

3.2. The cylinder-to-sphere rule. Before considering our next ex-ample, it will be helpful to recall the following result from Riemannian ge-ometry.

LEMMA 2.10. Let 0 < w < oc, and let 9 be a metric on the topologicalcylinder (-w, w) x S' of the form

g = cp (z)2 dz' + 0 (z)2 gcan,

where cp, (-w, w) -> I18+ and gcan is the canonical round metric of radius1 on S''. Then g extends to a smooth metric on S71+1 if and only if

(2.15)

(2.16) lim (z) = 0,

(2 17) lim , (z) = :1.

z--±w p (z)and

(2.18) urnd2k/

(z) = 0ds2k

for all k e N, where ds is the element of arc length induced by cp.

PROOF. Since the result is standard, we will only prove sufficiency. Theargument is essentially two-dimensional, so to simplify the notation we shallassume that n = 1 and

g = cp (z)2 dz2 +(z)2 d02,

where 0 E R/27rZ. It will suffice to consider the `north pole' z = w. Let

s (z) -fZ

o (C) d(,

and sets _ lim,z. + s (z) < oc. Let

r(z)=s-s,

30

and define

p (r) = P (r (z))V) (z)

rThen the metric may be written as

g = dr2 + r2p2 d62.

Introduce `Cartesian coordinates'

xrcos0,yrsin0.

Then r2 = x2 + y2 and we have

whence

dx =x

dr - y dOrdy = y dr + x dB,r

dr = xdx+ydyr rdO=-r2dx+ x2dy.

A simple calculation shows that21 22- 2

1 12d 2= C1 +P

2

pPd d 1 2 d 2g xJ - (r2 y .x x I yr2 ) + r2y + \I

JSo g extends to a smooth metric if

(r)2f (r) -

p(r)2

- 1

extends to an even function which is smooth at r = 0. By Taylor's theorem,

( )p(0) - 1 2P (0) P' (0) K dk (p2)

rk-2 x-1)O=f rr2

+r

+drk k!

(r+k=2 r=0

Hence f yields a smooth even function at zero if and only if p (0) = 1,p' (0) = 0, and all odd derivatives of p2 vanish at the origin. Noting that

dk(p2) k-1V) k) (7) (k-7)

drk= 2PP + cjkP P

j=1

for appropriate integers Cjk, we conclude by induction that f yields a smootheven function at zero if and only if p (0) = 1 and p(2k+1) (0) = 0 for all k > 0.To translate back to V) (z) = rp (r), we simply observe that

d kc -kdk-lp1 +rdrk

In particular, the observation1 dV - d - dpcp dz dr P + r dr '

2. SPECIAL AND LIMIT SOLUTIONS

shows p (0) = 1 and p' (0) = 0 if and only if

lim 0 (z) = 0 andz-W

lim V (z)Z-rW (0 (z)

3.3. The Rosenau solution. In Lemma 5.3, we will show that if gand h are metrics on a surface MZ conformally related by g = e"h, thenthe scalar curvature R. of g is related to the scalar curvature Rh of h byR9 = e_V (-Ohv + Rh). In the case that

for u :.M2 -4 R+, this formula becomes

-Oh (log u) + Rh(2.19) Rg =

U

Now if u is allowed to depend on time but h is regarded as fixed, g willsatisfy the Ricci flow equation for a surface,

aat9=-R-9,

if and only ifau _ 1 hlogu-Rhat

hU g'

hence if and only if a evolves by

au _T Ohlogu - Rh.

In the special case that (.M2, h) is flat, this reduces to

(2.20) at = Oh log u.

REMARK 2.11. There is an interesting connection between equation(2.20) and the porous media flow, which is the PDE

(2.21)aTU

= L1.u''.

When m = 1, this is just the ordinary linear heat equation. For all otherm > 0, it is nonlinear. When 0 < m < 1, equation (2.21) appears in variousmodels in plasma physics. When m > 1, it models ionized gases at hightemperatures; and when m > 2, it models ideal gases flowing isentropicallyin a homogeneous porous medium. (See [8].) If t _ mr, then equation(2.21) becomes

at u= 1 AU

In the formal limit m = 0, this equation may be applied to model thethickness u > 0 of a thin lubricating film if one neglects certain fourth-order

(See [19] ) Sinceeffects ..

um- 1lim = log u

m\O m

for u > 0, the calculation

li A I um - 1 = li m_2 I V[umu + ( 12]- 1)mm um um\,0 \ m m\,0

Au VU12

=Ologuu u2

shows that (2.20) is exactly the limit obtained for m = 0. This connectionbetween the porous media flow and the Ricci flow in dimension n = 2 wasfirst made by Sigurd Angenent. (See [128, 129].)

Now let h be the flat metric on the manifold M2 = lRxS1, where Sl isthe circle of radius 1. Give M2 coordinates x E R and 0 E Sl = R/27rZ.The Rosenau solution [111] of the Ricci flow is the metric g = u h definedfort<0by

(2.22) u (x, t) _2/3 sinh (-aAt)

cosh ax + cosh aAtfor parameters where a, /3, A > 0 to be determined. Because u is independentof 0, we have

82Oh log u = axe log U.

Hence the computations

(2.23) a u (x t) _ -2a/3Acosh aAt cosh ax + 1

at (cosh ax + cosh aAt) 2

and

(2.24)192

log u (x, t) = _a2 cosh aAt cosh ax + 1axe (cosh ax + cosh aAt)2

show that u satisfies (2.20) (hence that g = uh solves the Ricci flow on M2)if and only if

(2.25) 20A = a.

Note that the Rosenau solution is ancient but not eternal, since by equation(2.22),

lim u (x, t) = 0.t/O

By (2.19), the scalar curvature of g is given by

(2.26) R [g (t)] = -Ah log u - a2 cosh a_t cosh ax + 1

u 20 sinh (-aAt) (coshax + cosh aAt)

In particular, the solution has positive curvature for as long as it exists.Now regard M2 as the 2-sphere S2 punctured at the poles:

RxS1 ti S2\ { north and south poles } .

Since g (t) has positive scalar curvature for all t < 0, it is reasonable toinvestigate under what conditions g (t) extends to a smooth solution of theRicci flow on S2. We want to apply Lemma 2.10 with z = x and co = =V/1 Letting s (x, t) denote the distance from the `equator' x = 0 to thepoint x E R at time t, we estimate

X

s (x, t) = 20 sinh (-aAt) J 1 dxo cosh ax + cosh aAt

jjx<2 sinh (-aat) edx.

Hence the distance to the `poles' x = +oo is bounded at all times t < 0by a 0 sinh (-aAt) < oo. This shows that (2.15) is satisfied. It is easy tocheck that for all t < 0, we have

lim u (x, t) = 0,Ix I ---goo

hence that (2.16) is satisfied. Sincee u (x, t) sinh ax _

Xlim axu

(x, t) -a x1it cosh ax + cosh a.At a

we see that (2.17) is satisfied if and only if a = 1. Finally, we computea

=1 8 _ 1 a a sinh ax

as V1UV'U ax 2 ax

log u2 (cosh ax + cosh aAt)

and recall (2.24) to see thata2 _ 1 a2 a2 cosh aAt cosh ax + 1as2 2\ axe log u = -

2 20 sinh (-aAt) (cosh ax + cosh aAt)3/2

It follows that (2.18) is satisfied for k = 1; higher derivatives may be esti-mated similarly. Applying Lemma 2.10, we get the following result.

LEMMA 2.12. The metric defined by (2.22) for t < 0 extends to anancient solution of the Ricci flow on S2 if and only if

(2.27) a = 1 and 0 = 12A'

From now on, we assume (2.27) holds. Then it follows from (2.26) thatthe scalar curvature Rte (t) at the `poles' x = ±oo is strictly positive forall times t < 0:

R±,, (t) _ lim R = lim A (cosh At cosh x + 1)

xl-- 0C I x I oc sinh (-At) (cosh x + cosh At)= A coth (-At) > 0.

Moreover, the curvature R± (t) at the poles is actually the maximum cur-vature of (S2,g (t)) for all t < 0, since for all x > 0 we have

a R= 0 A(cosh At cosh x+1)ax ax sinh (-At) (cosh x + cosh At)

sinh x sinh (-At)(cosh x + cosh At) 2

If we compare the scalar curvature at an arbitrary point with its maximum,we find that

lim R (t) = lim(cosh At cosh x + 1) - 1

t/o R±,,) (t) t/o cosh (At) (cosh x + cosh At)

This confirms a result equivalent to what we shall prove in Chapter 5: anysolution of the unnormalized Ricci flow on a topological S2 shrinks to around point in finite time.

EXERCISE 2.13. The Rosenau solution provides an example of some rel-evance to our study of singularity models in Chapter 8: it is a Type IIancient solution which gives rise to an eternal solution if we take a limitlooking infinitely far back in time (as described in Section 6 of that chap-ter). In particular, there is a theorem which says that if one takes a limit ofthe Rosenau solution at either pole x = ±oo as t ---* -oo, one gets a copy ofthe cigar soliton studied above. Demonstrate this explicitly.

REMARK 2.14. The Rosenau solution is of interest here in part becauseit could potentially occur as a dimension-reduction limit of a 3-manifold sin-gularity. (Techniques of forming limits at singularities by parabolic dilationare presented in Section 3 of Chapter 8. The method of dimension reductionis introduced in Section 4 of Chapter 9 and will be discussed further in thesuccessor to this volume.) Recent work of Perelman [105] eliminates thispossibility for finite-time singularities. Indeed, if the Rosenau solution oc-curred as a dimension-reduction limit, one could by Exercise 2.13 adjust thesequence of points and times about which one dilated in order to get a cigarlimit. Perelman's No Local Collapsing Theorem excludes this possibility.

4. Immortal solutions

An immortal solution of the Ricci flow is one which exists on a max-imal time interval a < t < oc, where a > -oo. The example below appearsin Appendix A of [55]. It yields a self-similar solution on R2 which expandsfrom a cone with cone angle 27r C, where 0 < ( < 1. For t > 0, it formsa smooth complete metric on R2 with positive curvature that decays expo-nentially with respect to the distance from the origin. The reader is invitedto compare this solution with the expanding Kahler-Ricci solitons on Cndescribed in [24] that converge to a cone as t \ 0. (See also [40].)

4. IMMORTAL SOLUTIONS 35

Consider the one-parameter family of expanding metrics g (t) defined fort > 0 in the polar coordinate system (r, 9) on R2\ { any ray } by

(2.28) g (t) = t (f (r)2 dr2 + r2 d92) ,

where f is a positive function to be determined. We will study the metricsg (t) in local coordinates (z' = r, z2 = 0), instead of using moving framesas we did for the cigar soliton in Lemma 2.7 of Section 2. Again recallingthe formula

k W a arZ3 2 9 1\ axi 9je + axe 9i, - axe gij

we compute that the Christoffel symbols of g are

r11 = f'/f r12 = 0 r22 = -r/f2

r11=0r12=1/r r22=0

Then the standard formula

R jk aXi rjk axj rek + r m jk - rjmr k

shows that the only nonzero components of the Ricci tensor Rjk = Rzjk ofg are

(2.29) R11 = 8211 =

rf

(r)and R22 = R122 = f

rf (r)(r)3

.

Because Rc = Kg, where K is the Gauss curvature of g, this shows that

1 f'(2.30) K(t)=K[9(t)]= t f(r)

Now let X be the time-dependent vector field defined on 1R2\ {0} by

(2.31) X (t) =r a

t f (r) arNotice that the 1-form metrically dual to X is time-independent:

(2.32) = r f (r) dr.

The components of the Lie derivative Cxg of g with respect to X are

(2.33a) (.Cx9)Ii =Or (rf (r)) - ri1e1 = f (r)

(2.33b) (,Cx9)12 = V12 = 0(2.33c) (,x9)21 = 0

tr(2.33d) (x9)22 = V2s2 = 22 1 =r2

f (r)We want to determine f so that g (t) evolves by the modified Ricci flow

(2.34)atg

(t) = -2 Rc [g (t)] + Gx(t)9 (t)


(Compare with the Ricci-DeTurck flow analyzed in Section 3 of Chapter3.) If we can construct a solution g (t) of the modified Ricci flow (2.34),we will obtain an expanding self-similar solution g (t) of the Ricci flow asfollows. Observe that there exists a one-parameter family of diffeomorphismscpt : JR2 --' R2 defined for all t > 0 by

at ot (p) -X (p) , t)

(Because IR2 is not compact, long-time existence of cat must be checkedexplicitly, in contrast with Lemma 3.15.) Define an expanding self-similarfamily of metrics g (t) by

(2.35) g (t) - (Pt (g (t)) = t

Using the identity

(f (r)2 dr2 + r2 d02) .

a s J

(cat 1 ° cat+s) (ct I) * a s cat+s) = (Vt 1) * X (t) ,

5=0 5=0

we compute that

ag

_(t) =

a(

a(0a at t9 (t)) = as t+s9 (t + s))

05_

Vt (-'t9 (t)) + as (ct+s9 (t))5=0

= cpt { -2 Re [9 (t)] + Lx(t)9 (t) I + as 5=0[(WT, ° Ot+4 (49 (t)]

= -2 Re [9 (t)] + ot (Lx(t)g (t)) - 1),x(t)] (t))

= -2 Re [g (t)] .

Hence the self-similar metrics g (t) defined by (2.35) solve the Ricci flow.To construct a solution g (t) of the modified Ricci flow, we observe by re-

calling (2.29) and (2.33) that g (t) will solve (2.34) if and only if the followingsystem of equations is satisfied:

f (r)2 = at911= -2R11 + -2 f r) + f (r)

r2 = atg22 = -2R22 + 2V26 = -2rf ( + r2f (r)

3

3 f (r)

Remarkably, multiplying the second equation by f2/r2 gives the first, sothat this system is equivalent to the single first-order ODE

(2.36)r

(f2 - f3)f =2

Integrating this separable equation using partial fractions gives the relation

r2 1 f (r)(2.37)

4= C -

f (r)+ log If (r) - l

.

4. IMMORTAL SOLUTIONS 37

Notice that f (r) - 1 is possible only if r -> oo, so that we either have0 < f (r) < 1 or 1 < f (r) for all r. But combining equations (2.30) and(2.36) shows that the Gauss curvature of g (t) is

f.(2.38)

1 f

So in order to have a metric of positive curvature, we must find a solutionsatisfying 0 < f (r) < 1. Such a solution does in fact exist, and can begiven explicitly. Recall that the function w : (0, oo) --> (0, oo) defined byw (x) = xex is invertible; its inverse is the Lambert-W (product log) functionW : (0, oc) --> (0, oo). One can verify directly that

(2 39) f (r) - 1

4

is the solution of (2.37) satisfying f (0) _ E (0, 1).Following [55], we shall now demonstrate how one might arrive at the

representation (2.39) for f under the Ansatz that 0 < f (r) < 1. Define

F (r) _f

1r) -1,

noting that the Ansatz forces F (r) > 0 for all r _> 0. Then equation (2.37)is equivalent to

r2 4=C'-F(r)-logF(r),

where C' = C + 1, which we write as

2

C' - 4 = F (r) + log F (r) = log (F (r) eF(r))

Observing that C' = log (F (0) eF'(0)), we can put this into the form

F(r) = C'-r2/4 = r2(2.40) F (r) e - e - F (0) exp (F (0) - 4

By the Ansatz, we have F (r) > 0 for all r > 0. Thus we may apply W toboth sides of equation (2.40), obtaining

(F(O)exP/F(r)=W(F(0)-4J

whence equation (2.39) follows immediately.Note that limy,,, F (r) = 0, so that limy,,, f (r) = 1. Note too that

equation (2.38) implies that limr K (t) = 0 for all t > 0. This reflectsthe fact that the metrics g (t) are asymptotic at infinity to a flat cone.

FIGURE 2. A neckpinch forming

5. The neckpinch

The shrinking sphere we considered in Subsection 3.1 is the simplestexample of a finite-time singularity of the Ricci flow. In this section, weconsider what is its next simplest and arguably its most important singu-larity, the `neckpinch'.

One says a solution (Mn, g (t)) of the Ricci flow encounters a localsingularity at T < oo if there exists a proper compact subset K C Mnsuch that

sup JRml = 00K x [0,T)

butsup IRmI < oo.

(Mn\K) x [O,T)

This type of singularity formation is also called pinching behavior inthe literature. The first rigorous examples of pinching behavior for theRicci flow were constructed by Miles Simon [119] on noncompact warpedproducts IR x f S. In these examples, a supersolution of the Ricci flow PDEis used as an upper barrier to force a singularity to occur on a compactsubset in finite time. Another family of examples was constructed in [40].Here, the metric is a complete U (n)-invariant shrinking gradient Kahler-Ricci soliton on the holomorphic line bundle L-' over CCIPn-1 with twistingnumber k E {1, ... , n - 1}. As t / T, the CPn-1 which constitutes the zero-section of the bundle pinches off, while the metric remains nonsingular andindeed converges to a Kahler cone on the set (C'\ {0})/Zk which constitutesthe rest of the bundle. Both of these families of examples live on noncompactmanifolds.

5. THE NECKPINCH

FIGURE 3. The shrinking cylinder soliton

39

A neckpinch is a special type of local singularity. There exist quanti-tative measures of necklike behavior for a solution of the Ricci flow. (See[65], as well as Section 4 of Chapter 9.) However, the following qualitativecharacterization will suffice for our present purposes. One says a solution(M`+1 g (t)) of the Ricci flow encounters a neckpinch singularity at timeT < cc if there exists a time-dependent open subset Nt C .M'+1 such thatNt is diffeomorphic to a quotient of R x S' by a finite group acting freely,and such that the pullback of the metric g (t) INt to JR x S" asymptoticallyapproaches the `shrinking cylinder' soliton

(2.41) ds2+2(n-1)(T-t)gcan

in a suitable sense as t / T. (Here scan denotes the round metric of radius1 on S".)

From the perspective of topology, the neckpinch is perhaps the mostimportant and intensively studied singularity which the Ricci flow can en-counter, especially in dimensions three and four. In Chapter 9, we shall seesome reasons why this is so. For example, it is expected that one can per-form a geometric-topological surgery on the underlying manifold just priorto a neckpinch in such a way that the maximum curvature of the solution isreduced by an amount large enough to permit the flow to be continued onthe piece or pieces that remain after the surgery. (We plan to discuss suchsurgeries in a successor to this volume.)

From the perspective of asymptotic analysis, on the other hand, remark-ably little is known about singularity formation in the Ricci flow. For exam-ple, one cannot find in the literature examples of either formal or rigorousasymptotic analysis for Ricci flow singularities comparable to what has beendone for the mean curvature flow [7]. (Also see [1].) By contrast, excellentexamples of singularity analysis for related reaction-diffusion equations likeut = Au + uP can be found in [46, 47, 48] and [41], among other sources.

Because neckpinches are such important singularities, we shall studythem in considerable detail in this section. The first rigorous examples ofneckpinches for the Ricci flow were constructed by Sigurd Angenent andthe second author in [6]. In fact, these are the first examples of any sortof pinching behavior of the Ricci flow on compact manifolds. This sectionclosely follows that paper. The proofs are considerably more involved than

anything we have presented thus far, but illustrate many important ideasthat will be developed further in the chapters to come.

The main results of [6] may be summarized as follows.

THEOREM 2.15. If n > 2, there exists an open subset of the family ofSO(n+1)-invariant metrics on Sn+1 such that any solution of the Ricci flowstarting at a metric in this set will develop a neckpinch at some time T < oo.The singularity is Type-I (rapidly-forming). Any sequence of parabolic dila-tions formed at the developing singularity converges to a shrinking cylindersoliton (2.41) uniformly in any ball of radius o (/-(T - t) log(T - t)) cen-tered at the neck.

Any SO(n + 1)-invariant metric on Sn+1 can be written as

(2.42) g = cp2 dx2 +2 9can

on the set (-1, 1) x Sn, which may be identified in the natural way withthe sphere Sn+1 with its north and south poles removed. The quantity,'(x, t) > 0 may thus be regarded as the `radius' of the totally geodesichypersurface {x} x Sn at time t. It is natural to write geometric quantitiesrelated to g in terms of the distance

f(x)s (x) dx

from the equator. Then writing jx- and ds = cp dx, one can write themetric (2.42) in the nicer form of a warped product

(2.43) g=ds2+02gcanArmed with this notation, one can make a precise statement about theasymptotics of the developing singularity.

THEOREM 2.16. Lets (t) denote the location of the smallest neck. Thenthere are constants 6 > 0 and C < oo such that for t sufficiently close to Tone has the estimate

(2.44) 1 + o(1) < 0(x, t) < 1 + C (s - s)22(n --I) (T - t) - -(T - t) log(T - t)

in the inner layer Is - . < 2 -(T - t) log(T - t), and the estimate

(2.45)b (x, t)

< C,s - s log 5-9

T-t - -(T-t)log(T-t) f-(T-t)log(T-t)in the intermediate layer 2 -(T - t) log(T - t) < s - s < (T - t)(1-1)/2.

The estimates in the theorem are exactly those one gets when one writesthe evolution equation (2.47) satisfied by i/' with respect to the self-similarspace coordinate o- = (s - s) //T - t and time coordinate T = - log (T - t)and derives formal matched asymptotics. For this reason, it is expected thatthese estimates are sharp.

5. THE NECKPINCH 41

REMARK 2.17. In the special case of a reflection-symmetric metric onS'ti+1 with a single neck at x = 0 and two bumps, the solution exhibits whatwe call `single point pinching'; to wit, the neckpinch occurs only on thetotally-geodesic hypersurface {0} x Sn, unless the diameter of the solution(Sn+1 g (t)) becomes infinite as the singularity time is approached. (Thelatter alternative is not expected to occur.)

We shall justify this remark in Subsection 5.6.

5.1. How the solution evolves. The first task in studying neck-pinches is to compute basic geometric quantities related to the metric (2.42).One begins by observing that g will solve the Ricci flow if and only if cp and0 evolve by

(2.46) cpt = nS cp

and

(2.47) , = 7/1 - (n - 1)1-02

88. ,)

respectively. In order that g (t) extend to a smooth solution of the Ricciflow on Sn+1, it suffices to impose the boundary conditions

(2.48) limx-*±1

Indeed, if (2.48) is satisfied by the initial data, then the resulting solutionwill satisfy the hypotheses of Lemma 2.10 for all times t > 0 that it exists.

REMARK 2.18. The partial derivatives as and at do not commute, butinstead satisfy

[at, as] _ -nS as.

REMARK 2.19. Equation (2.46) will effectively disappear in what follows,because the evolution of So is controlled by the quantity ,,/b.

The Riemann curvature tensor of (2.43) is determined by the sectionalcurvatures

(2.49) Ko

of the n 2-planes perpendicular to the spheres {x} x Sn, and the sectionalcurvatures

(250) K =I7p2

of the (2) = n(n - 1)/2 2-planes tangential to these spheres. The Riccitensor of g is thus

(2.51) Rc = (nKo) ds2 + (Ko + (n - 1) K1) 9can,

and its scalar curvature is

(2.52) R = 2nKo + n (n - 1) KI.

5.2. Bounds on curvature and other derivatives. The first partof our analysis of a neckpinch singularity is to obtain bounds for the firstand second derivatives of O. To do so, it will be useful to consider the scale-invariant measure of the difference between the two sectional curvaturesdefined by

(2.53) a+V)2(KI-Ko).We collect the key results of this part of the analysis in our first proposition.

PROPOSITION 2.20. Let g (t) be a solution of the Ricci flow having theform (2.43) and satisfying the bounds 1',sj < 1 and R > 0 initially.

(1) For as long as the solution exists, I0.I < 1.(2) For as long as the solution exists, one has the bound

-2

< KI - Ko <

where a - sup Ia 0)(3) There exists C = C (n, g (0)) such that for as long as the solution

exists,

RmI<,02.

(4) The quantity (Ki)min is nondecreasing.(5) For as long as the solution exists, one has R > 0 and Vt < 0.(6) 02 is a uniformly Lips chitz- continuous function of time; in fact,

one has

I(2)tI <2(a+n).

(7) If g (t) exists for 0 < t < T, then the limit

(x, T) = t m (x, t)

exists for each x E [-1, 1].

We establish the claims above in the remainder of this subsection. De-noting the first derivative by v _ 0S, one first computes that

n2vv,+n 21(1-v2) v.(2.54) Vt=vSS+V

This equation has a happy consequence.

LEMMA 2.21. Assume g (t) is a solution to the Ricci flow having theform (2.43) and satisfying (2.48) for 0 < t < T. Then

1 < sup IV ( . , t) j < sup IV (', 0) I .

5. THE NECKPINCH 43

PROOF. Applying the maximum principle to (2.54), one concludes thatat any maximum of v which exceeds 1, one has

Vt<n-i (1-v2)v<0.V) 2

Similarly, at any minimum of v with v < -1, one has vt > 0. The resultfollows once we observe that (2.48) implies sup Iv 0)1 > 1.

Denoting the second derivative of / by w _ ass, one calculates that

)(2 55v w2 v2 n - 1

2 (4 5)2 +a. - n- 2w)Jw,s-wt=wss+(n-2

w

(2.55b) - 2 (n - 1) v2 (1 - v2),j3

Together, the evolution equations for v and w help us obtain a goodestimate for a.

LEMMA 2.22. Under the Ricci flow, the quantity a evolves by2

at =ass+(n-4)a,s-4(n-1)2a.PROOF. Noting that

a=Vw-v2+1,one computes that

as = 0sw + vws - 2vvs = z/>w3 - vw

and'/'ass = Y3w3 +Owss - vsw - vws = bwss - w2.

Then recalling equations (2.47), (2.54), and (2.55), one derives the equation

at = wit + 'zbwt - 2vvt

-=w{w-(n-1) 1 v2}

l wss+(n-2)vfJJJ-2

-(4n-5)

+(n - 1) - 2(n -1),2(1-,2)

_ 2)

-2v{ws+(n-2) -+(n-1)v(1 2v 1

11 v2w) v2 (1 - v2)= ' 'wss-w2+(n-4) vws-(5n-8) 2 -4(n-1)

2

2

= ass + (n - 4) as - 4(n - 1) a.

Applying the maximum principle proves that a is uniformly bounded.

COROLLARY 2.23. sup a+ supIa(-,0)1.

We can also show that the curvature is controlled by the radius.

COROLLARY 2.24. If g (t) is a solution of the Ricci flow of the form(2.43), then there exists C depending only on n and g (0) such that

IRmI < z

PROOF. Since v is bounded by Lemma 2.21, so is V)2K1 = 1- v2. Thensince a is bounded, it follows that 2K0 = 7p2K1 - a is bounded as well.Since

IRm12=2nKo+n(n-1)K?,

the result follows.

Now to simplify the notation, we define

K r -Ko = S and L _ K1 =1_o2

25

recalling that K0 and KI are the sectional curvatures defined in (2.49) and(2.50), respectively. Noting that the Laplacian of a radially symmetric func-tion f is given by

a2fAf = 0982 + n s as'

we compute the evolution of the sectional curvatures. One finds that Kevolves by

z

Kt =AK +2(n-1)KL-2K2-2(n-1)2(K+L).

The evolution equation for L is derived as follows.

LEMMA 2.25. Under the Ricci flow, the quantity L evolves by

Lt = AL +2SLS+2 [K2 + (n - 1)L2]

=AL-42' (K + L) + 2 [K2 + (n - 1)L2] .

PROOF. Using equations (2.47) and (2.54), one computes that

Lt = -2 2 (0S)t - 2LOt10

2

L222 ss - 2 (n - 1) 2 (K + L) + 2 I2 -L I K + 2 (n - 1) L2.

Then observing that

LS=-2S(K+L),

5. THE NECKPINCH 45

one calculates

L55 = -2 2''S +6L2(K+L)+2C 2

-L) K-2K2.

Combining these equations yields2

Lt=L,55,-2(n+2)'2(K+L) +2 [K2+(n-1)L2],

whence the result follows.

COROLLARY 2.26. Lmin (t) is nondecreasing; moreover, if Lmin (0) 0

then

Lmin (t) > _1 1I'min (0) - 2 (n 1) t

We can now show that the radius is strictly decreasing in time. Wewill make use of the observation that 't, a, and the sectional curvaturessatisfy the relations

C ) aKo-nR

LEMMA 2.27. Let g (t) be a solution of the Ricci flow having the form(2.43) and satisfying 1,08I < 1 initially. If the scalar curvature is positiveinitially, then it remains so, and one has

t <0for as long as the solution exists.

PROOF. That R > Rmin (0) is a general fact we shall prove in Lemma6.8. To show that Ot < 0, first note that the bound kbs1 < 1 forces K1 =(1 - os) /z/>2 to be nonnegative everywhere. There are now two cases. IfV3,g < 0, then one has

'bt=V)SB-(n-1)OK1<0.On the other hand, if bs,g > 0, then Ko = -4 ss/1 < 0 and hence

-Vt =iP [Ko+(n- 1)K1] =z/i (nR-Ko) >0.

We conclude this part of the analysis by showing that , 2 is a uniformlyLipschitz continuous function of time.

LEMMA 2.28. Let g (t) be a solution of the Ricci flow having the form(2.43) and satisfying

I

sl < 1 initially. Then

(02)t l < 2 (a + n) .

X (t)

FIGURE 4. A typical profile

PROOF. Noting that

loss - a + Zb2 - 1'

one hasOt=Y4ss-(n-1) (1 -%s) =a-n(1 -0s)

COROLLARY 2.29. If g (t) exists for 0 < t < T, then limt/T'cb(x, t) existsfor each x E [-1,1].

5.3. The profile of the solution. We call local minima of x H 0 (x, t)`necks' and local maxima `bumps'. We are interested in solutions whose ini-tial data has at least one neck. The second part of our analysis is to establishthe sense in which the profile of the initial data persists, in particular to showthat the solution will become singular at its smallest neck and nowhere else.

Let g (t) be a solution of the Ricci flow having the form (2.43). Wedenote the radius of the smallest neck by

rmin (t) - min f o (x, t) : Ox (x, t) = 0}

for each t such that the solution exists; if the solution has no necks, thenrmin will be undefined. We let x,, (t) denote the right-most bump, and callthe region right of x* (t) the `polar cap'.

The main results of this part of the analysis are as follows.

PROPOSITION 2.30. Let g (t) be a solution to the Ricci flow of the form(2.43) such that 10s 1 < 1 and R > 0 initially. Assume that the solutionkeeps at least one neck.

(1) At any time, the derivative z/;s has finitely many zeroes. The numberof zeroes is nonincreasing in time, and if O ever has a degeneratecritical point (to wit, a point such that Ws =ss = 0 simultane-ously) the number of zeroes of 0s drops.

5. THE NECKPINCH 47

(2) There exists a time T bounded above by rmin(0)2/(n - 1) such thatthe radius rmin(t) of the smallest neck satisfies

(n - 1)(T - t) < rmin(t)2 < 2(n - 1)(T -t).(3) The solution is concave (',ss < 0) on the polar cap.(4) If the right-most bump persists, then D = limt/T7P(x,, (t), t) exists.

If D > 0, then no singularity occurs on the polar cap.

We begin with the observation that the number of necks cannot in-crease with time, and further that all bumps/necks will be nondegeneratemaxima/minima unless one or more necks and bumps come together andannihilate each other.

LEMMA 2.31. Let g (t) : 0 < t < T be a solution of the Ricci flow ofthe form (2.43). At any time t E (0,T), the derivative v = 0, has a finitenumber of zeroes when regarded as a function of x E (-1, 1). This numberof zeroes is nonincreasing in time. Moreover, if ever has a degeneratecritical point, the number of zeroes of 'S drops.

PROOF. The derivative v = 'Ps satisfies (2.54), which can be written asa liner parabolic equation vt = v,ss + Qv, with

z

Q=(n-2)-+(n-1)1 v

Since _ one can in turn write (2.54) as

vt = cp-1 (cp-1vx), + Q (x, t) v = A (x, t) vxx + B (x, t) xv + C (x, t) v

for suitable functions A, B, C. Since v --> :F1 as x ±1, the Sturmiantheorem [5] applies.

We next derive upper and lower bounds for the rate at which a neckshrinks. These estimates show that a singularity will develop at the smallestneck in finite time, unless the solution loses all its necks first.

LEMMA 2.32. Let g (t) be a solution to the Ricci flow of the form (2.43)such that I Js 1 < 1 and R > 0 initially. Then

(n - 1) (T - t) < rmin (t)2 < 2 (n - 1) (T - t).In particular, the solution must either lose all its necks or else become sin-gular at or before

T _ rmin (0)2n-1

PROOF. Note that 0 t) is a Morse function except perhaps for finitelymany times, and that rmin (t) is a Lipschitz continuous function. We claimthat n-1 d n-1

rmin (t)< dtrmin (t)

2rmin(t)

holds for almost all times. The lemma follows from the claim by integration.

To prove the claim, fix some to such that z/i (-, to) is a Morse function, andlet its smallest critical value be attained at xo. Then the Implicit FunctionTheorem implies that there exists a smooth function x (-) defined for t nearto such that x (to) = xo and V ,x (x (t) , t) = 0. Taking the total derivative,one obtains

dtV) (x (t) , t) = Vt (x (t) , t) + 7PX (x (t) , t) dt (t)

_Ot(x(t)t)= (x (t) t) - n-1

ss

rmin (t)The first inequality in the claim follows from this when one recalls that

ass > 0 at a neck. To prove the other inequality, recall that R = 2nKo +n (n - 1) KI, where Ko ss/band K1 = (1 - 7P2) /o2. So at a neck,

n-1 R _ n-1 R0ss = -OKo =

2Kl

2n 2-0 R

1 2n*

Since the inequality R > 0 is preserved by the flow, the second inequalityfollows.

The final steps in this part of the proof show that no singularity occurson the polar cap: the region between the last bump and the pole. To do this,we first use the tensor maximum principle to show that the Ricci curvature ispositive there. We shall invoke the following slight modification of Theorem4.6.

LEMMA 2.33. Let (Nt, Wt, g (t) : 0 < t < T) be a smooth 1-parameterfamily of compact Riemannian manifolds with boundary. Let S and U besymmetric (2, 0)-tensor fields on Af such that S evolves by

as=OS+U*s,where U * S denotes the symmetrized product (U * S)Zj = UkSkj + SkUkj.Suppose that infPE.Nt S (p, 0) > 0 and that S (q, t) > 0 for all points q c- aNtand times t E [0,T). If (U * S) (V, ) > 0 whenever S (V, -) = 0, then

inf S(p,t) > 0.pENt, tE[0,T)

Once one observes that the hypotheses imply that S can first attain azero eigenvalue only at an interior point of .Nt, the proof is almost identicalto what we shall present in Chapter 4. We will use the proposition to provethat V is strictly concave on the polar caps.

LEMMA 2.34. If 038(x, 0) < 0 for x,,(0) < x < 1, then 0.,(x, t) < 0 forx*(t)<x<1 and allO<t<T.

PROOF. First we show that the Ricci tensor satisfies

(2.56) at Re = A Re +U * Rc,

5. THE NECKPINCH 49

where U is the (2,0) tensor given by

.U = (K1 -- Ko) [(n - 1) ds2+,02

9can]

T o verify (2.56) at any given point (x, P) E (-1, 1) x S', we adopt the con-vention that Roman indices belong to {0, ... n} while Greek indices belongto {1, ... , n}.

We choose coordinates {y1..... yr'} near P on S' such that g,,,,, hascomponents gao = S at P. Then we set y° = s so that {y°, y1, ... , y"}is a coordinate system near (x, P) on (-1, 1) x S''. The only nonvan-ishing components of the metric g in these coordinates are 900 = 1 andgcea = 'b2gaa. Moreover, all components of the Riemann tensor vanish inthese coordinates except Raooa = 'b2Ko and Rap,3a = *4K1 (a p). Sim-ilarly, all components of the Ricci tensor vanish except Roo = nK0 andRa0, = 02 [K0 + (n - 1) K1]. The evolution equation (2.56) now followsfrom formula (6.7). We have established (2.56) on the punctured sphere(-1, 1) x Sri; by continuity it remains valid at the poles {±1} x S'.

To apply Proposition 2.33, let Nt denote the topological (n + 1)-ball

Nt={(x,P):x>x*(t), PES'}endowed with the metric g (t). Observe that the sectional curvatures K0and K1 are strictly positive on

8Nt = {x* (t)} x S",

because 0 has a local maximum at x,, (t) and 0,, has a simple zero there byLemma 2.31. So Rc > 0 on D.N. tIf 0 0) is strictly convex for all x > x', (0),then Rc 0) > 0 on Nt. So if Rc ever acquires a zero eigenvalue, it mustdo so at some point p E int Alt and time t E (0, T). If Rc (V, V) I (p t) = 0for some vector V E TpNt, then (U * Rc) (V, V) = 0, because U and Rccommute. Hence Lemma 2.33 implies that Rc > 0 on Nt for as long as g (t)exists. The lemma follows immediately.

tityFinally, we prove that when y > 0 is chosen sufficiently small, the quan-

b = I = ')2-,7 I K1 - Ko l

remains bounded in a neighborhood ,t3 of the pole. Because the exponent 77breaks scale invariance, b may be regarded as a pinching inequality for thecurvatures on the polar cap. The bound on b thus lets us apply a parabolicdilation argument that shows that singularity formation on a polar cap underour hypotheses would lead to a contradiction. (We shall study parabolicdilations in greater generality in Chapter 8.)

LEMMA 2.35. Let g (t) be as in Proposition 2.30. If D > 0, no singularityoccurs on the polar cap

((x*(t), 1) x S' (1)) U {P+}.

PROOF. By Lemma 2.28, we may let Co be an upper bound for I('2)tl.Choose ti c (0, T) so that Co(T - ti) < D2/8. Then one has

')(x*(t), t)2 > D2 - Co(T - t) > 8D2

for all t E [t1iT). Now let x1 be the unique solution of 0(xi,ti)2 = 3D2 inthe interval [x*(ti),1]. Lemma 2.28 implies for all t E [ti,T) that

')(xl, t)2< 3D2 + Co(t - ti) < 8 D2 < 'p(x*(t), t) 2.

Thus one has x., (t) < xI < 1 for all t E [t1iT), and hence ,s < 0 and ass < 0on the interval [x1i1] for all t E [ti, T). It follows that the metric distance

d1(t) - s(1,t) - s(xi,t)from (xi, t) to the pole P+ is decreasing in time. Indeed,

d di (t) = l n 9 ds < 0.1

Next let x2 E (xi,1) be defined by (x2,ti)2 = 3D2. Then for t c [t1,T)one has

?G(X2,t)2 < 3D2+Co(T-ti) < 1D2- 8 2

and

*(XI' t)2 > 3D2 - Co(T - ti) > 5D2.

Thus )(XI, t)2 -VJ(x2, t)2 > D2/8. Hence by crudely estimating the quantity1P (X 1, t) + '(x2, t) from below by

V)(x2, t) ? 1/3D2 - Co(T - ti) > D214 = D/2,

we obtain2

&(xi, t) - 4'(x2, t) > D/2 = D/4.

Lemma 2.34 implies that for x E [x2,1), one has

' (xi, t) - 0(x, t) D/4s(xl, t) - s(x, t) ' s(xi, ti) - S(1, ti)

At this point we once again consider the quantity a defined in (2.53). Wefound that L(a) = 0, where L is the differential operator

2

L=at-83 - (n-4) 383+4(n-1)

One can compute that the quantity it __7 satisfies

L(u) _ (4 - 77) us + n21(4s - 77)u

_ (4 7l)j',-2,03 + n21(4V)s - 77)u.

5. THE NECKPINCH 51

We have 6 in the region Q2 = [x2,1) x [t1iT). If we take rt < 462 < 4,then we have L(u) > 0 in Q2. Then by the maximum principle, there existsC < oo such that l al < Cu in Q2. Indeed, b must attain its maximum Con the parabolic boundary of Q2. At the left end (that is at x = x2) wehave u = 0'7 _> (D/64)5, while Corollary 2.23 implies that Ial is boundedby sup I 0) At the other vertical side of Q2 (that is at x = 1) we havea = 0. Since a is smooth, this implies a = O(s(1, t) - s(x, t)). On the otherhand, because u = 0" and b3 = -1 at x = 1, we have limx/1 Ia/uI = 0 forall t < T. Finally, at t = t1, the quantity b is continuous for x2 < x < 1;while we have just verified that b --* 0 as x / 1. Thus b is bounded on theparabolic boundary of Q2, and hence bounded on Q2.

We now perform a parabolic dilation. Let 52 denote the portion ofSn+1 where x > X2. Then we have a solution g(t) to the Ricci flow on 132defined fort E [tl, T). By Lemma 2.34, this solution has Re > 0. Becauseof spherical symmetry, B2 is a geodesic ball in (Sn+1,g(t)) whose radius isbounded from above by dl (t). Since 0 = V) V (X2, t) > D/2 on the boundaryof 82, it follows from the estimate I7Ps1 < 1 that the radius of 52 is boundedfrom below by D/2.

Assume that the sectional curvatures of the metrics g(t) on X32 are notbounded as t / T. Then there is a sequence of points Pk E B2 and timetk E [t2iT) such that I Rm(Pk, tk) -* oo as k -+ oo. We may choose thissequence so that

sup IRm(Q,t)I = IRm(Pk,tk)QECi2

holds for t < tk. Writing xk for the x coordinate of Pk, we note that(xk, tk) -> 0; and from 1 ,0, > b we conclude that

lim dtk (Pk, P+) = 0,

where dt is the distance measured with the metric g(t).Define Ek = IRm(Pk,tk)I-1/2, and introduce resealed metrics

9k (t) = E9(tk + E2t).

k

Let C1 be the constant from Corollary 2.24 for which I Rm C1'-2 holds.Then we have

(xk, tk) :and because > S, we also have

C1dtk (Pk, P+)

SEk

The distance from Pk to the pole P+ measured in the resealed metric gk (0) _Ek2g(tk) is therefore at most C1/6. In particular, this distance is uniformlybounded.

Translating to the resealed metric, we find that gk(t) is a solution ofthe Ricci flow defined for t E (-Ek2tk, 0] on the region 52. The R.iemainn

curvature of gk is uniformly bounded by I Rm I < 1, with equality attainedat Pk at t = 0. One may then extract a convergent subsequence whoselimit is an ancient solution g,,(t) of the Ricci flow on R'+' x (-oo, 0] withuniformly bounded sectional curvatures. The limit solution has nonzerosectional curvature at t = 0 at some point P. whose distance to the originis at most C1/6.

We introduce the radial coordinate r = r(x, tk) = Ek1(s(1, tk) - s(x, tk) ).Then the metric gk(0) = Ek2g(tk) seen through the exponential map at thepole P+ is given by

gk(0) = (dr)2 + Tk(r)2g, with Tk(r) = Ek 10 (s(1, tk) - Ekr, tk)-

The metrics gk converge in C°O on regions r < R for any finite R, and thefunctions Tk hence also converge in C°° on any interval [0, R]. The scaleinvariant quantity a is given by a =ass - S + 1 = pTrr - 'r + 1, andit satisfies l al < CV7 < CE"T". Thus we find that the limit W = lira "ksatisfies

T-rr- r+ 1 =0.Hence for some A, p, one has

(rasinA(r-E.c), A<oo

'µ)- r - p A=oo

Since -Tr = E [S, 1] cannot vanish, the only valid solution is the onewith ) = oc and y = 0, that is ma(r) = r. But then the limiting metricgw = limgk(0) is (dr)2+r2g; to wit, g0 is the flat Euclidean metric. Becausethis is impossible, we conclude that the sectional curvatures of the metricsg(t) are in fact bounded from above for all x E (x2 (t) , 1) and t < T. SinceCorollary 2.24 bounds the sectional curvatures for x E (x,, (t), x2 (t) ), wehave proved Lemma 2.35.

5.4. Convergence to a shrinking cylinder. In the third part of ouranalysis, we derive estimates which indicate that a neckpinch asymptoticallyapproaches the shrinking cylinder soliton (2.41). By the construction inSubsection 5.5 below, we may assume that the solution keeps at least oneneck. Then letting x_ (t) and x+ (t) denote the left-most and right-mostbumps, respectively, we define the `waist'

W (t) _ [x- (t) , x+ (t)]for all times t such that the solution exists. The key to this part of ouranalysis is the quantity

F =: - KI log KI = i log L.

Notice that F is positive in a neighborhood of a neck. An application of themaximum principle will let us bound F from above when it is positive andK1 = L is sufficiently large. The value of this estimate is that the factor

5. THE NECKPINCH 53

log Ki breaks scale invariance. Our bound on F will thus show that Ko/KIbecomes small whenever Kl is large, in particular near a forming neckpinch.

The main results obtained in this part of the proof show that the singu-larity is Type I (rapidly forming) and estimate its asymptotics. (We shalldiscuss the classification of singularities in Section 1 of Chapter 8.)

PROPOSITION 2.36. Let g (t) : 0 < t < T be a maximal solution of theRicci flow having the form (2.43) such that 1b51 < 1 and R > 0. Assumethat the solution has at least one neck.

(1) A singularity occurs at the smallest neck at some time T < oo. Thissingularity is of Type I; in particular, there exists C = C (n, go)such that

IRml <C

T-t(2) There exists C = C (n, go) such that

K

L

[log L + 2 - log Lmin (0)] < C.

(3) Lets (t) denote the location of the smallest neck. Then there areconstants b > 0 and C < oc such that for t sufficiently close to T,one has the estimate

1< V

rmin

(s-s12<1+ CJ

- log rmin rmin

in the inner layer Is - 91 < 2rmin - log rmin, and the estimate

<C, sslog

s

rmin rmin - log rmin rmin - log rmin

in the intermediate layer 2rmin - logr min < s - s < rm;ri , where

rmin (t) = [1 + o (1)] 2 (n - 1) (T - t).

LEMMA 2.37. Let g (t) be a solution to the Ricci flow of the form (2.43)such that < 1 and R > 0 initially. Then there exists C = C (n, go) suchthat

jRmj _C

T-tPROOF. By Lemma 2.34, the curvature remains bounded on the polar

caps. On the waist W (t), we obtain the stated bound by combining Corol-lary 2.24 with Lemma 2.32

LEMMA 2.38. The quantity F = L log L evolves by

(2.57a) Ft = OF + 2log L - 1

( LlogL +2 - log L KL

S( tog L L3)\

gL

(2.57b) -2P(ts I2KLL+2QK,

where

(2.58) P=(n-1)logL-2L (log L-1)and

2

(2.59) Q=n-1- L (logL-1)-F.PROOF. Straightforward computation.

Rather than work with F directly, we shall consider a related quantityF, which is invariant under simultaneous rescaling of the metric g H Ag andtime t H At.

LEMMA 2.39. Let g (t) : 0 < t < T be a maximal solution of the Ricciflow having the form (2.43) such that IVs1 < 1 and R _> 0. For all t E [0, T)and x E W (t), the scaling-invariant quantity

F L [log L + 2 - log (Lmin (0))]

satisfies

(2.60) sup F t) < max n - 1, sup F 0W(t) W(O) )

PROOF. We first consider the case that Lmin (0) _> e2. Then by Corollary2.26, we have Lmin (t) > e2 for as long as the solution exists. We will applythe maximum principle to equation (2.57) in the region where F > n - 1,noting that K > 0 in this region as well. Our assumption that 0 < R =n [-2K + (n - 1) L] implies that

K<n21L.Using L > e2 > e, we conclude that the coefficient P in (2.58) satisfies

P> (n- 1)logL- (n - 1)(logL- 1) = n - 1.Thus when K > 0 and L > e2, equation (2.57) implies the differentialinequality

Ft < AF + 2(lO1gi)

LFS + 2QK.

But if F > n - 1, then the coefficient Q in (2.59) satisfies2Q<-L (logL-1)<0.

Hence

Ft < AF + 2logL - 1L log L

L8F,.

We conclude that if Lmin (0) > e2, then sup t) cannot increase wheneverit exceeds n - 1.

5. THE NECKPINCH 55

To complete the proof we must deal with possibility that Lmin (0) < e2.In this case, we consider a rescaled solution

I

of the Ricci flow. Denoting its sectional curvatures by Ko = -k andKl = L, we have Lmin = A-"'Lmin and K = A-1K. Thus the choiceA = e-2Lmin (0) > 0 implies that L (0) > e2, whence the preceding ar-guments apply to the metric g. We conclude therefore that max{n - 1, F}is nonincreasing, where F = (K/L) log L. Translated back to the originalmetric g (t), this implies that

maxfn - 1, K log (L.in

e2 L)L (0) JJJJJJ

does not increase with time, as claimed.

The bound for F we obtained above is an example of what is called a`pinching estimate' for the sectional curvatures, and is a foreshadowing ofsimilar results we will see in Chapters 6 and 9.

Our next result will let us compare the actual radius zJ (x, t) near aneck with the radius 2 (n - 1) (T - t) of the cylinder soliton which is itssingularity model.

LEMMA 2.40. Let g (t) be as in Proposition 2.36. For t E [0, T), choosex0 (t) E W (t) so that

0 (x0 (t) , t) = rmin (t)Define

a=s(x,t) -s(xo(t),t),Then there are constants 5 > 0 and C < oo such that for t sufficiently closeto T, one has

(2.61) 1(x, t) C v ) 2

rmin (t)1

- log rmin (t) rmin (t)

for Ja < 2rmin -logrmin, and

(2.62)b(x,t)

< Ca

loga

rmin (t) rmin (t) - log rmin (t) rmin (t) - log rmin (t)

for 2rmin (t) - log rmin (t) < a < rmin (t)1-b

PROOF. Let e denote a small positive number to be chosen below. Weregard t as fixed, and consider the neighborhood of the neck xo(t) in which' < e and IV;5I < E. In this region, one always has L _> (1 - E2)/E2, so that

there exists a constant C < oc such that

12 - log Lmin (0)I < CL.

Taking C larger if necessary, we may assume by Lemma 5.4 that i log L < Cas well. This implies that

1-1s C,02 log

1-2

and thus

Hence

17pss C 2 s

og (1 - b) - 2 log /

_4'ss C 1

1 - V)2 - -2 log & 1 - 1109(1-0.2)g

Since we have restricted attention to the region where < e and IV, Ithe denominator of the second factor on the right-hand side obeys the bound

1 -l log(1 - o .2

> 1 - 1 log(1 - e2)2 logo - 2 logE

So by choosing e small enough, we can ensure that

boss < C1 - 4/JS - - logo'

We now further restrict our attention to the region to the right of theneck. There 08 > 0, which allows us to choose the radius 0 as a coordinateand thereby regard all quantities as functions of 0. Then we have

_ d 2 20 o., 0 < Cdlog

(log(1 - s)) 1-02 os -logo'Integrating this differential inequality from the center of the neck (where0 =rmin and os = 0) to an arbitrary point, one gets

d (log u) =C log l

lggn .- log(1 - KGs) :5 C

Ju=rmin - log u

Using the calculus inequalities x < - log(1 - x) and log x < x - 1, one thenobtains

2 log rmin (l;r(2.63) C log C - 1 .

Since we are assuming ?PsI < E, this last inequality will only be useful if theright-hand side is no more than E2. Henceforth we assume that

"( 1

(2.64) rmin <- 4 < (\ rmin

),-2/2crmin.

Then using the fact that e-e2/C < 1 - E2/2C for small e, one finds that(2.64) and (2.63) imply that 10SJ < e and < E, as required.

5. THE NECKPINCH 57

Now we integrate once again to get

-\/Ca>du

Ir//_'

min__1

Substitute u = rminv. Then it follows from the inequalitiesthat 1 < v < 2/5/rmin. By (2.64), this implies that 0 < logyThus we have

rmin<u<V,< -2GE, logrmin

(2.65a)VCa > f b/rm;n - log rmin - log y

dvrmin J1 log v

1 /')/rmin dv(2.65b) 2 - log rmin J

log v1

To put this inequality into a more useful form, we shall use the followingfact, which is left to the reader to verify.

CLAIM 2.41. The function Z : [0, oo) --+ [1, oo) defined by

1Z(e) dv

= J1 log v1

is monotone increasing and has the asymptotic behavior that

Z( as0and

Z [1 + o (1)] logy as /oo.In light of the claim, estimate (2.65) can be recast as

< Z 2v/Car - rmin - log rmin

Estimate (2.61) follows from this and the expansion of for small .

For larger oa we get

< C alog

or,

rmin rmin - log rmin rmin - log rminwhich is exactly (2.62). This estimate will be valid in the region where(2.64) is satisfied and a > 8rmin - logrmin. Using C -log rmin = rm n),

we conclude that (2.62) will hold if /rmin <- (1/rmin)e2/2C+o(1) 11

By Lemma 2.37, the neckpinch forms a Type I (rapidly forming) singu-larity. One can therefore construct a sequence of parabolic dilations nearthe developing neck which converge to a singularity model that is an ancientsolution of the Ricci flow. (See Subsection 4.1 of Chapter 8.) The lemmaabove shows that the singularity model must in fact be the cylinder solution(2.41). It follows that

rmin (t) _ [1 + o (1)] 2 (n --1)(T - t).

In this way, one obtains part (3) of Proposition 2.36 as an immediate corol-lary of Lemma 2.40.

5.5. Neckpinches happen. In this part of the analysis, we show thatthere exist initial data P = 0 (0) meeting our hypotheses. In particular,we construct simple examples obtained by removing a neighborhood of theequator of a standard sphere and replacing it with a long thin neck. Theseexamples satisfy T = A + Bs2 near the equator (for appropriate constantsA and B) and blend smoothly into the standard sphere metric on the polarcaps. Our construction justifies

PROPOSITION 2.42. There exist initial metrics

g = ds2 + T29can

for the Ricci flow on Sn+I which satisfy JWs J < 1, have positive scalar cur-vature, and possess a neck sufficiently small and a bump sufficiently largeso that under the flow, the neck must disappear before the bump can vanish.Hence these solutions exhibit a neckpinch singularity in finite time.

Our method will be to remove a sufficiently large neighborhood of theequator of the round metric ds2 + (cos s)2 9can on Sn+1 and replace with asufficiently narrow neck. Let A > 0, and let B satisfy

0<B<1/2 ifn=20<B<1 ifn>3

Define the functionW (s) + A + Bs2.

LEMMA 2.43. The metric

ds2 + W (s)2 9can

on R x Sn has positive scalar curvature, and the scale-invariant measure ofits curvature pinching

a = W(s)W"(s) - W'(s)2 + 1

obeys the bounds1-B<a<1+B.

PROOF. One computes thatW(s)2

R = -2W(s)W"(s) + (n - 1){1- W'(s)2}n

= n - 1 - 2B + (3 - n)B2s2

W(s)2.

Since Bs2 < A + Bs2 = W (S)2, we find that when n > 3,

W(s)2 R>n-1-2B+3-n=2(1-B).n

5. THE NECKPINCH 59

For n = 2 we getW(5)2 .R>n-1-2B=1-2B.-n

To estimate a, we write

a=WWss-WS +1= 2(W2)83-2Ws +1,

which implies thatB2s2a=B-2A+Bs2+1,

hence that 1-B<a<1+B. 11

Now for A and B chosen as above, we define

A,B (s) =min {W (s) cos s} W)

if ISI !5 SA,Bos s if SA,B < I < 7r/2

where SA,B is the unique positive solution of cos s = A + Bs2. The

function '.3A,B is piecewise smooth an satisfies A < 1 for all s E[-7/2,7r/2] \ {SA,B}. Moreover, it is easy to check that the metric ds2 +A,Bgcan has positive scalar curvature.

We now smooth out the corner that A,B has at SA,B. First we constructa new function Z%IA,B which coincides with ?GA,B outside a small intervalIE + (SA,B - e, SA,B + e) and has a?A,B constant in I. This constantmay be chosen so that 4B is C'. Since A,B is increasing for 0 < s < SA,Band decreasing thereafter, we have dSA,B < 0 in I. Moreover, we alsohave dsA,B < 1 in I. Hence the metric ds2 +A B91-an will have R > 0everywhere.

Now the function IA B is C', and its second derivative d bA,B hassimple jump discontinuities at SA,B ± e. So we may smooth it in arbitrarilysmall neighborhoods of the two points SA,B ± e in such a way that thesmoothed function 'OA,B is CO°, satisfies R > 0 everywhere it is defined, andcoincides with "A,B outside of I2E

Finally, we observe that is possible to perform the construction above insuch a way that the initial neck is small enough that the solution will en-counter a neckpinch singularity before its diameter shrinks to zero. Indeed,it is not hard to check that one may choose A > 0 sufficiently small so thatfor all A E (0, A), the smoothed function "'I'A,B will coincide with -%A,B inthe interval Isl < a. Moreover, its derivatives d OA,B will be bounded forall I s I> a uniformly in A E (0, A) .

We have proved the following.

LEMMA 2.44. There exists a family of initial metrics

9A,B = ds2 + 2GA,B(s) 9c.

such that each metric in the family has positive scalar curvature, satisfiesI

ddsA,B < 1, has a neck of radius rmin (0) = A and has a bump of heightno less that Av" . Moreover, the scale-invariant measure a of its curvaturepinching is bounded uniformly for all A E (0, A).

Lemma 2.32 implies that the solution of the Ricci flow starting from.A,B must lose its neck before time

rmin (0)2 ATA n-1 n-1

On the other hand, the solution will have a bump at some x* (t); and sincelad is uniformly bounded for all solutions under consideration, the height ofthis bump will be bounded from below by

(x,,(t),t)2 > AIB-Ct>

A E (0, A) is small enough, the neck must disappear before the bump canvanish. This concludes our proof of Proposition 2.42

5.6. Single point pinching. Finally, we justify Remark 2.17. Weconsider the special case of a reflection-invariant metric on S'1+I with a singlesymmetric neck at x = 0 and two bumps. (See Figure 2.) For solutionsof this type whose diameter remains bounded as the singularity time isapproached, the neckpinch singularity will occur only on the totally-geodesichypersurface {0} x S. We shall prove this claim by constructing a familyof subsolutions v for v =g.

Recall that x*(t) denotes the location of the right-hand bump. (Forreflection symmetric data, this is the unique point in (0, 1) where 'omax(t) isattained). Recall too that D = limt/T,%(x*(t), t) denotes the final heightof that bump. By Proposition 2.42, we may assume that D > 0. Define thefunction

p(t) = nIt

J D

(,,s1 2 ds dt,

noting that p is monotone increasing in time, so that p(T) = limt/T p(t)exists.

Now let s* (t) = s (x* (t), t) denote the distance from the equator to theright-hand bump. By Lemma 2.21, one has j '.9 < 1; and because atWmax-(n - 1) /"maze one also has V) (s*(t),t) > D for all t E [0,T). Together,these results imply that s*(t) > D for all t E [0,T). Let XD(t) denote theunique point in (0, x*(t)) such that s(XD(t), t) = D. Notice that for anys, the hypothesis of reflection symmetry about x = 0 lets one integrate byparts to obtain the identity

x ,s (x) 28s _ nss as dx = n S + ds

atJ0

TX Jo

5. THE NECKPINCH 61

Since Lemma 2.31 implies that 0,5. > 0 when 0 < s < s*(t), one may thenestimate at any x E [xv(t), x*(t)] that

,

(2.66) s* (t) > s (± t) > nj

t fs(ue) ) 2

ds dt > p(t).

In particular, p(t) is bounded above by the distance from the equator to thebump.

We are now ready to prove that the singularity occurs only on the equa-torial hypersurface.

PROPOSITION 2.45. If the diameter of the solution g(t) remains boundedas t / T, then 0(s,T) > 0 for all 0 < s < D/2.

To establish this result, let to E (0, T) and 8 > 0 be given. For E > 0 tobe chosen below, define

11(8, t) = E{s - [p(t) - p(to - 8)]}.

By (2.66), the finite-diameter assumption implies that p(T) < no, hencethat

sup [p(t) - p(to - 6)]to-6<t<T

becomes arbitrarily small when to - 8 is sufficiently close to T. Proposition2.45 is thus an immediate consequence of

LEMMA 2.46. If p(T) < no, then for any to E (0, T) and 6 E (0, to),there exists E > 0 such that v = 0, satisfies

v (s, t) > v(s, t)

for all points 0 < s < D/2 and times to < t < T.

PROOF. Since 0, to) < 0 and v(s, to) > 0 for all s E (0, s* (to)), onemay choose El such that if 0 < E < El, then v(s,to) > v(s,to) whenever0 < s < D/2. So if the lemma is false, there will be a first time t E (to, T)and a points E (0, D/2) such that v(s, tl = v_(s, 0. At (s, 0, one then has

(2.67) vt < vt = Eas

Pat

as well as v = v, v,3 = v. = E, and vs,s > v_S,s = 0. Hence

vt=vgg+(n-1) 1-v2 v>E(n-2)v+(n-1)(1-v_2)v.q/,2 02

Whenever 0 < E < E2 = f/D, one has v < es < 1// for all 0 < s < D/2and t E [to, T). Then because v > 0 for s E (0, D/2) C [0, s* (t)], one

estimates at (s, t) that

vas , n-2 2)vt - vt < E atp v -(n-1)(1-v-

< E(as , n-1 v

-at -

P2 ,2

=En -JsD()Zds -n21_zv n-1

<'

En - 2,Choose E3 < (n - 1)/[2n'0max(0)]. Then if 0 < E < min {El, E2i E3}, theconsequence t < 0 of Lemma 2.27 implies the inequality v_t - vt < 0. Thiscontradicts (2.67), hence proves the result.

6. The degenerate neckpinch

In contrast to the contents of Section 5, the discussion in this section isheuristic and not fully rigorous. Nonetheless, degenerate neckpinches havebeen rigorously demonstrated [7] for the mean curvature flow of a surfacein lR3. Although an analogous result is still lacking for solutions of theRicci flow, the mean curvature flow is similar in so many respects that it isstrongly conjectured that degenerate neckpinches exist for the Ricci flow.

6.1. An intuitive picture of degenerate neckpinches. Let us con-sider how a degenerate neckpinch should arise. Imagine a family of rota-tionally symmetric solutions

{(Sn,ga (t)) : a E [0,1]}

of the Ricci flow parameterized by the unit interval . When a = 0, letthe initial metric have the profile described in Subsection 5.5. This is asymmetric dumbbell with two equally-sized hemispherical regions joined bya thin neck. (See Figure 2.) We proved in Section 5 that there exist suchinitial conditions which lead to a neckpinch singularity of the Ricci flow atsome time To < oo. On the other hand, when a = 1, let the initial metric bea round metric on the n-sphere. As we saw in Subsection 3.1, such a metricremains round and shrinks to a point at some time Tl < no.

Now suppose that for a close to 1, the initial metric ga (0) is a lopsideddumbbell where the neck is fat and short and one of the hemispheres issmaller than the other. In this case, if a is close enough to 1, the smoothingeffects of the Ricci flow are conjectured to be strong enough to allow thesmaller hemisphere to pull through before the neck shrinks. (See Figure5.) In this case, the metric should eventually acquire positive curvatureeverywhere and shrink to a round point. (This is of course known if ga (0)is a metric of positive Ricci curvature on the 3-sphere; see Chapter 6.)

6. THE DEGENERATE NECKPINCH 63

FIGURE 5. A lopsided dumbell shrinking to a round point

As a increases from 0 to 1, we may thus imagine that the initial metricsga (0) are dumbbells in which the necks become shorter and fatter while oneof the hemispheres becomes progressively smaller. Such metrics can in factbe constructed having positive scalar curvature. Then for each a E [0, 1],the solution will exist up to a time Ta < co when a singularity forms.(By Theorem 6.45, a finite-time singularity occurs at some time T if andonly if the curvature becomes unbounded as t / T. And by Lemma 6.53,a finite-time singularity is inevitable if the scalar curvature ever becomeseverywhere positive.) By the continuous dependence of a well-posed PDE onits initial conditions, one expects that there is some parameter & E (0, 1)such that the neck pinches off for all a E [0, &). On the other hand, oneexpect for all a E (&, 1] that the necks will not pinch off but that the metricswill approach constant curvature while shrinking to a point. (Depending onthe initial family of metrics ga (0), there may of course be more than onesuch bifurcation point in (0, 1) at which the solutions change qualitatively.)For the solution ga (t), we expect that the neck pinches off at Ta < ooexactly at the same time that the smaller hemisphere shrinks to a point. Asillustrated in Figure 6, this should result in a cusplike singularity known ata degenerate neckpinch. (Another discussion of the intuition behind thisexample may be found in Section 3 of [63].)

The degenerate neckpinch is the standard conjectural model for whatis known as a slowly forming singularity. As we shall see in Chapter 8, afinite-time singularity of a solution (M', g (t)) to the Ricci flow is classifiedas Type I and called `rapidly forming' if it occurs at the natural parabolicgrowth rate suggested by the example of the round sphere in Subsection 3.1,hence if there exists C < oc such that

sup IRml . (T - t) < C < oo.M'1 x [0,T)

FIGURE 6. A cusp forming

On the other hand, the singularity is classified as Type Ha if

sup IRml (T - t) = oo.Mn x [0,T)

Type IIa singularities are called `slowly forming'. (We will explain thisterminology below.) The heuristic picture we have described motivates thefollowing question about the formation of slowly forming singularities.

PROBLEM 2.47. Given a compact smooth 3-manifold M3 and a one-parameter family of initial metrics {ga : a E [0, 1] } such that the Ricci flowstarting at gi forms a Type I singularity model which is a quotient S3/I'1 ofthe 3-sphere, while the Ricci flow starting at go forms a Type I singularitymodel which is a quotient (S2 x ]R) /Fo of the cylinder, does there exist

E (0, 1) such that the flow starting at ga forms a Type IIa singularity?

It is expected that at any /9 E (0, 1) where there is a transition in thesingularity model, a Type IIa singularity should form.

6.2. An intuitive picture of slowly forming singularities. Be-cause a Type IIa singularity has the property that

sup IRmH (T - t) = oo,Mn x [0,T)

the terminology `slowly forming' may initially seem counterintuitive. On thecontrary, it is in fact perfectly logical, as we now explain.

Let us first explore why supMn x [o,T) IRmj (T - t) is a meaningful quan-tity for finite-time singularities. Let (M'1, g (t)) be a solution of the Ricciflow defined on a maximal time interval a < t < T, and define

K (t) _ sup IRm (x, t) I .

xEMn

6. THE DEGENERATE NECKPINCH 65

In Theorem 6.45, we shall prove that T < oo only if

lim K (t) = oo.t/'T

Because the metric g scales like (length)2 and Rm = (R fij) is dimensionless, the quantity Rm scales like (length) 2. We will see in Section 3 ofChapter 3 that the Ricci flow is equivalent to a parabolic PDE. Since para-bolic PDE equate time with (length)2, the quantity (T - t) K (t) is naturallydimensionless.

We shall discuss short-time and long-time existence theory for the Ricciflow in Chapters 3 and 7, respectively. (See also Sections 7 and 8 of Chapter6.) In Corollary 7.7, we will prove a short-time existence result which impliesthat

(2.68) T-t> C

- K(t)'where c > 0 depends only on the dimension n. In other words, the naturalquantity (T - t) K (t) is always bounded from below. One classifies finite-time singularities, therefore, by whether or not it is bounded from above.

One says the singularity at time T < oo is rapidly forming if theestimate (2.68) is sharp in the sense that there exists C E [c, oo) such thatfor all t < T one has

c <T - t< CK (t) - K (t)

A rapidly forming singularity is evidently Type I, because (T - t) K (t) < C.On the other hand, if for every C E [c, oo) there exists a time tc < T suchthat

C(2.69) T - tc>

K (tc)'

one says the singularity is slowly forming. Equation (2.69) reveals whythis terminology makes sense: there is more time remaining until extinctionthan the maximum curvature predicts.

The conjectural picture sketched earlier in this section is very usefulfor developing intuition for understanding slowly-forming singularities. Re-gardless of their role in Hamilton's program for obtaining a geometric andtopological classification of 3-manifolds via the Ricci flow, degenerate neck-pinches and indeed any slowly forming singularities are interesting in theirown right. Indeed, a rigorous proof of the existence of degenerate neck-pinches for the Ricci flow and an asymptotic analysis of their behaviorswould be highly valuable in forming productive conjectures concerning cer-tain analytic (as opposed to topological) properties of singularity formation.



Although no examples of slowly forming (Type Ha) singularities on com-pact manifolds are known at the time of this writing, a rigorous example[35] has recently been constructed on 1R2.

CHAPTER 3

Short time existence

A foundational step in the study of any system of evolutionary par-tial differential equations is to show that it enjoys short-time existence anduniqueness. In this chapter, we prove short-time existence of the Ricci flow.Because the flow is quasilinear and only weakly parabolic, short-time exis-tence does not follow from standard parabolic theory. Hamilton originallyused the sophisticated machinery of the Nash-Moser implicit function theo-rem to establish short-time existence in [58]. Our proof follows the elegantmethod suggested by Dennis DeTurck in [36].

1. Variation formulas

Given any smooth family of metrics g (t) on a smooth manifold M7 , onemay compute the variations of the Levi-Civita connection and its associatedcurvature tensors. These calculations will be applied later to derive theevolution equations for these quantities under the Ricci flow. (See Chapter6.) We compute the variation formulas now because the variation of the Riccitensor determines the symbol of the Ricci flow equation when we regard theRicci tensor Rc (g) as a second order partial differential operator on themetric g. (These variation formulas may also be found in [20].)

We first derive the variation formula for the Levi-Civita connection.Although a connection is not a tensor, the difference of two connections isa (2,1)-tensor. In particular, the time-derivative of a connection is a (2, 1)-tensor. In this section, we shall assume that

a(3.1) tgij = hij,where h is some symmetric 2-tensor.

LEMMA 3.1. The metric inverse g-1 evolves by

(3.2) a gij = -9ikgj'hktat

PROOF. Since the Kronecker delta satisfiesSi =

9ike- At,equation (3.1) implies that

0 = a gik) At + gakhk2

67

68 3. SHORT TIME EXISTENCE

Hence(9

atg(atgtk) gktgj' = _gikgJehke

LEMMA 3.2. The variation of the Levi-Civita connection r is given by

k

= 2gke (Vihje +Vjhie - V,hij)(3.3)atr'

PROOF. Recall that

r k = I k1 (aigje + ajgie - aegij)

in local coordinates { xZ } , where ai Hence

a k I a kP

atrij = 2 atg (agj2 +ajgie -agj)

+2gke(ai(a

In geodesic coordinates centered at p E Mh, one has r (p) = 0. It followsthat aiAjk = ViAjk at p for any tensor A; in particular, aigjk (p) = 0 for alli, j, k. Thus we obtain

(p)=

1 W(v2 a a a )(p)atr2j 2g VQatgij

Since both sides of this equation are components of tensors, the result holdsin any coordinate system and at any point.

Since the Riemann curvature tensor is defined solely in terms of theLevi-Civita connection, we can readily compute its evolution.

by

LEMMA 3.3. The evolution of the Riemann curvature tensor Rm is given

(3.4)

ViVjhkp +ViVkhjp - ViVphjkaRQ - I nep

at 'J" 2 -Vi vi hkp - V jV khip + V jV phik

PROOF. In local coordinates {xi}, we have the standard formula

rjp.

Thus we compute

ai I

a (r) rzp + r,k (atr p) - a rk) rip - r k (atrjp)

1. VARIATION FORMULAS 69

As in Lemma 3.2, we use geodesic coordinates centered at p c .M11 to cal-culate that

atR jk (p) = Vi (rk) () - oj (atrik) (p),

and then observe that this formula holds everywhere. The present lemmafollows from substituting (3.3) into this equation.

REMARK 3.4. By commuting derivatives, one can also write the evolu-tion of the Riemann tensor in the form

a1 ep

ViV hip+VjVphik-V Vphjk-VjVkhipRlfj- 9

2ot -R9khgp - Rq phkq )

LEMMA 3.5. The evolution of the Ricci tensor Re is given by:

(3.5)at

Rjk = 2gpq(VgVjhkp+VgVkhjp-VgVphjk-VjVkhgp).

PROOF. This follows from contracting on i = £ in Lemma 3.3.

REMARK 3.6. Recall that the divergence (3.19) of a (2, 0)-tensor is givenby

(Sh)k = - (div h) k = -g'l Vihjk,and denote the Lichnerowicz Laplacian [93] of a (2, 0)-tensor by

(3.6) (ALh)jk - Ohjk + 2g9pR4jk hrp - ggpRjp hqk - ggpRkp hjq.

(The Lichnerowicz Laplacian is discussed in Section 4 of Appendix A). De-noting the trace of h by

H _ tr sh = gpghpq,

one can write the evolution of the Ricci tensor in the form

atR'k -2 [Lhk + VjVkH + V (5h)k + Vk (5h)] .

LEMMA 3.7. The evolution of the scalar curvature function R is givenby:

(3.7) atR = -OH + VpVghpq - (h, Re) .

PROOF. Using Lemmas 3.1 and 3.5, we compute that

atR =(ask) Rik +9jk (Rk)

_ _9ij9ke (ViVjhke - VZVkhie +

0REMARK 3.8. One can also write the evolution of the scalar curvature

in the invariant form

atR = -AH + div (div h) - (h, Re).


The general formula for the evolution of volume is given by the followingobservation.

LEMMA 3.9. The volume element dµ evolves by

atdµ

2 (g23 atgZj) dµ2dµ.

PROOF. If {xi} 1 are oriented local coordinates, then

dµ=

det g dx I A A dxn.

COROLLARY 3.10. The total scalar curvature fm, Rdµ evolves by

dt (fMnR dµ) = fM (2 RH - (Rc, h)) dµ.

We will also need to understand the evolution of length.

LEMMA 3.11. Let -yt be a time-dependent family of curves with fixedendpoints in (Mn, g (t)), and let Lt (yt) denote the length of -yt with respectto the metric g (t). If

then

(3.8) d Lt (yt) = 1 f h (S, S) ds -J

(VSS, V) ds,dt 2 -rt It

where S is the unit tangent to yt, V is the variation vector field, and ds isthe element of arc length.

PROOF. Consider the variation

y : (a, b] x (-e, E) Mn,

regarded as the family of curves -yt (u) = y (u, t). Assume y (a, ) =X E Nlnand y (b, ) . y E Mn. Define the vector fields

U = y* (a/au) and V = y* (a/at)

along y, and set

s (u) _ fu(U, U) 1/2 du.

aThen the unit tangent to yt is

S (U, U)-1/2 U.

2. THE LINEARIZATION OF RICCI

Since [U, V] = 0, we have

=fbg(U,U)1/2

du

1jbg(u,U)_1/2(u,u)dlL

=

b

+ f g (U, U)-1/2 g (U, V V) dua

f(S, S) ds + (S, VSV) ds

((sv)_jvssv)2 b h (SS) ds + ds)a

2. The linearization of the Ricci tensor and its principal symbol

2.1. The symbol of a nonlinear differential operator. Let V andW be vector bundles over M", and let

L : C°° (V) --> C° (W)

be a linear differential operator of order k, written as

L (V) = 1: LaaaV,lal<k

where La E hom (V, W) is a bundle homomorphism (namely, a linear mapwhen restricted to each fiber) for each multi-index a. If ( E COO (T*MT),then the total symbol of L in the direction C is the bundle homomorphism

v [L] (() - L. (njC°')lal<k

Note that a- [L] (() is a linear map of degree at most k in (. The principalsymbol of L in the direction ( is the bundle homomorphism

& [L] (() E L. (III (a') .lal=k

The principal symbol of a differential operator L captures algebraically thoseanalytic properties of L that depend only on its highest derivatives. Thefollowing basic property of the total symbol will be useful: if X is anothervector bundle over .M' and

M:C°°(W)--*C°°(X)is a linear differential operator of order .E, then the symbol of M o L in thedirection ( is the bundle homomorphism

(3.9) Q [M o L] (() = o, [M] (C) o v [L] (() : V - + X


of degree at most k + t in (.We now regard the Ricci tensor Re (g) as a nonlinear partial differential

operator on the metric g,

Rc9. Re (g) : C°° (S2T*M") - C°° (S2T*Mn) .

Here C°° (S2T*Mn) denotes the space of positive definite symmetric (2, 0)-tensors (in other words, Riemannian metrics) and C°° (S2T*Mn) denotesthe space of symmetric (2, 0)-tensors. By Lemma 3.5, the linearization

D (Rc9) : C°° (S2T*Mn) -> C°° (S2T*.Mr )

of the Ricci tensor is given by

[D (Rcg) (h)]jk = 19pq (VqV jhkp +VgVkhjp - VgVphjk - VjVkhgp)

Let ( E C°° (T*.Mi) be a covector. The principal symbol in the direction( of the linear partial differential operator D (Rcg) is the bundle homomor-phism

r Q[D(Rcg)](S):S2T*.A4n--3 S2T*Mn

obtained by replacing the covariant derivative V% by the covector (j, namely

1 1 Cq(jhkp + (q(khjp[&

p(3.10) [D (Rcg)] (() (h)]jk = 2g

-(q(phjk - (j(khgpA linear partial differential operator L is said to be elliptic if its principal

symbol & [L] (() is an isomorphism whenever (0 0. A nonlinear operator Nis said to be elliptic if its linearization DN is elliptic. There is a rich existenceand regularity theory for linear elliptic operators. However, as we shall seebelow, the fact that the Ricci tensor is invariant under diffeomorphism,

(3.11) Re (cp*g) = co* (Re (g)) ,

implies that the principal symbol & [D (Rcg)] (() of the nonlinear partialdifferential operator Rc9 has a nontrivial kernel.

2.2. The Bianchi identities. To increase our understanding of theconsequences of the diffeomorphism invariance of the Riemann curvaturetensor, we shall show that it implies the first and second Bianchi identities.We follow a proof of Kazdan [81], which is implicit in work of Hilbert. Wewill see that the Bianchi identities are a consequence of the Jacobi identityfor the Lie bracket of vector fields, which in turn follows from the diffeomor-phism invariance of the bracket.

Let X, Y, and Z be arbitrary vector fields on a manifold .Mn. Let cotbe the one-parameter group of diffeomorphisms generated by X such thatcoo = idMn.. Diffeomorphism invariance of the Lie bracket implies that

ca [Y, Z] = [cOt X , cot Y]

2. THE LINEARIZATION OF RICCI 73

Differentiating at t = 0, we obtain

[X, [Y, Z]] = dt`at [Y, Z] = [d i ZJ + dt Z)J= [[X, Y] , Z] + [Y, [X, Z]] ,

which is equivalent to the Jacobi identity.To illustrate the basic idea, we first consider the diffeomorphism invari-

ance

R 9] = pt (R [9])of the scalar curvature function R. Linearizing this identity, we get

(3.12) DR9 (Ixg) =,CxR = VxR.

If the variation of g is h, then formula 3.7 of Lemma 3.7 may be written as

DR9 (h) = _giigW (ViVjhke - DiVkhje+ Rikhje) .

Substitutinghij = (.Cxg)ij = V X3 +VjXX

and commuting covariant derivatives yields

DR9 (,Cx9) = -2AV XZ - 2RijVZXj + V VjViXj + VjVjVjXZ

= 2XiVjRij.

Combining this result with equation (3.12), we obtain

2X2VjRii = XiViR.

Since X is arbitrary, this proves that the diffeomorphism invariance of Rimplies the contracted second Bianchi identity:

(3.13) VjRij = 2 ViR.

Similarly, recalling formula (A.2) from Section 2 of Appendix A, weobserve that the diffeomorphism invariance (3.11) of the Ricci tensor impliesthat

(3.14) [D (Rc9) (,CXg)]jk = (Lx Rc)jk = (X,V Rc)+RikVjXi+RjiVkXi.

On the other hand, Lemma 3.5 implies

=[D (Rc s) (h)]jk =2

(V V j hki + V''Vkhji - Ahjk - V jV khi) .

When h = .Cxg, this becomes

[D (Rc9) (Cx9)]jk = RijVkXi +RikVjXi+1

2(XZVjRik +XiVkRiij)

.+2

(V'RejkiXi + vtR$kjiXi )


Combining this equation with (3.14) and again using the fact that X isarbitrary, we obtain the following consequence of the second Bianchi identity

(3.15) ViRjk = 2 (vi ekji + V'Rejki + V jRik + VkRj) .

Notice that this yields the contracted second Bianchi identity when onecontracts on the indices j, k.

Finally, we observe that the diffeomorphism invariance of the Riemanncurvature tensor implies that

[D (Rm s) (rx9)j jk = (L x Rm) fj k

where by formula (A.2), the components of the Lie derivative of the Riemanntensor are

(3.16a)

(3.16b)

(Lx XPVPR k + R$jkViXP + R PkpjXP

+R M. VkXP -RPkVPXE.

On the other hand, Lemma 3.3 implies that

2 [D (Rm9) (h)Jjk = ViVjhk + ViVkh - VjVEhjk

- VjVihf - VjVkhf +VjVEhik

Substituting h = Lxg and commuting derivatives, we find that

2 [D (Rm9) (Gx9)) jk = ViVjVtXk - ViVEVjXk - VjViVQXk

- VjVEViXk +ViVkVEXj - ViVtVkXj-VjVkVtXi + VjVEVkXi - ViVjVkXQ

+ViVkVjXE + VjViVkXE - VjVkViXE

+ 2ViV jVkXE - 2VjViVkXE.

Thus if we rewrite (3.16) as

(Lx Rm) jk = 91P

=9

X gVgRijkp - V iXgRj - VjXgRRki

-VkXgRjp - VgXPRjk}

-Vi (RkpjXq) - V j (R;kiXq) - VkXgR2jp

-OgXPRj + Xq (VqR"jkp+ V iRkPj + V j RPki)


and compare terms, we get

0 = [D (Rm s) (Lxg)]e k - (Lx Rm) k

Vi I[(Rjpk - Rkpj - RJkp) `YqI

2gep +v j [(RiPk - Rkpi - Rikp ) X q]

-2Xq (VgRijkp +ViRkpjq + VjRpkiq)

Now let p E Mn be arbitrary. Since X is an arbitrary vector field, wemay first prescribe X (p) = 0 and ViXj (p) = gij (p), obtaining

0 = - (Rjeki - Rktji - Rjkei) + (Rikkj - Rkeij - Rikej)Recalling the symmetries Rijke = -Rjike = -Rijkk = Rkeij, we see that thisimplies the first Bianchi identity

(3.17) 0 = Rijke + Rikej + Rijjk

Then if we choose X to be an element of a local orthonormal frame field anduse the first Bianchi identity to cancel terms, we obtain the second Bianchiidentity

(3.18) 0 = VgRijke + ViRjgke + V jRgike

2.3. The principal symbol of the differential operator Rc (g).We shall now explore the principal symbol & [D (Rc g)] of the Ricci tensor,regarded as a nonlinear partial differential operator on the metric g.

Recall that the formal adjoint of the divergence

(3.19a) bg : Coo (S2T*Mn) -' C°° (T*Mn)

(3.19b) (Sgh)k -g'jVihjk

with respect to the L2 inner product

(3.20) (V, W) = fM (V, W) dyg

is the linear differential operator

(3.21a)

(3.21b)

69 : C°° (T*Mn) -, C°o (S2T*Mn)

(b*X)jk _1

2(VjXk + VkXj) =

1

2(Lxog)jk9

In other words, SyX is just a scalar multiple of the Lie derivative Lxpg ofg with respect to the vector field Xa which is g-dual to X. The total symbolof R in the direction (is the bundle homomorphism

o [8;] (() : T*Mn _i S2T*Mn

that acts by

(3.22) (0-1o; (b) (X)) jk = 2 (( Xk + SkXj)


Consider the composition

D (Rc9) o 69 : C°° (T*Mn) -> C°° (S2T*Mn) .

This is a priori a third-order differential operator, so its principal symbol

& [(D (Rc9) o as)] (() : T*.Mn -> S2T*Mn

is the degree 3 part of its total symbol. But as we observed in Section 2.2,the invariance (3.11) of the Ricci tensor under diffeomorphism implies that

[(D (Re9) o 59) (X) ] jk =1

2[GxA (Rcg)]Jk

Since the right-hand-side involves only one derivative of X, we see that itstotal symbol is at most of degree 1 in c. In other words, the principal (degree3) symbol & [(D (Re9) o bg)] is in fact the zero map. Since by (3.9), onehas

0 = & [(D (Rc9) o ss)] (S) _ & [D (Rcg)] (() o & [ay] (C)

it follows thatim (& [Sg] (()) C ker (& [D (Rc9)] (()) ,

hence that & [D (Re9)] (() has at least an n-dimensional kernel in each(n(n + 1)/2)-dimensional fiber:

(3.23) dim (ker (& [D (Rc9)] (())) > n.

REMARK 3.12. In Section 2.2, we proved that the diffeomorphism in-variance of the Ricci tensor implies the contracted second Bianchi identity.They are actually equivalent, since commuting derivatives and applying thecontracted second Bianchi identity reveals that

[(D(Re9)o 5)(X)]jk= 1VVj(VkXp+VpXk)

+ IV Vk (VjXp + V Xj)

- 40 (VjXk +VkXj) + 2VjVk (8 X)

= 1xP (VjRkp + VkRjp + V9Rgjkp + V9Rgkjp)

+ 2(RvkxP+Rvx)

= 2 (XpvpRjk + RPVkXp + RkVjXp)

= 2 [Gxn (Re 9)] jk

Now consider the linear operator

B9 : C°° (S2T*Mn) -+ C°° (T*Mn)


defined by

1 \(3.24) [B9 (h)]k 92 (vihk - 2V khii)

The total symbol of B9 in the direction ( is the bundle homomorphism

S2T*Mn --> T*Mn

given by

(a [B9] (()(h))k = 92' ((ihjk - 2(khii) .

Notice that a [B9] (() is of degree 1 in (. Writing the contracted secondBianchi identity (3.13) in the form

B9(Rc9) = 0

shows that Re 9 belongs to the kernel of B9. Linearizing, we obtain

(D (B9)) [Rc (g + h)] + B9 [((D Rc9) (h))] = 0.

Since the differential operator D (B9) is of order 1, its degree 3 symbol iszero. But it is easy to check that the linear operator B. o D (Re g) is of order3. It follows that the principal (degree 3) symbol

Q [B9 o D (Rc9)] (() = & [B9] (() a Q [D (Rcg)] (()

must be the zero map, hence that

im (Q [D (Rcg)] (()) C ker (& [Bg] (()) C S2T*Mn.

The considerations above have shown that for all (, the following shortsequence constitutes an algebraic chain complex:(3.25)

0T*MnP[b S2T*Mn&[D(Rc9)](C)5,2T*Mna[Bs](C)T*Mn

) 0.

We shall now show that it is in fact a short exact sequence by provingthat the nontrivial kernel of & [D (Rc9)] (() (hence the failure of D (Rc9)to be elliptic) is due only to the invariance (3.11) of the Ricci tensor underdiffeomorphism. We begin by studying the kernel of v [B9] ((). Given anonzero 1-form (, define

AC _ {(®X + X (9 (- ((,X)g : X ET*M'} C S2T*.Ml

and

KC _ ker (a [B9] (()) C S2T*Mn.

Clearly, dim AC = n in each fiber where ( 0. In any such fiber, thecalculation

( [Bg](()(h),X)=2((®X+X(9 (-((,X)9,h)


proves that K( = A-, hence that dim K( = n (n - 1) /2. Now we furtherconsider the kernel of & [D (Reg)] ((). For all h E KS, it follows easily from(3.10) that

[& [D (Re g)] (C) (h)] jk = - 2 I(I2 hjk,

hence that& [D (Rc.9)] (0

2I(12 idKS .

In particular, & [D (Rcg)] K( --- K( is an automorphism in eachfiber where ( 0, which proves that

dim (im (& [D (Rcg)] (())) > dim K( -n (n -

2

1)

By (3.23), this implies that

(3.26) dim (ker (& [D (Rcg)] (c))) = n.

Before we conclude this discussion, it is worth comparing the global andinfinitesimal viewpoints it affords. The facts that

imb9lkerbg

under the global inner product (3.20), and that

C°°(S2T*Mn)=imb9®ker6g

are classical. (See [38] and [15].) On the other hand, define

h _ im (& [8 ] (()) = ker (& [D (Rcg)] (()) .

Then in each fiber where ( 0, we have

S2T*Mn = 1"( + K(.

However, this decomposition is not orthogonal with respect to the innerproduct induced by g on each fiber. This is because the operator& [D (Re g)] (C) is not self-adjoint: indeed, if h, k E S2T*.Mn, we have

(& [D (Rcg)] (C) (h) , k) - (h, & [D (Rcg)] (k))

= 2 ((trek) h - (trgh) k, c

3. The Ricci-DeTurck flow and its parabolicity

The short exact sequence (3.25) show that the nonlinear differential op-erator Rc g is not an elliptic operator on the metric g. Because of this, wecannot immediately apply standard theory to conclude that a unique solu-tion of the Ricci flow exists for a short time. In spite of this fact, the Ricciflow does enjoy short-time existence and uniqueness:

THEOREM 3.13 (Hamilton). If (Mn, go) is a closed Riemannian mani-fold, there exists a unique solution g (t) to the Ricci flow defined on somepositive time interval [0, E) such that g (0) = go.

3. PARABOLICITY OF THE RICCI-DETURCK FLOW 79

REMARK 3.14. We shall see in Corollary 7.7 that the lifetime of a max-imal solution is bounded below by

c

maxMn JRm [9o] 190'

where c is a universal constant depending only on n.

Hamilton's original proof [58] of this result relied on the Nash-Moser in-verse function theorem and was lengthy and technically difficult. However,he showed that the degeneracy of the equation is due only to its diffeo-morphism invariance. Soon after, DeTurck found a much simpler proof ofTheorem 3.13. DeTurck showed that it is possible to modify the Ricci flowand thereby obtain a parabolic PDE by a clever trick: one modifies the right-hand side of the equation by adding a term which is a Lie derivative of themetric with respect to a certain vector field which in turn depends on themetric. Remarkably, one then can obtain a solution to the original Ricci flowequation by pulling back the solution of the modified flow by appropriatelychosen diffeomorphisms.

To motivate how the DeTurck trick is done, we rewrite the linearizationof the Ricci tensor as follows:(3.27)

-2 [D (Rc 9) (h)] jk = O hjk

- V j(gpgV ghpk) - V k (gpgV ghpj) + V jV k (gPghgp)

+ 2g'Rgjk hrp - g9PRjp hkq - g'Rkp hj4

Define a 1-form V = V (g, h) by V = B9 (h) where B9 is given by (3.24), sothat

1Vk _ gpgVPhgk - 1Vk(gpghpq).

We may then express the linearization of the Ricci tensor as

(3.28) -2 [D (Rc9) (h)]jk = 0 hjk - VjVk - VkVj +Sjk,

where the symmetric 2-tensor S = S (g, h) is defined by

Sjk = 2gg7Rgjk hrp - ggPRjp hkq - gaPRkp hjq

Note that S involves no derivatives of h. The 1-form V may be rewritten as

Vk = 19pq (VPhgk +Vghpk - Vkhpq) = 9pg9kr [D (r9) (h)]Pq ,

whereD (I'9) : C°° (S2T*Mn) , C°° (S2T*A4 n (@ TM )

denotes the linearization of the Levi-Civita connection and is given by

[D (r9) (h)]'. = as

whena g=h.

Loo


Now let r be a fixed torsion-free connection. (For instance, we couldtake I' to be the Levi-Civita connection of a fixed background metric g.)The considerations above lead us to define a vector field W = W (g, f) by

(3.29) Wk = gpq (Fpkq - rPq)

Since the difference of two connections is a tensor, W is a globally well-defined vector field (independent of the coordinates used to describe it lo-cally). Because W involves one derivative of the metric g, the map

P = P(F) : C°° (S2T*Mn) -> Coo (S2T*.Mn)

that corresponds to taking the Lie derivative of g with respect to W, namely

P(g) - £wg,is a second-order partial differential operator. The linearization of P is

(3.30) [DP(h)]jk = VjVk +VkVj + Tjk,

where Tjk is a linear first-order expression in h. Comparing the equationabove with equation (3.28) leads us to consider a modified Ricci operator,namely

Q _ -2 Rc +P : C°° (S2T*Mn) --- C°O (S2T*Mn) .

From (3.28) and (3.30), the linearization of Q satisfies

DQ(h) =Oh+'U,where Ujk = -2Sjk + Tjk is a linear first-order expression in h. Hence theprincipal symbol of DQ is given by

(3.31) a [DQ] (()(h) = I(I2 h,which implies in particular that Q is elliptic.

DeTurck's strategy for constructing a unique short-time solution g (t) ofthe Ricci flow

(3.32a) at g = -2 Rc (g) ,

(3.32b) g (0) = 90

on a closed manifold .Mn proceeds in four steps:STEP 1. One defines the Ricci-DeT urck flow by

(3.33a)aatgij = -2Rij + ViWj + V jWi ,

(3.33b) g (0) = go,

where the time-dependent 1-form W is g-dual to the vector field (3.29). Inparticular,

(3.34) Wj = gjkWk= gjkgpq (Fq - rpq)

depends on g (t), its Levi-Civita connection F (t), and the fixed backgroundconnection F. It follows from (3.31) that the Ricci-DeTurck flow is a strictly

parabolic system of partial differential equations. It is a standard result thatfor any smooth initial metric go, there exists E > 0 depending on go such thata unique smooth solution g(t) to (3.33) will exist for a short time 0 < t < E.

STEP 2. One observes that the one-parameter family of vector fieldsW (t) defined by (3.29) exists as long as the solution g (t) of (3.33) exists.Then one defines a 1-parameter family of maps cot : M' ---> M' by

(3.35) at cot (p) = -W (cOt (p) , t)

coo=idMn.

Notice that cot (p) is constructed by solving a non-autonomous ODE at eachp E M. Because M'1 is compact, it follows from Lemma 3.15 (below) thatall cot (p) exist and remain diffeomorphisms for as long as the solution g (t)exists, namely for t E [0, e).

STEP 3. One observes that the family of metrics

(3.36) (t) - cptg (t) (0 < t < E)

is a solution to the Ricci flow (3.32). Indeed, we have g (0) = g (0) = go,because cpo = idMn. Then we compute that

a(cot+sg (t + s))at (cpeg (t)) =

8 s=o

= `ot (a (t)) + a (wt*+sg (t))-o

co (-2 Rc (g (t)) + £w(t)g (t))

+ - [(SOt 1 ° cot+s) cotg (t)]CA

aS s=0

= -2 Rc (coag (t)) + cPi ('Cw(t)g (t)) -,C[(,_ 1),w(t)] (wtg (t))

= -2Rc(4*g(t)).The equality on the penultimate line follows from the identity

a(cOt` ° cot+s) = (cot-') ( Cpscot+s J = (`pt 1) * W (t)

19s=o s=o

Hence g (t) - cotg (t) is a solution of the Ricci flow for t E [0, e). Thiscompletes the proof of the existence claim in Theorem 3.13.

STEP 4. All that remains is to show that g (t) - cot*t g (t) is the uniquesolution of the Ricci flow with initial data g (0) = go. This will be easierto do after we have the machinery of the harmonic map heat flow at ourdisposal. Consequently, we shall postpone the proof of uniqueness: see Step4 of the existence and uniqueness proof outlined in Section 4.4.

3.1. Existence of the DeTurck diffeomorphisms. In this subsec-tion, we establish the existence of a family {cpt} of diffeomorphisms solvingthe ODE (3.35).

LEMMA 3.15. If {Xt : 0 < t < T < oo} is a continuous time-dependentfamily of vector fields on a compact manifold M7`, then there exists a one-parameter family of diffeomorphisms {cot : ,/Vl" -* .M' : 0 < t < T < oo} de-fined on the same time interval such that

(3.37a) att (x) = Xt [(Pt (x)]

(3.37b) coo (x) = x

for allxGM" and tE [0, T).PROOF. We may assume there is to E [0, T) such that cos (y) exists for

all 0 < s < to and y E M". Let t1 E (to,T) be given. We shall show that cptexists for all t E [to, tl]. Since t1 is arbitrary, this implies the lemma. Givenany xo E M", choose local coordinate systems (U, x) and (V, y) such thatTO E U and coto (xo) E V. As long as x E U and cpt (x) E V, the equation(3.37a) is equivalent to

at [Y o cot o x-1 (p)] = Y. [aattt

[x-I(p)]

= (y*Xtoy-1) (yocptox-1 (p))

for p E x (U) such that cot o x-1 (p) E V. Setting zt = y o cpt o x-1 andFt = y*Xt o y-1, we get

aatzt = Ft (zt)

where zt and Ft are time-dependent maps between subsets of lRn. Thus wesee that (3.37a) is locally equivalent to a nonlinear ODE in ll8". Hence for allx E U such that cot,, (x) E V, a unique solution to (3.37a) exists for a shorttime t E [to, to + 6). Since the vector fields Xt are uniformly bounded on thecompact set M1' x [to, t1], there exists an > 0 independent of x E M" andt E [to, tl] such that a unique solution cot (x) exists for t E [to, to + e]. Sincethe same claim holds for the flow starting at cot+E (x), a simple iterationfinishes the argument.

REMARK 3.16. The lemma may fail to be true if M" is not compact.For example, if M1 = I[8 and Xt (u) = u2, then we have the ODE

atcPt(x) [cot (x)]2

coo (x) = x

whose solution is cot (0) = 0 if x = 0 and1

r X-1 -tif x 0. If x < 0, the maximal solution exists for t E (x-1, oo). If x > 0, themaximal solution exists fort E (-oo, x-1). In this case, we note that onlycpo is a diffeomorphism of 1[8, whereas for t > 0 we have the diffeomorphism

cot : (-oo, t-1) -f (-t-1, Cc) .

(To see this, note that if x > 0, then t < x-1 implies that x < t-1.)

3.2. DeTurck's notations. This subsection (which is not needed forthe sequel) describes DeTurck's original formulation of the Ricci-DeTurckflow.

Define the algebraic Einstein operator

G : C°° (S2T*.Mn) - C°° (S2T*Mn)

by

C (v)ij - vij - 2 (tr 9v) 9ij,

noting that G takes the Ricci tensor to the Einstein tensor:

G(Rc) = Rc-2Rg.

Recall the divergence S and its formal L2 adjoint S* introduced in (3.19) and(3.21), respectively, noting that S* of a 1-form X is just a scalar multiple ofthe Lie derivative of g with respect to the vector field metrically dual to X.Using these operators, we can rewrite the linearization of the Ricci tensor(3.27) as

[D (Rc9) (h)]jk = -2 [Ahk + 2Rpjkghpq - Rjhtk - Rkhje] +Yjk,

where

1 t 1 t 1

Yjk = -V jV htk + 2 VkV htj - 2 VkVj (tr(vh9h)

= 2Vj 1 Vk (trsh)) + 217k, - 2Vj (tr,h))

=2VjVI 1 (tr 9h) 9tk) + 2 VkVt Chtj - 2 (tr 9h) 9tj/

_ - [5* (S [G (h)])]jk

Thus by recalling the Lichnerowicz Laplacian defined in (3.6), we obtain thefollowing result.

LEMMA 3.17. The variation of the Ricci tensor has the form

D (Rc9) (h) = -2 (OLh) - [5* (S [C (h)])].

We now reconsider the Ricci-DeTurck flow (3.33) and rewrite the 1-forms W (t) by computing at any point p E Mn in a coordinate systemchosen so that (F9(t)) (p) = 0. Since we have

(V 9(t) )i Q = ai"


at p for any tensor Q, it follows from (3.34) that the identityk k

Wj (t) = gjk (t)9pq (t) ((Fg(t)) i j - (r9)pq

_ -9jk (t) 9pq (t) (r9)pq(V- -2gjk(t)9pq (t) gke ((g())p1q + (g(t))ggpe - (Vg(t))e9pq)

= 9jk (t) 9ke (5 [G (9)])e

holds at p. Thus if one defines

9-I : C°° (T*Mn) --+ C' (T*Mn)by

one can write

(g-'Q)j 9jk (t) 9ktQe,

W(t) =g-'(8[G(9)]).

In particular,

DiWj + V WW = 2 (8*W)ij = 2 (b* [9-' (b [C (9)])] )ij

Hence the Ricci-DeTurck flow (3.33) can be written in the alternate form

(3.38a)atg = -2 {Rc -b* [g-I (8 [C(§)])] }

(3.38b) 9 (0) = go.

4. The Ricci-DeTurck flow in relation to the harmonic map flow

It is a surprising and useful observation that equation (3.35) for thediffeomorphisms cpt may be interpreted in terms of the harmonic map heatflow.

4.1. The harmonic map heat flow. Let (M', g) and (N'", h) betwo Riemannian manifolds, and let f : Mn -+ N' be a smooth map. Thederivative of f is

df = f* E COO (T*.Mn ® f *TN") ,

where f *TN' is the pullback bundle over Mn. Using local coordinates{xi} on Mn and {ya} on A(, we denote the Levi-Civita connection of gby r (g) and that of h by 1'(h)',,. Then

adf = (df ); (dxj ®aa

= axjj (dxj ®aaa)y y

The induced connection

V : CO° (T*Mn (9 f *TN') --> COO (T*.Mn (D T*Mn ® f *TN'")

is determined by f(f *r)Zo a (rh o f )aQ

4. RELATION TO THE HARMONIC MAP FLOW 85

Hence

V (df) =

where

(Vdf) =Vidjfa

i,9,a

(V df) dx® ® dx3 (9aaya

= a (f) k

axi axe (rg) axk) + Ofp (rh ° f)ay ax

The harmonic map Laplacian with respect to the domain metric gand codomain metric h is defined to be the trace

°g,h f = trg (V (df)) E C°° (f 'TAI")

°g,h f = (gijVidj f ry)a

aye

In components,

(3.39) (°g,hf )ry = [axiaxj - (rg)j axk + ((rh)a ° f) aa

axa]

Given fo : Mn --> N'n, the harmonic map flow introduced by Eellsand Sampson is

(3.40a)

(3.40b)

at=at °g,hf,

f (O) = fo.

This is a parabolic equation, so a unique solution exists for a short time.When f is a diffeomorphism, the Laplacian °g,h f may be rewritten in

a useful way:

LEMMA 3.18. Let f : (Mn, g) -> (N, h) be a diffeomorphism of Rie-mannian manifolds. Then

(°9,hf)ry (x) _ (f (X))

PROOF. We first compute the Christoffel symbols of the pull-back of ametric in local coordinates {x'} on M' and {y' } onNn. Let co : A4' -,V'


be a diffeomorphism. Then if is is a metric on N', we have

'I] j7Xk

1\ axe aya

1a2IP3

ay

a(pa awa a( )* (axiaxi aya

+r(K)c@axi axe by7

a2cP13a

(,P

I)k r ! 7 a(w-1)k

axiaxj ay+ aa axi axe ay7

Hence

Since

r 7awa &Q

axiaxj ()aa axi axi

aa aW13axi axi

we multiply by (co*rc)ZJ to obtain

(3.41)

raxk

-rcaAr(K)aQ = (axiaxi - r ij(w*r,) axk )

a

axk

Putting cp = f and i = (f -1) * g in (3.41) and substituting into (3.39), weget

(3.42) (of)7 =((f1)*g) [-r ((f1)*g)ap +r(h)a

whence the lemma follows.

COROLLARY 3.19. Taking M' = Nm and f to be the identity, we have

(Ag,h id)7 = g`0 (_i' (g)a,Q + r(h)7aa I .

A particular case of this result is of some independent interest:

COROLLARY 3.20. If (M', g) is a manifold of strictly positive Ricci cur-vature, then

id : (Mtm, 9) -+ (M', Rc (9))

is a harmonic map.

PROOF. Note that h + Rc (g) is a metric, and let A denote the globally-defined tensor field

A=F(h)-r(g).

At the origin of a normal coordinate system for g, we haveA j =

2

(Rc_1)ke (V RjE+V1Rt - VeRj)

Since both sides of this identity are components of tensors, it holds every-where. Applying the contracted second Bianchi identity to the result of theprevious corollary, we get

(Ag,hid)k=9Z3A = 2(Rc_1)k1

[gij (ViRje+VjRe-VzRij)] =0.

REMARK 3.21. If Rc < 0, then the result holds for the target manifold(M', - Rc (g)).

4.2. The cross curvature tensor. Corollary 3.20 raises some inter-esting albeit tangential questions. We shall introduce these here only briefly,because they are not needed for the remainder of this volume. However, weplan to provide a more complete discussion in the next volume. (Also see[32] for more details.)

PROBLEM 3.22. Given a Riemannian manifold (M', g), does there exista metric -y 0 g on M' such that id : (Mn, -y) -+ (M'n, g) is a harmonicmap?

Assuming the sectional curvatures of g have a definite sign, the answerin dimension three is given by the cross curvature tensor. If we choose alocal orthonormal frame in which the sectional curvatures are i1 - R2332,K2 = R1331, and !c3 - R1221, then the Ricci tensor corresponds to the matrix

K2 + !c3

Rc= k1+rc3n1 + K2

and the cross curvature tensor c = c (g) of (M3, g) corresponds to thematrix

(K2K3C = KI K3

Jcl 2

More explicitly, one can write c in terms of the Einstein tensor

E = Rc - 2 Rg,

which has eigenvalues -Kl, -t 2i and -?c3. We have

Ctij= (detE) (E-1)y? = 1ipq,,jrsE''prEgs

det g J 2

pqk rsP1L8µ µ pgRkjrs

where pijk are the components of the volume form with indices raised andnormalized so that µl23 = 1 in an oriented orthonormal frame.


EXERCISE 3.23. Verify that

i 1(c-1) j V Cjk = 2 (C-1)i3 VkCij.

Use this to prove that if the sectional curvatures of (M3, g) are negativeeverywhere (or positive everywhere), then c9 is a metric and

id : (M3, Cg) -* (M3, g)

is a harmonic map.

Since we are interested in curvature flows, one may consider the cross-curvature flow. The cross curvature flow, is defined on a 3-manifold ofnegative sectional curvature by

aatg=c(g),

and on a 3-manifold of positive sectional curvature by

aag = -c (g)

When M3 is closed, it can be proven that a solution exists for a shorttime. The main open problem is to determine the long-time existence andconvergence properties of the cross curvature flow. When the initial metrichas negative sectional curvature, the following monotonicity formulas havebeen proven by Hamilton and the first author.

PROPOSITION 3.24. As long as a solution to the cross curvature flowexists, one has

atVol (E) > 0

and

d /' 1 - det E 113dt M3 3 (gijEi3) detg) dt S 0.

Note that Vol (E) is scale-invariant in g. The integrand

3 (g2jEij) - (detE/detg)113

is the difference between the arithmetic and geometric means of -'ti, -K2,and -r13. It vanishes identically if and only if the sectional curvature isconstant. The monotonicity of the integral is surprising, because the metricg is expanding, and the integral scales like g112.

CONJECTURE 3.25. If (M3, go) is a manifold of negative sectional cur-vature, the cross curvature flow with initial condition go should exist for allpositive time and converge.


REMARK 3.26. Ben Andrews [4] has observed that when the universalcover of the initial Riemannian manifold may be isometrically embedded asa hypersurface in Euclidean or Minkowski 4-space, the Gauss curvature flowof that hypersurface effects the cross curvature flow of the induced metric;in this case, he can prove convergence results.

4.3. The DeTurck diffeomorphisms and the harmonic map flow.Recall that the diffeomorphisms got defined by (3.29) and (3.35) satisfy

a

atwt = -W o W = gpq (-rpq + rpq)

If r is the Levi-Civita connection of a metric g, then it follows from Lemma3.18 and equation (3.36) that

9P9 (-rpq + rPqk

=g)P9

(-rpqk + rpq) Wt

Therefore

a

Thus we come to the interesting observation that the DeTurck diffeomor-phisms satisfy the harmonic map heat flow:

LEMMA 3.27. The diffeomorphisms Sot used to obtain the solution g (t) ofthe Ricci flow (3.32) from the solution g (t) of the Ricci-DeTurck flow (3.33)satisfy the harmonic map flow with domain metric g (t) and codomain metric9

4.4. An alternate approach to existence and uniqueness. In[63], Hamilton introduced a variant of DeTurck's argument for short-timeexistence and uniqueness of the Ricci flow on a closed manifold M. Thismethod uses both the Ricci-DeTurck flow and the harmonic map flow, andprovides a particularly elegant proof of uniqueness. Hamilton's strategy isas follows:

STEP 1. Fix a background metric g on M.STEP 2. If a solution to the Ricci flow (3.32) exists, denote it by g (t).

In this case, let cot denote the solution of the harmonic map heat flow

aatt = 9(t),9 Ipt

with respect tog (t) and g. Note that, a priori, we only know that the mapscot exist and remain diffeomorphisms for a short time if g (t) exists.

STEP 3. Observe that if g (t) exists, then

g (t) - (cot)* 9 (t)

is a solution of the Ricci-DeTurck flow (3.33). Indeed, using (3.35) wecompute that

at9 (t) at((1)*(t))

_ (cPt 1)* ((t)) + [(1)*(t)]

_ (cot 1) * (-2 Rc [9 (t)J) + G(ovt)<(W [W(t)]) [9 (t)]

_ -2 Rc [g (t)[ + GW(t) [g (t)]

But since (3.33) is parabolic, we know that a unique solution g (t) does exist.And once we have g (t), we can obtain the diffeomorphisms Wt by solvingthe non-autonomous ODE (3.35) at each point, as in Step 2 of DeTurck'smethod. So

(t)=cPe9(t)does exist after all, and solves the Ricci flow.

STEP 4. We now prove that a solution of the Ricci flow is uniquelydetermined by its initial data. Suppose gl (t) and 92 (t) are two solutionsof the Ricci flow (3.32) on a common time interval. Let (col)t denote thesolution of the harmonic map flow with respect to gl (t) and g. Let (`p2)tdenote the solution of the harmonic map flow with respect to 92 (t) andThen

91 (t) _ (((ol)t)*91 (t) and 92 (t) _ (t)are both solutions of the Ricci-DeTurck flow (3.33). Because gl (0) = 92 (0)and (3.33) enjoys unique solutions, we have 91 (t) = 92 (t) for as long as bothexist. But then both (col)t and (W2)t are solutions of the ODE

at (Wi) t (p) _ -W ((cOi) t (p) , t) (i = 1, 2)

generated by the same vector fieldWk = 9Pq (rq r)

Pq

Hence (col)t = (;02)t for as long as they are both defined, which implies inparticular that

91 (t) = (W I )t 91 (t) = (col)t 92 (t) = 92 (t) .

5. The Ricci flow regarded as a heat equation

Heuristically, the Ricci flow may be interpreted as a nonlinear heat equa-tion for a Riemannian metric. In this section, we develop this point of view,which is related to DeTurck's trick. The first step is to consider the Riccitensor in local harmonic coordinates.

DEFINITION 3.28. Local coordinates {xz} are called harmonic coor-dinates if each coordinate function xi is harmonic:

0 = AXi = 9ik (aiak - r,ka) xi = -gikrjik.

5. THE RICCI FLOW AS A HEAT EQUATION 91

Existence of harmonic coordinates is a straightforward consequence ofstandard existence theory for elliptic partial differential equations.

THEOREM 3.29 (Local solvability of elliptic PDE). Let

F

be a C°° function, and let x0 E R'. If there exists a function v E Ck(U)defined in a neighborhood U of x0 such that

F(x, v, av,... , akv) (x0) = 0

and if F is elliptic at v (that is, if the linearization of F at v is elliptic),then there exists a neighborhood V of x0 and a function u such that

F(x, u, au,.. ., aku)(x) = 0

for all x E V and

a-, u(xo) = a3v(xo)

for all multi-indices j such that j < k.

COROLLARY 3.30. Given a point p E M, there exist harmonic coordi-nates defined in some neighborhood of p.

PROOF. Let {xi} denote geodesic coordinates centered at p. Becauseq (p) = 0, we have

oxi(p) = [gii(ajOj - r jak)xi, (p) = 0.

By the theorem above, there exist a neighborhood V of p and functions {ui}such that Aui = 0 holds in V, with ui(p) = 0 and ajui(p) = ajxi(p) =As a consequence, {u'} are harmonic coordinates in some neighborhoodW C V of P.

Standard elliptic regularity theory implies that if the metric g is in aHolder class Ck,a (respectively CW) in the original coordinates, then theharmonic coordinates themselves are in the class Ck+1,' (respectively CW).In particular, the map from the original coordinates to the harmonic co-ordinates is of class Ck+l,, (Cw). Since transforming a tensor involves atmost first derivatives of the map, it follows readily that g itself has optimalregularity in harmonic coordinates.

LEMMA 3.31. If a metric g is of class Ck,a (CW) in a given coordinatechart, then any tensor of class Ck,, (Co-') in that chart belongs to the sameHolder class in harmonic coordinates. In particular, g itself is of class Ck,a

(CW).

The insight into the Ricci flow which harmonic coordinates provide stemsfrom the following observation.


LEMMA 3.32. In harmonic coordinates, the Ricci tensor is given by

-2R,,j = A (gij) + Qjj (g-1 8g) ,

where A(g,j) denotes the Laplacian of the component gij of the metric re-garded as a scalar function, and Q denotes a sum of terms which are qua-dratic in the metric inverse g-1 and its first derivatives ag.

PROOF. The Ricci tensor is given in local coordinates by

-2Rjk = -2Rggjk = _2 (agrgk - ajrgk + r rqp --8q [9qr (ajgkr + (9k9jr - argjk)]

+ aj [9qr (aggkr + ak9gr - arggk)] + r'krgp - rp rq .

Using Lemma 3.1, we write this as

-2Rjk = gqr (agargjk - agakgjr + ajak9gr - ajar9gk) + 9-1 * g-1 * ag * ag

_ (gjk) - 9grak (rgr9sj) - 9graj (rgrysk) + g-1 * g-1 * ag * a9,

where g-1 * g-1 * ag * ag denotes a sum of contractions whose exact formulais irrelevant. In harmonic coordinates, the second and third terms of thelast line vanish, and we obtain

-2Rjk = A (gjk) + 9-1 * g-1 * ag * ag.

COROLLARY 3.33. In harmonic coordinates, the Ricci flow takes the formof a system of nonlinear heat equations for the components of the metrictensor:

a 1

(3.43)at- gij (9ij) + QZj (9 a9) .

REMARK 3.34. The reader is cautioned that equation (3.43) is not tenso-rial. Since the metric is parallel, it is of course clear that Ag - 0. Moreover,one should not in general expect coordinates which are harmonic at timet = 0 to be harmonic at times t > 0.


Hamilton's original proof of short-time existence [58] used the Nash-Moser implicit function theorem. [57] DeTurck's proof [36, 37] is a sub-stantial simplification. Other presentations of DeTurck's proof may be foundin §6 of [63] and in [112].

The Lichnerowicz Laplacian acting on symmetric 2-tensors (which aswe saw above is essentially the linearization of the Ricci flow operator) isformally the same as the Hodge-de Rham Laplacian acting on 2-forms. SeeRemark A.3 in Appendix A.

CHAPTER 4

Maximum principles

Maximum principles are among the most important tools in the studyof second-order parabolic differential equations. Moreover, they are robustenough to be effective on manifolds. For this reason, they are particularlyuseful for the study of the Ricci flow. In this chapter, we review some basictheorems of this type. For pedagogical reasons, the chapter is organizedas follows: we begin with the most elementary results and then introduceprogressively more general ones. We conclude by stating some powerfuland advanced theorems and then presenting a brief discussion of strongmaximum principles.

1. Weak maximum principles for scalar equations

1.1. The heat equation with a gradient term. The heat equationis the prototype for parabolic equations. One of the most important prop-erties it satisfies is the maximum principle. On a compact manifold, themaximum principle says that whatever pointwise bounds hold for a smoothsolution to the heat equation at the initial time t = 0 persist for times t > 0.

PROPOSITION 4.1 (scalar maximum principle, first version: pointwisebounds are preserved). Let u : Mn x [0,T) ---> IR be a C2 solution to the heatequation

au-_Du

at s

on a closed Riemannian manifold, where Dg denotes the Laplacian withrespect to the metric g. If there are constants Cl _< C2 E R such thatCl < u (x, 0) < C2 for all x E Mn, then Cl < u (x, t) < C2 for all x E M'and t E [0,T).

The proposition follows immediately from a more general result in whichone allows a gradient term on the right-hand side. Let g (t) : t E [0, T) bea 1-parameter family of Riemannian metrics and X (t) : t E [0, T) a 1-parameter family of vector fields on a closed manifold A4'. We say a C2function u : M' x [0, T) --4R is a supersolution to the heat-type equation

8t= Ag(t)v + (X, Vv)

atat (x, t) E M- x [0, T) if

(4.1) at (x, t) > (O9(t) u) (x, t) + (X, Vu) (x, t) .

93

94 4. MAXIMUM PRINCIPLES

THEOREM 4.2 (scalar maximum principle, second version: lower boundsare preserved for supersolutions). Let g(t) : t E [0, T) be a 1-parameterfamily of Riemannian metrics and X (t) : t E [0,T) a 1-parameter familyof vector fields on a closed manifold Mn. Let u : M' x [0, T) - JR bea C2 function. Suppose that there exists a E R such that u (x, 0) _> afor all x E Mn and that u is a supersolution of the heat equation at any(x, t) E Nl' x [0, T) such that u (x, t) < a. Then u (x, t) > a for all x E Mnand t E [0,T).

The idea of the proof is eminently simple: in essence, one just appliesthe first and second derivative tests in calculus.

PROOF. If H : Mn x [0, T) --> JR is a C2 function and (xo, to) is a pointand time where H attains its minimum among all points and earlier times,namely

H (xo, to) = min H,Mn x [o,to]

then

(4.2a)

(4.2b)

(4.2c)

aHat -

VH (xo, to) = 0,OH (xo, to) > 0.

(.m to) < 0

Consider the function H defined by

H (x, t) _ [u (x, t) - a] + Et + e,

where e is any positive number. Note that H > e > 0 at t = 0. Using (4.1),we find that H satisfies

(4.3) aH > OH+ (X, OH) + E

at any point where u < a. To prove the theorem, it will suffice to prove theclaim that H > 0 for all t E [0, T). To prove that claim, suppose that H < 0at some (xl, tl) E Mn x [0,T). Then since Mn is compact and H > 0 att = 0, there is a first time to E (0, ti] such that there exists a point xo E Mnsuch that H (xo, to) = 0. Then since

u (xo, to) = a - etc - E < a,

combining (4.2) with (4.3) implies that

0> OH(xo, to) > OH (xo, to) + (X, VH) (xo, to) + e > e > 0.

This contradiction proves the claim and hence the theorem. 0

1. SCALAR EQUATIONS 95

1.2. The heat equation with a linear reaction term. More gen-erally, one may add in a reaction term. We first consider the case where thereaction term is linear. In particular, if g (t) is a 1-parameter family of met-rics, X (t) is a 1-parameter family of vector fields, and 0 :M" x [0, T) ---* 118is a given function, we say u is a supersolution to the linear heat equation

8v= Ag(t)v + (X, vv) +,Ov

atat any points and times where

(4.4) at > O9(t)u + (X, Vu) + o u.

PROPOSITION 4.3 (scalar maximum principle, third version: linear re-action terms preserve lower bounds). Let u : M' x [0, T) --* R be a C2 su-persolution to (4.4) on a closed manifold. Suppose that for each T E [0,T),there exists a constant CT < oo such that ,0 (x, t) < C.,- for all x E Mh andt E [0, T]. If u (x, 0) > 0 for all x E M'ti, then u (x, t) > 0 for all x E Mnand t E [0,T).

PROOF. Given r E (0, T), define

J (x, t) - e-CTtu (x, t) ,

where C.,- is as in the hypothesis. One computes that

at >O9(t) J + (X, V J) + (13 - Cl) J.

Since ,0 - CT < 0 on M' x [0,,r], one has

at (x, t) > (Og(t) J) (x, t) + (X, V J) (x, t)

for all (x, t) E M'' x [0, T] such that J (x, t) < 0. By Theorem 4.2, oneconcludes that J > 0 on M7 x [0, r). Hence u > 0 on M' x [0, T). SinceT E (0, T) was arbitrary, the proposition follows. O

1.3. The heat equation with a nonlinear reaction term. Nowwe treat the case where the reaction term is nonlinear. In particular, weconsider the semilinear heat equation

(4.5) at = O9(t)v + (X, Vv) + F (v)

where g (t) is a 1-parameter family of metrics, X (t) is a 1-parameter familyof vector fields, and F : JR --> R is a locally Lipschitz function. We say u isa supersolution of (4.5) if

at > Og(t)u + (X, Vu) + F (u)

and a subsolution if

at <- pg(t)u + (X, Vu) + F (u) .

THEOREM 4.4 (scalar maximum principle, fourth version: ODE givespointwise bounds for PDE). Let u : Mn x [0, T) -> R be a C2 supersolutionto (4.5) on a closed manifold. Suppose there exists C1 E R such that thatu (x, 0) > C1 for all x E Mn, and let cpl be the solution to the associatedordinary differential equation

dtsatisfying

Then

Pl(0)=CI.

u (x, t) > VI (t)for all x E Mn and t E [0, T) such that cPl (t) exists.Similarly, suppose that u is a subsolution to (4.4) and u (x, 0) < C2 for allx E M. Let c02 (t) be the solution to the initial value problem

dco2 = F (cP2)dt1P2(0)=C2.

Then

u (x, t) < cp2 (t)

for all x E Mn and t E [0, T) such that c02 (t) exists.

PROOF. We will just prove the lower bound, since the upper bound issimilar. We compute that

at(u-cpl)>A(u-cpi)+(X,V(u-W1))+F(u)-F(Wi).The assumptions on the initial data imply that u - cpi > 0 at t = 0. Weclaim that u - cpl > 0 for all t E [0, T). To prove the claim, let r E (0, T)be given. Since M1 is compact, there exists a constant Cr < oo such thatJu (x, t) I < CT and Icpi (t) < C.. for all (x, t) E .A4' x [0, T]. Since F is locallyLipschitz, there exists a constant LT < oc such that

IF (v) - F (w)l < L,. Iv - wl

for all v, w E [-CT, CT]. Hence we have

at(u-cPi)(u-cPi)+(X,V(u-col))-LTsgn(u-cpi)-(u-(Pi)on Mn x [0, T], where sgn E {-1, 0, 1} denotes the signum function.Applying Proposition 4.3 with 3 -L.,- sgn (u - coi ), we obtain

u - cpi > 0

on Mn x [0, T] . This proves the claim. The theorem follows, since T E (0, T)was arbitrary.

REMARK 4.5. In what follows, we shall freely apply Theorem 4.4 withoutexplicit reference by simply invoking the (parabolic) maximum principle.

2. TENSOR EQUATIONS 97

2. Weak maximum principles for tensor equations

The maximum principle is extremely robust: it applies to general classesof second-order parabolic equations and even to some systems, such as theRicci flow. The following result is a simple example of the maximum prin-ciple for systems. Recall that one writes A > 0 for a symmetric 2-tensor Aif the quadratic form induced by A is positive semidefinite.

THEOREM 4.6 (tensor maximum principle, first version: non-negativityis preserved). Let g (t) be a smooth 1-parameter family of Riemannian met-rics on a closed manifold M. Let a(t) E COO (T*Mn ®s T*.M7`) be asymmetric (2, 0)-tensor satisfying the semilinear heat equation

8t a > Og(t)a + Q,

where 0 (a, g, t) is a symmetric (2, 0) -tensor which is locally Lipschitz in allits arguments and satisfies the null eigenvector assumption that

0 (V, V) (x, t) = (/3 VzVj) (x, t) > 0

whenever V (x, t) is a null eigenvector of a (t), that is whenever

(ajj Vj) (x, t) = 0.

If a (0) > 0 (that is, if a (0) is positive semidefinite), then a (t) > 0 for allt > 0 such that the solution exists.

This result is in a sense a prototype for more advanced tensor maximumprinciples that we shall encounter later. So before giving the full proof, wewill describe the strategy and key concepts behind it.

IDEA OF THE PROOF. Recall that one proves the scalar maximum prin-ciple (for example, Theorem 4.2) by a purely local argument at a point andthe first time when the solution becomes zero. The tensor maximum prin-ciple essentially follows from the scalar maximum principle by applying thetensor to a fixed vector field. To illustrate this, suppose that a > 0 for all0 < t < to, but that (xo, to) is a point and time and v E TXOMn is a vectorsuch that

aZjv3 (xo, to) = 0.

Then azj WiWj (x, t) > 0 for all x E Mn, t E [0, to], and tangent vectorsW E TXM . One wants to extend v to a vector field V defined in a space-time neighborhood of (xo, to) such that V (xo, to) = v and

(4.6a)at (xo,to) = 0,

(4.6b) VV (xo, to) = 0,

(4.6c) AV (xo, to) = 0.

This may be accomplished by parallel translation (with respect to g (to)) ofv in space along geodesic rays (with respect to g (to)) emanating from xo,

and then taking V to be independent of time. To see that the Laplacian of Vvanishes at (xo, to), choose any frame {ei E Tx,Mn} 1 which is orthonormalwith respect to g (to) and parallel translate it in a spatial neighborhood alonggeodesic rays emanating from xo. With respect to this local orthonormalframe, the Laplacian of V is

n

AV (x0, to) _ [Ve2 (Vey V) - VVe2ey V] (x0, to)i=1n

[v0_ vov] (xo, to) = o.i=1

Then at any point in the space-time neighborhood of (xo, to), one has

at(aZj VZ0) (aii) ViV3

= (Acij +oij) Viva.

Now since (aijViVj) (xo, to) = 0 and (aijViVj) (x, to) > 0 for all x in aspatial neighborhood of xo, one has

A (aijViVi) > 0.

But equations (4.6) imply that at (xo, to),

A (aijVZVj) = (Aaij) V1V3.

Combining these observations with the assumption

(/ v vi) (xo,to) > 0shows that

atVj) = A (aijv Vi) +QijvZvi > 0

at (xo, to). Hence, if aijViVj ever becomes zero, it cannot decrease further.

Now we give the rest of the argument.

PROOF OF THEOREM 4.6. Given any T E (0, T), we shall show thatthere exists 6 E (0, -r] such that for all to E [0, T - b], if a > 0 at t = to, thena _> 0 on M x [to, to + 6]. The theorem follows easily from this statement.

Fix any to E [0, 1' - J]. For 0 < a < 1, consider the modified (2, 0)-tensorAE defined for all x E .M7 and t E [to, to + 6] by

where S > 0 will be chosen below. As in the proof of the scalar maximumprinciple, the term e5g makes AE strictly positive definite at t = to because

AE (x, to) = a (x, to) + ebg (x, to) > 0,

and the term e (t - to) g will make

aat E> ata

2. TENSOR EQUATIONS 99

for t E [to, to + 8] when we choose S > 0 sufficiently small, depending on

amax

at.titre X [o,T]

The evolution of AE is given by

a A, = as+eg+a[8+(t-to)] a9a t at at

Since AA, = Aa, we have

atAE > AAE + ,Q + eg + E [8 + (t - to)] at'

which we rewrite as

(4.7a) at AE > AA6 + Q (A, g, t) + [Q (a, g, t) -,3 (AE, g, t)]

(4.7b) + Eg + E [8 + (t - to)] at .

We first choose So > 0 depending on g (t) for t E [0, T] to be small enoughso that on M72 x [to, to + So], we have

a 1

at 9 - 48o

This implies in particular that

(4.8) eg+E[80+(t-to)] at > lag

on MI x [to, to + 80]. Since /3 is locally Lipschitz, there exists a constant Kdepending on the bounds for a and g on Mn x [0,.T] (but not on e) whichis large enough that

/3 (a, g, t) - 0 (AE, g, t) > -Ke [5o + (t - to)] g >- -2KESog

on Mn x [to, to + So]. Then if we choose 8 E (0, So) so small that

18<4K'

we have

(4.9) Q (a, 9, t) - /3 (AE, g, t) > - 2 eg.

Hence combining (4.8) and (4.9) to (4.7) shows that

(4.10) at AE > AA, +,3 (AE, g, t)

on Mn x [to, to + 6]. We claim that AE > 0 on Mn x [to, to + 8]. Suppose theclaim is false. Then there exists a point and time (xl, tI) E .Mn x (to, to + 8]and a nonzero vector v E Tx1Mn such that AE > 0 for all times to < t < t1,but

((A) ij V') (xi,ti) = 0.


Extend v to a vector field V defined in a space-time neighborhood of (xl, t1)by the method described above. Then (4.10) and the null-eigenvector as-sumption imply that at (xi, t1), we have

0 > cat((A,)j V'Vj) _

at i;

> [(A)ij + Oij (AE, g, t)] Vi vi

= A ((A)jjvvi) +Qij (Ae, 9, t) ViVi > 0.

This contradiction proves the claim. Then since b > 0 depends only on

maxMnX[0,r]

0at9

and K, and is in particular independent of e, we can let E \ 0. Theorem4.6 follows.

REMARK 4.7. The proof above corrects a minor oversight in the originalargument of Section 9 of [58], which failed to consider the term.

3. Advanced weak maximum principles for systems

Theorem 4.6 was proved in [58]; it may be regarded as the tensor ana-logue of Theorem 4.2. There is a tensor analogue of Theorem 4.4 whichshows how a tensor evolving by a nonlinear PDE may be controlled by asystem of ODE; it was proved in [59]. We shall state the result here butpostpone our discussion of its proof.

The set-up is as follows. Let Mn be a closed oriented manifold equippedwith a smooth 1-parameter family of metrics g (t) : t E [0, T] and their Levi-Civita connections V (t). Let it : £ -> Mn be a vector bundle over M" witha fixed bundle metric h. Let

V(t) : C°°(£) C°°(£®T*Mn)be a smooth family of connections compatible with h in the sense that forall vector fields X E TM', sections cp, ' E C°° (£), and times t E [0, T] onehas

X (h h h

Define0 (t) : C°° (£ ® T*,Mf) --> C°° (£ ® T*.Mn (g T*.A4n)

for all X E T.Mn, E T*.Mn, and cp E C°° (£) by

txThen the time-dependent bundle Laplacian A (t) is defined for all co EC°° (£) as the metric trace

Aco = tr9V (00)

3. ADVANCED MAXIMUM PRINCIPLES 101

Let F : S x [0, T] -> S be a continuous map such that F (., -, t) : £ ---+ 9 isfiber-preserving for each t E [0, T], and F (-, x, t) : £, --> £x is Lipschitz foreach x E .M' and t c [0, T]. Let K be a closed subset of E. Then we canstate the following result from [59]. Although it is quite general, we still callit a `tensor maximum principle' because we shall apply it in the case thatg (t) is a solution of the Ricci flow and £ is a tensor bundle.

THEOREM 4.8 (tensor maximum principle, second version: ODE givespointwise bounds for PDE). Under the assumptions above, let a (t) : 0 < t <T be a solution of the nonlinear PDE

ata Aa + F (a)

such that a (0) E K. Assume further that:

K is invariant under parallel translation by ' (t) for all t E [0, T];andK,x _ K n 7r-1 (x) is a closed convex subset of £x = 7r-1 (x) for allxE.M71.

Then if every solution of the ODE

dt a= F (a)

a (0) E Kx

defined in each fiber S. remains in 1C, the solution a (t) of the PDE remainsin K.

There are two important generalizations of this result which will beuseful for us. Both are proved in [33] (though some special cases appearin Hamilton's work). In the first, one allows the set K to depend on time,obtaining a time-dependent maximum principle for systems.

THEOREM 4.9 (tensor maximum principle, third version: ODE controlsPDE in time-dependent subsets). Adopt the assumptions above, allowingK (t) to be a closed subset of £ for all t E [0,T]. Let a (t) : 0 < t < Tbe a solution of the nonlinear PDE

ataAa+F(a)

such that a (0) E K (0). Assume further that:

the space-time track UtE[0,T] (K (t) x {t}) is a closed subset of S x[0, T];K (t) is invariant under parallel translation by V (t) for all t E[0, T); andK,, (t) - K (t) n 7r-(x) is a closed convex subset of E__ for allx E .M' and t E [0,T].

t a = F (a)

a (0) E Kx (0)

defined in each fiber EE remains in K., (t), the solution a (t) of the PDEremains in K (t).

The final generalization we will state is an enhanced version of the resultabove. It is motivated by applications in which the ODE solution mightescape from a certain subset A C K, but where we may be able to assumethat the solution of the PDE avoids A. Accordingly, we call A (t) C K (t)the avoidance set.

THEOREM 4.10 (tensor maximum principle, fourth version: ODE controlsPDE outside avoidance sets). Adopt the assumptions above, where K (t) is aclosed subset of E for all t E [0, T]. Let a (t) : 0 < t < T be a solution of thenonlinear PDE

at a = Da + F (a)

such that a (0) E K (0). Assume further that:the space-time tracks UtE[o,T] (K (t) x {t}) and UtE[o,T] (A (t) x {t})are closed subsets of E x [0, T];K (t) is invariant under parallel translation by V (t) for all t E[0,T);Kx (t) (t) n 7r-(x) is a closed convex subset of Ex for allxEM' andtE[0,T]; anda (t) avoids A (t), meaning that a (t) 0 A (t) for any t E [0, T].


dta=F(a)a (0) E K, (0) \A, (0)

defined in each fiber Ex either remains in K,, (t) for all t E [0, T] or elseenters A (to) at some time to E (0, T], the solution a (t) of the PDE remainsin K (t).

In the successor to this volume, we will present detailed proofs of all theresults stated in this section.

4. Strong maximum principles

It is well known that any nonnegative solution u of the heat equation

ut Au

has u > 0 for all t > 0 unless u = 0. (See [110].) In particular, if oneknows at t = 0 that u > 0 everywhere and that u > 0 at one point, one can

conclude that u is strictly positive for all positive time. This is an example ofa strong maximum principle. The availability of strong maximum principlesfor parabolic equations is one of the principal advantages of using evolutionequations in geometry. Heuristically, strong maximum principles show thata pointwise bound which holds at some time t can propagate to a globalbound at any time t + E.

Strong maximum principles apply to both scalar and tensor quantitiesevolving under the Ricci flow. We will study their general formulation inthe next volume. But we shall benefit from them in subsequent chaptersof this monograph. For example, if one has a solution of the Ricci flowwith initially nonnegative scalar curvature, the strong maximum principlefor scalar equations implies that R > 0 for all times t > 0 that the solutionexists - unless all g (t) are scalar flat metrics. (See for instance Section 7 ofChapter 5.) The strong maximum principle for tensors is also highly useful.We shall sketch a key example here; rigorous arguments will appear in thechapters which follow. If (M3, g (t)) is a solution of the Ricci flow withnonnegative sectional curvature at t = 0, then the eigenvalues v < IL < A ofthe curvature operator satisfy exactly one of the following patterns

0=v=µ='\0=v=tc<A0<v<tc<A

for all points and times t > 0 such that the solution exists. (The Lie algebrastructure of so (3) rules out the case 0 = v < p < A.) In the first case,the manifold is flat; in the second case, it splits as the product of a one-dimensional factor with a positively curved surface; and in the third case,the manifold has strictly positive sectional curvature everywhere. Because(as we shall see in Corollary 9.7) any solution of the Ricci flow constructedas a limit of parabolic dilations about a finite-time singularity in dimensionn = 3 has nonnegative sectional curvature, this rigidity - a consequenceof the strong maximum principle for tensors - is very important for theanalysis of singularities.


Maximum principles (weak and strong) are among the most importanttechnical tools used to study the Ricci flow. They have numerous applica-tions. For example, the scalar maximum principle implies that the boundR > Rmin (0) for the scalar curvature is preserved (Lemma 6.8). The ten-sor maximum principle implies that nonnegative Ricci curvature and hencenonnegative sectional curvature is preserved in dimension n = 3 (Corollary6.11) and that nonnegativity of the curvature operator is preserved in alldimensions (Corollary 6.27).


A nice presentation of the scalar maximum principle for generalized heatequations may be found on pages 101-102 of [561. The maximum princi-ple for tensor equations (Theorem 4.6) first appeared in [58]. The tensormaximum principle for systems (Theorem 4.8) appeared in [59]. Maximumprinciples for systems with time-dependent convex sets (Theorem 4.9) andavoidance sets (Theorem 4.10) are studied in [33]. Certain maximum prin-ciples of these types are used implicitly in some of Hamilton's work.

A reader interested in discussions of maximum principles for more gen-eral classes of equations is encouraged to consult [110] and [121].

CHAPTER 5

The Ricci flow on surfaces

One of the triumphs of nineteenth-century mathematics was the Uni-formization Theorem, which implies that every smooth surface admits anessentially unique conformal metric of constant curvature. This result maybe interpreted as the statement that every 2-dimensional manifold admits acanonical geometry. Looked at another way, it may be regarded as a classi-fication of such manifolds into three families - those of constant positive,zero, or negative curvature. In this sense, the Uniformization Theorem maybe regarded as a forerunner, at least in spirit, of current efforts to classifythe geometries of closed 3-manifolds.

In this chapter, we study the Ricci flow on closed 2-dimensional mani-folds. We shall show that a solution of the normalized Ricci flow exists for alltime and converges to a constant-curvature metric conformal to the initialmetric. The existence of a constant curvature metric in each conformal classis a classical fact which is equivalent to the Uniformization Theorem. So inthis sense, the Ricci flow may be regarded as a natural homotopy betweena given Riemannian metric and the canonical metric in its conformal classwhose existence is guaranteed by the Uniformization Theorem.

If (M2, g) is a compact Riemannian surface (to wit, a 2-dimensionalRiemannian manifold) we denote its average scalar curvature by

r_ (JM2R)/(fM2 dttl

By the Gauss-Bonnet theorem, r is determined by the Euler characteristicX (M2) of the surface, hence is independent of the metric g. The objectiveof this chapter is to prove the following result.

THEOREM 5.1. If (M2, go) is a closed Riemannian surface, there existsa unique solution g (t) of the normalized Ricci flow

a _atg=(r-R)g

g (0) = go.

The solution exists for all time. As t --* oo, the metrics g (t) convergeuniformly in any CI`-norm to a smooth metric of constant curvature.

One of the objectives of this chapter is to introduce important tech-niques. Therefore, in the interest of pedagogical clarity, we shall not always

105

106 5. THE RICCI FLOW ON SURFACES

attempt to present the most efficient proofs known. The organization of thechapter is as follows. In Sections 1 through 4, we study the evolution ofa metric on an arbitrary surface and derive certain a priori estimates thatapply to all 2-dimensional solutions (M2, g (t)) of the Ricci flow. Then weestablish Theorem 5.1 by considering the three natural cases. In Section 5,we prove exponential convergence to a metric of constant negative curvaturein the case that X (M2) < 0. In Section 6, we prove convergence to a flatmetric on surfaces satisfying X (M2) = 0. In the remainder of the chapter,we treat the far more difficult case that X (M2) > 0, following the strategyoutlined in Section 7.

REMARK 5.2. The Ricci flow still has not provided a complete proof ofthe Uniformization Theorem if X (M2) > 0. As we shall see in Section 7, theproof of Theorem 5.1 for this case uses the Kazdan-Warner identity to showthat the only Ricci solitons on S2 are the metrics of constant curvature. Theproof of the Kazdan-Warner identity requires the Uniformization Theorem.(In [60], Hamilton gives another proof that the only solitons on S2 aretrivial. But this proof uses the fact that S2\ { point } is conformal to R2,hence also requires Uniformization.)

1. The effect of a conformal change of metric

Let us recall how the curvature, Laplacian, and volume element of a Rie-mannian manifold (M', q) transform under a conformal change of metric.

To study the curvature, we shall use moving frames. Let {ei}Z 1 be alocal orthonormal frame field: a local orthonormal basis for TM' in an openset U C .Mn. The dual orthonormal basis of T*.Mf (induced coframe field)is denoted by {w}

1and defined by wi (ej) = b for all i, j = 1, . . . , n. The

metric is then given byn

g=twi®w2.i=1

The connection 1-forms wi E S21 (U) are the components of the Levi-Civitaconnection with respect to {ei} 1 and are defined by

Vxei = wi (X) ej,

for all i,j = 1, ... , n and all vector fields X E COO (TMnI u) . They areantisymmetric: wi = We write the first and second structure equationsof Cartan in the form

dwi = w3 A w

Rm z = S2i = &,)j - w A wk.

(Note the Einstein summation convention.)

1. A CONFORMAL CHANGE OF METRIC 107

FIGURE 1. A 2-sphere whose curvature is of mixed sign

In the case of a surface M2, we have dwl = w2 A w2, dw2 = wl A wi, and

Rm2

= dw2-

In particular, the Gauss curvature is given by

K - (R (el, e2) e2, el) = Rm 2 (el, e2)

LEMMA 5.3. If g and h are metrics on a surface M2 conformally relatedby

g=e2uh

then their scalar curvatures are related by

R9 = e-2u (-2Ahu + Rh) .

PROOF. Let If,, f2} be an orthonormal frame field for the metric h. Let{ ,q1, 7721 be its dual coframe field, and denote the corresponding connection1-form by 772. Then

el - e-".fl e2 e-uf2


is an orthonormal frame field for g andl "1 2 - u

We r' w e 77

is an orthonormal coframe field for g. To compute the connection 1-form w2for {ei } with respect to g, we begin by calculating

dwl =eU(d771+duA7]1) =e''(r/2A7]2+f2(u)772A771)

dw2=eu(d772+duA7l2) =eu(771A7JI+f1(u)771A172).

Combining these equations with the general formula

(5.1) w2 = dwl (e2, el) wl + dw2 (e2, e1) w2,

we obtain

w2 = [772 (el) + e-' f2 (u)] w1 - [771 (e2) + e-' fl (u)] w2

=772+f2(u)171-fl(u)7]2.Then by applying this result and the second structure equation, we can writethe curvature 2-form of g as

Rm[9]2=dw2

Rm[h]2+d[f2(u)]A771-d[f1(u)]A172+f2(u)d711-f1(u)d7]2

= Rm [h]2 + f2f2 (u) 772 A 771 - flfl (u) 771 A 772

+ f2 (u) 772 (f1) 772 A 771 - fl (u) 771 (f2) 771 A 772,

where we again used the identity d77k = 771 (fk) 771 A 772 implied by (5.1).Because Ohu = V fl fl u + V f2 f2 u, this proves

Rm[g]2 =Rm[h]1 - (Ahu)771 A72

and hence

Rg = 2Kg = 2 Rm [g]2 (el, e2) = 2e-2u (Kh - Ohu) = e-2u (Rh - 20hu)

To describe how the Laplacian changes, we begin with a general obser-vation.

LEMMA 5.4. Let g (t) be a smooth 1-parameter family of metrics on M.Then

atA9(t)(.qij) Vivi _9k

(giivi (9)2V4

(giigjj))Vk

PROOF. If u is an arbitrary smooth function, we have

at(Du) = at [9ij (aza, -- r'ak) u]

=(gii) ViV;u - gt3 (r) Vku + o (atU)

2. EVOLUTION OF THE CURVATURE 109

where

921atr =('7i

(gj) +V3 (Yi1?) Vi (gij= gke (giivi

C

- 2 pe (9ii9))11

This formula simplifies nicely in the special case of a conformal defor-mation.

COROLLARY 5.5. If g (t) is a smooth 1-parameter family of metrics on.Mn such that

atg = fgfor a scalar function f : N[' ---+ R, then

ate -f0+(2In particular, if n = 2, then

aA_-f°.PROOF. It suffices to calculate

gke (giivi (9ie)2Ve (giL_gjj))

=(1. - 2) vkf.

For later use, we also record the following immediate consequence ofLemma 6.5.

COROLLARY 5.6. If (Mn' g (t)) is a solution of the normalized Ricci flow

2ratg = -2Rc+n g,

where r (t) denotes the average scalar curvature, then

(5.2)at

dm = (r - R) dµ.

2. Evolution of the curvatureIn this section, we compute the PDE for the evolution of the scalar cur-

vature R and study the corresponding ODE obtained formally by ignoringthe Laplacian term. This analysis will let us apply the maximum principleto obtain a lower bound for R.

LEMMA 5.7. If g (t) is a smooth 1-parameter family of metrics on aRiemannian surface M2 such that ag/at = f g for a scalar function f, then

a R-A1f -Rf.

PROOF. Let h be a fixed metric on M2 and let g be conformally relatedto h by g = euh. By Lemma 5.3, the scalar curvatures of g and h are relatedby

R9 = e-u (-Ahu + Rh) .

If ag/at = f g, then au/at = f, and differentiating the equation above withrespect to time yields

atRy -(aat u l e-u (-Ahu + Rh) - e-ulh (aatu) = - fR9 - Aqf.

COROLLARY 5.8. Under the normalized Ricci flow on a surface, we have

(5.3)at R=OR+R(R-r),

where the average scalar curvature r is constant in time.

This type of evolution is known as a reaction-diffusion equation.The Laplacian term promotes diffusion of R, whereas the quadratic reactionterms promote concentration of R. If the right-hand side contained onlythe Laplacian term, the equation would be the heat equation, albeit withrespect to a time-dependent metric; and one would expect R to tend to aconstant as t -> oc. On the other hand, if the right-hand side contained onlythe R (R - r) term, the equation would be an ODE; and the solution wouldblow up in finite time for any initial data satisfying R (0) > max {r, 0}.The answer to the question of how the scalar curvature evolves under thenormalized Ricci flow depends on which term dominates. We shall later seethat it is the diffusion term which determines the qualitative behavior of theequation.

As we discussed in Chapter 4, one may ignoring the Laplacian term inthe PDE (5.3) in order to compare its solutions with those of the ODE

(5.4) its=s(s-r), s(0)=so,

where the function s = s (t) plays the role of R. When r 0 0 and so 0,

the solution of this ODE isr

1- (1-r)ert

\ so

when r = 0, we have s (t) = so/ (1 - sot); and when so = 0, the solution iss (t) = 0. It follows that whenever so > max {r, 0}, there is T < oo given by

T- f -rlog(1-r/so)>0 ifr00l 1/s0>0 ifr=0

such thatlim s (t) = oo.t-iT

3. CURVATURE ESTIMATES USING RICCI SOLITONS 111

On the other hand, the ODE behaves much better when so < min {r, 0}, inwhich case we have

s(t)-r>so-r.The brief analysis above shows that we cannot obtain a good upper

bound for the curvature under the normalized Ricci flow simply by applyingthe maximum principle to equation (5.3). Nevertheless, our formulas for s (t)do allow us to derive useful lower bounds. Set Rmin (t) _ infxEM2 R (x, t).Then we have the following estimates.

LEMMA 5.9. Let g (t) be a complete solution with bounded curvature ofthe normalized Ricci flow on a compact surface.

If r < 0, thenrR - r >

1(I_

- r > (Rmin(0) - r) e''t- Rmr o ) ert

T.-7.70

. If r = 0, thenRmin (0) 1

R >_1 - Rmin (0) t

> tIf r > 0 and Rmin (0) < 0, then

R >r

I/> Rmin (0) a-rt

1 - I 1 - jamn (0 ) ert

Notice that in each case, the right-hand side tends to 0 as t --> no.

In summary, we have uniform lower bounds for the curvature under thenormalized Ricci flow. If the average scalar curvature r is negative, thenRmin (t) approaches its average r exponentially fast. If r is positive andRmin (t) ever becomes nonnegative, it remains so for all time. If r is positivebut Rmin (t) is negative, then Rmin (t) approaches zero exponentially fast.

Since the upper bounds for the curvature which one can derive directlyfrom the maximum principle blow up in finite time, we shall use other tech-niques in the next section to obtain a uniform upper bound for the curvaturewhen r < 0 and an exponential upper bound when r > 0.

3. How Ricci solitons help us estimate the curvature from above

Let 9Rv (Mn) denote the space of metrics of volume V on a manifoldM'. There is a natural right action of the group 0o (M') of volume-preserving diffeomorphisms of M' on 9nv given by (g, co) ' cp*g. If M2 isa surface, then 9RA/2o is the space of geometric structures of area A in agiven conformal class.

Ricci solitons (equivalently, self-similar solutions of the Ricci flow) maybe regarded as fixed points of the normalized Ricci flow acting 93ZA/;t7o.These special solutions motivate us to consider certain quantities that mayguide us in developing estimates for general solutions. It turns out to beparticularly useful to study functions which are constant in space on Ricci


solitons. In particular, we shall obtain upper bounds for the scalar curvatureby estimating such a quantity. These upper bounds are uniform except whenr > 0, in which case they are exponential. (Ricci solitons and self-similarsolutions were introduced briefly in Section 1 of Chapter 2. In a chapter ofthe planned successor to this volume, we will make a more systematic studyof Ricci solitons in all dimensions.)

DEFINITION 5.10. A solution g(t) of the normalized Ricci flow on asurface.M2 is called a self-similar solution of the normalized Ricci flow ifthere exists a one-parameter family of conformal diffeomorphisms cp(t) suchthat

g(t) = P(t)*g(0)

Differentiating this equation with respect to time implies that

(5.5) atg = Gxg,

where X (t) is the one-parameter family of vector fields generated by cp(t).By definition of the normalized Ricci flow, equation (5.5) is equivalent to

(5.6) (r - R)gij = DiXj + V jXi.

If X = - V f for some function f (x, t), we obtain

(5.7) (R - r)gij = 2V Vj f.

DEFINITION 5.11. We say a metric g (t) on a surface M2 is a Riccisoliton if it satisfies equation (5.6). We say g (t) is a gradient Riccisoliton if it satisfies equation (5.7).

Tracing (5.7) yields the equation

(5.8) Of =R-r.This equation is solvable even on non-soliton solutions, because

fM20.

Thus for any solution g (t) of the normalized Ricci flow, the solution f of(5.8) is defined to be the potential of the curvature . On a compact(closed) surface M2, the potential is unique up to addition of function c (t)of time alone, because the only harmonic functions there are constants.

Denoting the trace-free part of the Hessian of the potential f of thecurvature by

we observe that the gradient Ricci soliton equation (5.7) is equivalent toM = 0. Recalling that Rik =

2gik and taking the divergence of M, we


obtain

(divM)i V Mji = Vjvi0jf - 1

ViVjVjf

= Rikvk f + 2 Vi0 f = 2 (RV if + V R) .

Thus when R > 0, the gradient Ricci soliton equation (5.7) implies thatV (log R + f) = 0, hence that log R + f is constant in space. More generally,regardless of the sign of R, we have

0 = 2(div M)i = V1R + (R - r) Di f + rVi f

= ViR+2V VjfVj f +rVif = Vi (R + (Of I2 + rf) .

Hence on any Ricci gradient soliton, there is a function C (t) of time alonesuch that

F+R+1Vf12+rf =C.Since F is constant in space on Ricci gradient solitons, we expect quantitiesrelated to it to satisfy nice evolution equations. In fact, we may alwaysassume that the potential function f itself satisfies a nice equation.

LEMMA 5.12. Let fo (x, t) be a potential of the curvature for a solution(M2, g (t)) of the normalized Ricci flow on a compact surface. Then thereis a function c (t) of time alone such that the potential function f - fo + csatisfies the evolution equation

(5.10) tf =0f + rf.

PROOF. By Corollary 5.5, we have TO = (R - r)0. Thus when wedifferentiate (5.8) with respect to time and use equation (5.3), we obtain

(R - r)2 + A I fo) = -tR = t fo + R(R - r),

which implies that

(o) =A(Afo+rfo)Since the only harmonic functions on a closed manifold are constants, thereis a function y (t) of time alone such that

8tfo = Afo+rfo+y.The lemma follows by choosing c (t) _ -eT't fo e-rT ry (T) dr.

REMARK 5.13. Our normalization for f differs from that introducedin [60]. Compare the result above with Definition 4.1 and the evolutionequation derived in Section 4.2 of that paper.

Applying the maximum principle to equation (5.10) yields a useful esti-mate.

COROLLARY 5.14. Under the normalized Ricci flow on a compact sur-face, there exists a constant C such that

IfI < Cert.

A consequence of this when r < 0 is that the metrics g (t) are uniformlyequivalent for as long as a solution exists.

PROPOSITION 5.15. Let (.MM2, g (t)) be a solution of the normalized Ricciflow on a compact surface with r < 0. Then there exists a constant C > 1depending only on the initial metric g (0) such that for as long as the solutionexists,

1g(0) C g(t) Cg(0).

PROOF. It follows from (5.8) and (5.10) that

tg= (r-R)g= (Af)g= (f_rf).Integrating this equation with respect to time implies

[f(x,t)/t 1

g(x, t) = exp If - f (x, 0) - r J f (xr) dTr] g (x, 0) .0

Applying the corollary, we conclude that for any fixed vector V,

g (V, V)log I (X,t) C (ert + 1)

.

9 (VI V) (X,o)

Now define

(5.11) H=R-r+wfl2.We expect H to satisfy a nice evolution equation, both because f itself doesand because H + r f = F - r is constant in space on Ricci gradient solitons.

PROPOSITION 5.16. On a solution (M2, g (t)) of the normalized Ricciflow on a compact surface, the quantity H defined in (5.11) evolves by

(5.12) aH=OH-2IM12+rH,

where M is the tensor defined in (5.9).

PROOF. Using (5.8), we rewrite equation (5.3) in the form

(5.13) at (R -r) =AR+R(R-r) = A (R-r)+(Af)2+r(R-r).

To compute the evolution equation forI

V f l2, we use equation (5.10) andrecall that Rg = 2 Rc on a surface,(ghj)obtainiing

a VfI2 = at (92JDjVjf) = V fVjf +2 (vf) Vif

_ (R-r)IVf12+2(V (VAf + rVf,

=rlof12+2(oof,Vf)(5.14) =Alofl2-2loofl2+rIVf12.Combining (5.13) and (5.14) gives the result, because

MI2 =IVVf12 - 2

(Af)2.

Applying the maximum principle to equation (5.12) yields a useful esti-mate.

COROLLARY 5.17. On a solution of the normalized Ricci flow on a com-pact surface, there exists a constant C depending only on the initial metricsuch that

tR-r<H<Ce'Combining this result with Lemma 5.9 lets us estimate R both from

above and below.

PROPOSITION 5.18. For any solution (M2, g (t)) of the normalized Ricciflow on a compact surface, there exists a constant C > 0 depending only onthe initial metric such that:

If r < 0, thenr-Cer't<R<r+Ce't

Ifr = 0, thenC <R< C.

Ifr > 0, then1+Ct - -

-Ce-''t < R < r + Ce't

In summary, we now have time-dependent upper and lower bounds forthe scalar curvature that are valid for as long as a solution exists. As weshall see later in this volume, the long-time existence of solutions is a con-sequence of these estimates. For now, we shall assume Corollary 7.2, whichtells us that long-time existence of solutions follows from appropriate a pri-ori bounds on their curvature. To guide the reader's understanding, wesummarize the relevant arguments here.

STEP 1. The smoothing properties of the Ricci flow (Theorem 7.1) revealthat bounds on the curvature imply a priori bounds on all its derivatives

for a short time. We call these Bernstein-Bando-Shi (BBS) derivativeestimates. They originate in Bernstein's ideas [16, 17, 18] for provinggradient bounds via the maximum principle, and were derived for the Ricciflow in [9] and [117, 118]. (We begin our analysis of BBS estimates inSections 5, 6, and 9. We discuss their general n-dimensional formulationand provide further references in Chapter 7.)

STEP 2. Short-time existence results (Theorem 3.13 and Corollary 7.7)tell us that the lifetime of a maximal solution (Ma, g (t)) is bounded belowby

c

maxMtn IRm [9o]Iyo'

where c is a universal constant depending only on n. (See also the doubling-time estimate of Lemma 5.45.)

STEP 3. Long-time existence results (Theorem 6.45) then imply thatthe flow cannot be extended past T < no only if

sup IRm (x, t)I= no.lim (1EA4nt/TBecause of these facts, the bounds in Proposition 5.18 allow us immedi-

ately to apply Corollary 7.2 to get the following long-time existence resultfor solutions on surfaces.

PROPOSITION 5.19. If (M2, go) is a closed Riemannian surface, a uniquesolution g(t) of the normalized Ricci flow exists for all time and satisfies9(0)=go.

In Sections 5 and 6, we will derive bounds for all derivatives of thecurvature on surfaces of nonpositive Euler characteristic, thus enabling adirect proof of long-time existence in those cases. In Section 9, we apply theBBS method to bound the gradient of the scalar curvature on a surface ofpositive Euler characteristic. Finally, in Chapters 6 and 7, we will prove moregeneral results that yield bounds on all higher derivatives of the curvatureand establish maximal-time existence (under appropriate hypotheses) oncompact manifolds in all dimensions.

4. Uniqueness of Ricci solitons

The main objective of this section is to prove that the only gradient Riccisolitons on surfaces are the metrics of constant curvature. This fact will beimportant when we consider the long-time behavior of the normalized Ricciflow on a surface of positive Euler characteristic. We start with a result thatholds in all dimensions.

4.1. Uniqueness of steady and expanding solitons. If (M', g) isa compact Riemannian manifold, let us denote its volume by

V=J dµMn

4. UNIQUENESS OF RICCI SOLITONS 117

and its average scalar curvature by

_ &, RdpP V

If g is a Ricci soliton, then (2.3) implies that

-2 Re = Lxg + 2)g,

whence we get the divergence identity

nA+R=dXby tracing. Integrating this identity shows that

(nA + p) V = 0,

hence that p = -nA. It follows that the average scalar curvature of a solitonthat is respectively expanding, steady, or shrinking must be respectivelynegative, zero, or positive. This is not surprising, because applying Lemma3.9 to the corresponding self-similar solution implies that

tlogV(t)=-p(t).

PROPOSITION 5.20. Any expanding or steady self-similar solution of theRicci flow on a compact n-dimensional manifold is Einstein.

PROOF. We consider the corresponding solution (.Mn, g (t)) of the nor-malized Ricci flow, (See Section 9 of Chapter 6.) Because g (t) flows bydiffeomorphisms, there exists a one-parameter family of 1-forms X (t) suchthat

la2

g = Lxagatg =

Re -n

Thus by Corollary 6.64, we have

- (VR, X) = -GxR = atR = OR + 2 IRc12 - 2pR,

which implies that the identity

(5.15) AR+(VR,X)+2IRcl2-2nR=0

holds identically in time.Now assume g (t) is an expanding or steady self-similar solution. Then

p (t) < 0 as we observed above. Replacing Rc by Re -Rg, one finds thatequation (5.15) is equivalent to

(5.16) A(R - p)+(V (R- p),X)+2Rc-ngl2

+

n(R - p) = 0.

By (5.16), at any point x E Mn such that R (x, t) = Rmin (t), we have22 Rng +

n(R - p) < 0.

Since R (x, t) < p (t) < 0, this is possible only if

Rc-pgn

=02

at (x, t). Tracing then implies that

Rmin (t) = R (x, t) = p (t)hence that R - r is constant in space. Substituting back into (5.16), weconclude that

Rc (., t) = p(t) g (., t)

holds identically, hence that g (t) is Einstein.

4.2. Surface solitons and the Kazdan-Warner identity. Now weprove the main assertion of this section. Since the argument uses a partic-ular formulation of the Kazdan-Warner identity [82], we follow it with adiscussion of that result.

PROPOSITION 5.21. If (M2,g (t)) is a self-similar solution of the nor-malized Ricci flow on a Riemannian surface, then g(t) = g(0) is a metric ofconstant curvature.

PROOF. By Proposition 5.20, we may assume that r > 0. By passing toa cover space if necessary, we may thus assume that M2 is diffeomorphic toS2. Contracting the Ricci soliton equation (5.6) by Rg-1 yields

2R (r - R) = 2R div X,

and hence

XdA.-f(R - r)2 dA= 1

Since X is a conformal Killing vector field, integrating by parts and applyingthe Kazdan-WarnerfS2fS2identity (5.18) implies that

(OR, X} dA=0.

Hence R = r, and the proposition is proved.

We begin our discussion of the Kazdan-Warner identity by stating itsusual formulation. (See for example page 195 of [113].) Let g be an arbitrarymetric on a topological S2. We shall denote the standard round metric onS2 by g and will use bars to indicate geometric quantities associated to g.Let cp be a first eigenfunction of the rough Laplacian 0 on S2. (In otherwords, cp is a spherical harmonic.) By the Uniformization Theorem, one maywrite g = e2ug. Then one can write the standard Kazdan-Warner identityin the form

p (K) e2u dA = 0,(5.17)fS2

c

4. UNIQUENESS OF RICCI SOLITONS 119

where K = R/2 denotes the Gauss curvature of g.We now derive the form of (5.17) that we used in proving Proposition

5.21. The real vector space of conformal Killing vector fields of (S2, g) is6-dimensional. Three of the dimensions arise from vector fields of the formV , where cp is a spherical harmonic. The other three dimensions come fromthe Killing vector fields for g. We claim that if Y is a Killing vector field forg, then

fY(K) e2u dA = 0.2

By the dimension count 6 = 3 + 3, the claim implies that for any conformalvector field X, one has

(5.18) J 2 (X, VK) dA = 152 X (K) dA = 0,

since dA = e2u dA. Note that conformality is well defined, because X isconformal with respect to g if and only if X is conformal with respect to g.

The claim is derived from an integration by parts. Since k = 1, Lemma5.3 implies that

K=e-2u (1-Du).

Recalling that divY = 0 whenever Y is a Killing vector field for g, wecompute that

152Y (K) e2u dA =

1,2(Y, OKg )g e2u dA

_ -2 r K (Y, Vu)g e2u dA.JS2

Now

1s22 (1 - Du) (Y, Vu)g dAK (Y, Vu)g e2u dA = Js

Js2(Au) (Y, Ju)g dA,

because

152(Y,Vu)g dA= - fS2 u divYdA=0.

Integrating by parts again and using the fact that VY is antisymmetric, weobtain

152(Au) (Y, Vu)g_ dA = - 152 ViujuYdA - f Jiu0juV'Y3 dA

z

2 152\Y'V IVulg> dA

=2 J 2 I 'u I9 (divY) dA = 0.

5. Convergence when X (.M2) < 0

In this section, we will prove the following case of Theorem 5.1.

THEOREM 5.22. Let (.M2, go) be a closed Riemannian surface with aver-age scalar curvature r < 0. Then the unique solution g (t) of the normalizedRicci flow with g (0) = go converges exponentially in any C"-norm to asmooth constant-curvature metric g,,, as t --> oc.

By Proposition 5.19, the solution g (t) exists for 0 < t < oo. By Proposi-tion 5.15, the metrics g (t) are all uniformly equivalent. And by Proposition5.18, there exists a constant C > 0 depending only on go such that R isexponentially approaching its average in the sense that

(5.19) IR - rl < Cert

So to prove the theorem, it will suffice to show that all derivatives of R aredying exponentially.

LEMMA 5.23. On any solution (,M2, g (t)) of the normalized Ricci flow,JVRJ2 evolves by

(5.20) !t IVR12 = A IVRI2 - 2 IVVR12 + (4R - 3r) IVRI2.

PROOF. Using (5.3) and the Ricci identity for a surface

VA=AV- 1RV,

we get

a (VR) = V (AR+ R(R - r)) = AVR+ 3RVR - rVR.

Hence

at JVR12 = 8t(gZ3VZRVjR)

= (R - r)IVR12+2(AVR+3RVR-rVR,VR).

Now the lemma follows from the fact that

A JVRI2 = 2 (AVR, VR) + 2 IVVRI2 .

0

COROLLARY 5.24. If (M2,g (t)) is a solution of the normalized Ricciflow such that r < 0, then there exists Cl > 0 such that

(5.21) JVR12 < C1ert/2

PROOF. By (5.19), one has

at1VR12 < A IVRI2 - 2 JVVRI2 + (r + 4Cert) JVR12.

5. CONVERGENCE WHEN X (M2) < 0 121

So for all t > 0 large enough, one gets

atIVRI2 < 0 IVR12

+ 2IVRI2

,

whence the result follows from the maximum principle.

Before treating the general case, we provide another example, in orderto illustrate the role played by the evolution of the Levi-Civita connection.

LEMMA 5.25. On any solution (M2, g (t)) of the normalized Ricci flow,IVVRI2 evolves by

at IVVRI2 = A IVVRI2 - 2 IVVVRI2 + (2R - 4r) IVVRI2

+ 2R (OR)2 +2 (VR, V VRI2) .

PROOF. In dimension n = 2, the standard variational formula (3.3)becomes

atrk29ke [Vi (9ii) + V j VQ

2(VjRJ3 + V -R5 - VkRgij)

We find by commuting derivatives that

ViVjOR = OViVjR - 2RViVjR + (RR +VRJ gij - VRVR.Thus we get

atViV;R = Vivj (R) - (r) VkR

= AViVjR+RLXRgij+2ViRVjR-rViVjRand hence

atIVVR12 =

at(giigkViVjRVkVtR)

= O IVVRI2 - 2 IVVVRI2 + (2R - 4r) IVVRI2

+2R(AR)2+2VR,VJVRI2).

COROLLARY 5.26. If (.M2, g (t)) is a solution of the normalized Ricciflow such that r < 0, then there exists C2 > 0 such that

IVVRI2 < C2ert/2.

PROOF. By (5.19), there exists to > 0 such that R < 0 for all t > to.Applying (5.21) at such times, we find that there exists Cl' > 0 such that

atIVVRI2 < 0 JVVRI2 - 4r VVRI2 + Ciert/2 IVVRI.

Define co > IVVRI2 by

cp- IVVR12-3rIVRI2.

Then there exists t1 > to large enough such that for all t > t1, one has4R - 3r <

4r and hence

tcp < pcp + 2r VVRI2 + (4R - 3r) (-3r IVRI2) + Ciert/2 IVVRI

< Acp+ 3rcp+Ciert/2 IVVRI

< Ac0 + 3 rcp + Ci ert/2 VrV-

< L1cp + 3rcp + C2ert,

where C2 > 3 (Ci )2 / I rI. By the maximum principle, there exists C2 > 0such that v < C2 ert/2, which implies the result.

We are now ready for the general case.

PROPOSITION 5.27. Let (M2, g (t)) be a solution of the normalized Ricciflow on a closed surface with r < 0. Then for each positive integer k, thereexists a constant Ck < oo such that for all t E [0, 00),

sup vkR (x, t)xEM2

2 < Ckert/2

PROOF. The proof is by complete induction on k; we may suppose theresult is known for 0 < j < k - 1. There is a commutator of the sort

Lk/2J

VkAR - LaVkR = E (vjR) ®g (Vk-jR) ,

j=0

where L-1 is the greatest-integer function, and X ®g Y denotes a finite linearcombination of contractions of the tensors X and Y taken with respect tothe metric g (t). Similarly, there are formulas

Lk/21vk (R2) = E (Vi R) ®g (Vk-iR)

j=0

and

(at r) ®g (Oj R) = (V R) ®g (vj R)

Thus by recursive application of the identity

att(ViVj...VgR)-pi fitpj...VgR) - (ii,) pP...p9R,

6. CONVERGENCE WHEN X (M2) = 0 123

we obtainLk/2i

at(vk) = zvkR + E (DAR) ®g (VR) - r (VkR)

j=0

and hence2

atVat (gip...gj9V .V RVj.VgR)

=O VkR2-2 Vk+1R2-(k+2)rVkR2

+(VkR)

®g[Lk/2i

(viR) ®(Vk-,R)]By induction, there exists a time to _> 0 and a constant C such that fort > to one has

at pkR 2 < D p R 2 - (k + 2) r VkR 2 + Cert/2 (1 + OkR) .

The induction hypothesis also implies that

atVk 1R2=O V2 VkR2-(k+1)rjVk-1R2

+ (Vc_1R) ®g [[(k_1)/2] (QjR) ®g (Vk_1_1R)]

< O Vk-1R 2 -2 VkR 2 + Cert12.

Define ? > I VkRl2 by

.p OkR2- (k+1)rpk-1R2

Then there exist C' and C" such that

l8t`I?

< o(D + kr VkR 2 + C'ert/2 (1+ V kRl )

A-15 + 2r(, + Ciiert

As above, the conclusion now follows readily from the maximum principle.O

Our proof of Theorem 5.22 is now complete.

6. Convergence when x (M2) = 0

In this section, we will prove the following case of Theorem 5.1.

THEOREM 5.28. Let (M2, go) be a closed Riemannian surface with aver-age scalar curvature r = 0. Then the unique solution g (t) of the normalizedRicci flow with g (0) = go converges uniformly in any Ck-norm to a smooth

, as t --> oo.constant-curvature metric g.

FIGURE 2. A surface of Euler characteristic zero

By Proposition 5.19, the solution g (t) exists for 0 < t < oo. By Propo-sition 5.15, the metrics g (t) are all uniformly equivalent. To prove the the-orem, we shall show that the scalar curvature and all its derivatives vanishin the limit as t -4 oo.

We begin by deriving an estimate for the decay of the scalar curvature.Recall from Section 3 that the potential of the curvature f satisfies

Af = Rand is normalized by a function of time alone so that

atf Of.

LEMMA 5.29. Let (M2, g (t)) be a solution of the Ricci flow on a closedsurface with r = 0. Then there exists C < oo depending only on g (0) suchthat for all t E [0, oc), the potential of the curvature satisfies

sup lof (x, t)12XEM2 1+t*

PROOF. It follows from Proposition 5.16 that

at lVf12 = A JVf12 -2 IVV f12 .

By the maximum principle, there is Co = Co (g (0)) such that I V f 12 < Co forall time. To improve this estimate, we apply another BBS technique. (Themethod of obtaining gradient bounds via maximum-principle arguments willbe discussed further in Chapter 7. Please see the references cited therein.)Noticing that

andat (t 1Vf12) <A (t 1of12) + 1of12

aft=A(f2)--21of12,

6. CONVERGENCE WHEN X (M2) = 0 125

we obtain the inequality

19

Y (t I°f 12 +f2) < o (t I V f12 + f2) .

Hence there is C1 = C1 (g (0)) such that t I V f 12 + f 2 < C1; in particular, wehave I V f 12 < C1 It. The lemma follows from combining these estimates.

PROPOSITION 5.30. Let (M2, g (t)) be a solution of the Ricci flow on aclosed surface with r = 0. Then there exists C < oc depending only on g (0)such that for all t E [0, oc),

sup (ixt)I + IVf (x,t)I2) <C

xEM2 I+ tThe method of proof is a hybrid of BBS techniques and certain innova-

tions of Hamilton. (See for instance Propositions 5.16 and 5.50.)

PROOF. Recalling that R2 = (f)2 < 2 I VV f 12, we compute

at (R+2 I Vf 12) = 0 (R+2IVf12) +R2 -4IVVf12

(R+2pvfJ2) R2

The motivation to consider this quantity is the favorable -R2 term on theright-hand side. Not only do we have R + 2 I V f I2 < Co (g (0)) by the maxi-mum principle, but we can also apply the BBS method to obtain

at [t (R +2IVf12)J<0[t(R+2IVf12)1-tR2+R+2IVf12

0 [t (R+21vf12)] 2R2+R+2Iof12

- 2(R+2IVf12)2+2t1Vf12

(R+ IVf 2).

Since t I V f I2 < C1 by Lemma 5.29, there is C' < oo so large that at anypoint and time where t (R + 2 I V f 12) > C', we have R > 0 and hence

at[t(R+21v112)] < 0 [t (R+ 2 Iofl2)]

2R2 2(R+21vf12)2

+(1+2C1)R+2(1+C1) lVf12

0 [t (R+2 lVfl2)]2t [t (R +2IV(12)

] 2

[R(1+2cl)]2+.Thus there is C > C' large enough that t (R + 2 I V f 12) > C implies

Y[t(R+2 lVf 12)] <0[t(R+21Vf12)]

By the maximum principle, R + 2 I V f I2 < C/t for all positive times. Be-cause Proposition 5.18 and Lemma 5.29 uniformly bound R + IVf12, theproposition follows easily.

We now proceed to derive suitable bounds on the derivatives of thecurvature. For pedagogical reasons, our method will again emphasize clarityof exposition rather than efficiency. The reader interested only in seeing thegeneral case may proceed directly to Proposition 5.33.

LEMMA 5.31. Let (M2, g (t)) be a solution of the Ricci flow on a closedsurface with r = 0. Then there exists C < oo depending only on g (0) suchthat for all t c [0, cc),

sup VR(x,t)12XEM2

- (1+t)3

PROOF. When r = 0, equation (5.20) takes the form

at(VRI2 = A (VRI2 - 2IVVRI2 +4RIVRI2 < OIVRI2 +4RIVRI2.

Let a be a constant to be chosen later, and consider the quantity

c0 t4 VR12 + at3R2,

which satisfies co 0) = 0. Observing that

and

we compute

at(t4 IVRI') < A (t4 IvR2) + 4t3 (tR + 1) (VRI2

at(t3R2) = 0 (t3R2) - 2t3 I VRI2 + 2t3R3 + 3t2R2,

a cp < Ocp + 4t3 (tR + 1 - a/2) I VRI2 + a (2t3R3 + 3t2R2) .

By Proposition 5.30, one may choose a < oo such that tR + 1 < a/2.Therefore

at` < A`0+C,

where C = a (4a3 + 4a2). Hence by the maximum principle,

t4VRI2+at3R2=cp<Ct.

This method can clearly be generalized to higher derivatives of the cur-vature. To motivate the proof for the general case, we will next estimateIVVRI.

6. CONVERGENCE WHEN x (M2) = 0 127

LEMMA 5.32. Let (M2, g (t)) be a solution of the Ricci flow on a closedsurface with r = 0. Then there exists C < oo depending only on g (0) suchthat for all t E [0, oo),

xsup2

IVVR(x,t)j2 <(1+t)4

C

PROOF. Recall from Lemma 5.25 thataat IVVR12 = A IVVR12 - 2 IVVVR12 + 2R IVVR12

+2R(OR)2+2CVR,V IVRI2).

Let R > 0 be a constant to be chosen later, and define

0 -t5 IVVR12+/3t4JVR12,

noticing that zfi (-, 0) - 0. Using the facts that (OR)2 < 2 1VVRI2 and

(VR, V JVRI2) = 2 (VVR) (VR,VR),

it is easy to estimate

at0 <Az/,+(6tR+5-20)t4JVVR12

+4 (t3 JVRI2) t4 JVVRI2+4/3(tR+1) (t3 JVRI2) .

By Proposition 5.30, there is C1 < oc such that tR < C1. By Lemma 5.31,there is C2 < oo such that t3 I VRI2 < C2. So one can choose 8 = J3 (C1)large enough and then C = C (Cl, C2) such that

19ate<-°'C.

Hence we have < Ct by the maximum principle. CJ

We now treat the general case.

PROPOSITION 5.33. Let (M2, g (t)) be a solution of the Ricci flow on aclosed surface with r = 0. Then for each positive integer k, there exists aconstant Ck < oo depending only on g (0) such that for all t E (0, oo),

sup VkR(x,t)2

I < ckk+2

XE.M2 (1 + t)

PROOF. The proof is by complete induction on k; we may suppose theresult is known for 0 < j < k - 1. As in Proposition 5.27, we compute that

aat VkR

2

= at(gip ... gjq Vi ... VPR Vj ... VqR)

= 0 I VkRI2 -2 I Vk+1RI2

+ (VkR) ®y[[k/2J

(viR) ®9(viR)]

Let N be a constant to be determined, and set

,D tk+3 I VkR12 + Ntk+2 Vk-1R 2

Then there are constants a, bj, and cj such that we can estimate

4) < A(D + tk+2at

VkR 2 [atR + (k + 3 - 2N)]

+ tk+2lVkR12.Lk/2J

b j+2 IojRl2 . tk-j+2 lok-jRl2

[(k-1)/2J

+ N E cj tk+1 Dk-1R I2 . tj+2 ViRl2 - tk-j+1 IVk-j-1RI2

j=02

+ N (k + 2) tk+1 IDk-1R

By the inductive hypothesis, there is a sufficiently large constant N depend-ing only on g (0), and positive constants A, B, C, D depending on g (0) andN such that

2

(fit< 0-D - Atk+2 VkR + Btk+2 vkRl2 + C < AID + D.

As in Lemma 5.32, this suffices to yield the result.

Now that our proof of Proposition 5.33 is complete, Theorem 5.28 followsreadily.

7. Strategy for the case that X (M2 > 0)

We have established Theorem 5.1 for surfaces such that r < 0. In theremainder of this chapter, we tackle the far more difficult case that r > 0.For the convenience of the reader, we now summarize the path we will taketo prove that the normalized Ricci flow on a surface of positive Euler char-acteristic converges exponentially to a metric of constant positive curvature.

7.1. The case that R 0) > 0. We first consider the special casethat the scalar curvature is initially nonnegative. By the strong maximumprinciple, one then has Rmjn (t) > 0 for any t > 0 unless R - 0 everywhere,which is possible only if the initial manifold was a flat torus. Hence (byrestarting the flow after some fixed time e > 0 has elapsed) we may assumethat R 0) > 0.

Recall that the trace-free part of the Hessian of the potential f of thecurvature is the tensor M defined in (5.9) as

where by (5.8), one has A f = R - r. In Section 3, we observed that thetensor M vanishes identically on a Ricci soliton. And in Section 4, we sawthat the only self-similar solutions of the Ricci flow on a compact surface

7. STRATEGY FOR THE CASE THAT X (M2 > 0) 129

are the metrics of constant curvature. Taken together, these observationssuggest one might be able to show that g (t) converges to a metric of constantpositive curvature by proving that M decays sufficiently rapidly. To explorethis idea, we compute the evolution equation satisfied by M.

LEMMA 5.34. On a solution (M2, g (t)) of the normalized Ricci flow,the tensor M evolves by

at MOM + (r - 2R) M.

PROOF. By Lemma 3.2, the Levi-Civita connection of g evolves by

atr 3 - 2gkl[Vi (Jl) +Vi ()

1kl 9ie+V1R 9ij)= 2 9

= 2 (_vi.s - VjR.SZ +

Using this formula and recalling equations (5.3) and (5.10), we calculatethat

atMij = at(ViVif_(R_r)9ij)

/of a k 1 a a=ViVj I at - atria Vkf - 2

(R)gij_(R_r)9ij= ViVj (Of + rf) +

2(vR. + V -R. - VkR gij) Vkf

- 2 [OR + R (R - r)] gij + 2 (R - r)2 gij

=ViVjOf +1

2(ViRVjf +VifVjR- (VR,Vf)9ij)

- 2 (AR) gij + rMij.

Next we use the fact that

RRijkl =

2(9il9jk - gikgje)

on a surface to compute the commutator [VV, 0], obtaining

V VjAf = VivjVkvkf(Rvtf)= vivkvjvkf - of

= Vkvivjvkf - RekjVeVkf - RieVjVlf-RjpViVPf - VRjV'f

= ODiVjf - Vk (Rkvtf) - RikjVPVkf

- RieVjV f - RjeV V'f - V RjpV$f

= DDiVjf - 1

2(V RVjf +VifVjR- (VR,Vf)gij)

/- 2R I Divjf - 1

2(Of)9ij .

Combining these results, we get

a Mij=Ovivjf - 2 (OR)9ij+(r-2R)Mij

= 0 (ViVjf - 2 (R - r) gij J + (r - 2R) Mij.

COROLLARY 5.35. On a solution (M2,g (t)) of the normalized Ricciflow, the norm squared of the tensor M evolves by

(5.22) at IMI2 = 0 IMI2 -2IVMI2 - 2R IMI2 .

PROOF. Recalling Lemma 3.1 and using the result above, we obtain

19

8t IMI2 8t(gikgiMjjMk)

=2(M,AM+(r- 2R) M)+2(R-r)IMI2=AIMI2-2IVMI2-2RIMI2.

Equation (5.22) is the key result that motivates the following strategy.If we can prove that R > c for some constant c > 0 independent of t, wewill obtain an estimate of the sort

IMI < Ce-`l.

Then we can consider the modified Ricci flow

(5.23) atg=2M=2VVf -(R-r)g=(r-R)g+£vfg.

7. STRATEGY FOR THE CASE THAT X (M2 > 0) 131

As we saw in Section 4 of Chapter 2, the solution to (5.23) differs from thesolution to the normalized Ricci flow

atg-(r-R)

only by by the one-parameter family of diffeomorphisms cot generated bythe time-dependent vector fields V f (t). Since M2 is compact, Lemma 3.15implies that these diffeomorphisms exist as long as the potential functionf (t) does. And since the quantity IM12 is invariant under diffeomorphism,the estimate I M 12 < Ce-ct will hold on the solution to the modified Ricciflow (5.23). Moreover, as we did for the curvature in Sections 5 and 6, onecan obtain estimates for all derivatives of M, which prove that the solutiong (t) to the modified flow converges exponentially fast in all Ck to a metricgoo such that M,,,, vanishes identically. By equation (5.7), this will imply thatg,,, is a gradient soliton. Then Proposition 5.21 (which uses the Kazdan-Warner identity, hence the Uniformization Theorem) will imply that g(, isa metric of constant positive curvature. It will then follow that there existpositive constants Ck, Ck for each k E N such that the solution g (t) of themodified flow satisfies

VkRI < Cke-ckt.

But by diffeomorphism invariance, the same estimates must hold for thesolution of the unmodified flow. In this way, we will be able to concludethat the normalized Ricci flow starting at a metric of strictly positive scalarcurvature converges exponentially fast to a constant curvature metric.

In order to obtain uniform positive bounds for R, we proceed as follows.We begin by developing two important technical tools.

STEP 1. The surface entropy of a compact 2-manifold (M2, g (t)) ofpositive curvature evolving by the normalized Ricci flow is defined as

N(g(t)) _ RlogRdA.

In Section 8, we will prove that N (g (t)) is strictly decreasing unless g (t)is Einstein; we shall later use this result to obtain uniform bounds on thescalar curvature of a solution to the normalized Ricci flow on a surface ofpositive Euler characteristic. (If R (0) changes sign, there is a modified en-tropy formula; in this case, in this case, we will prove that N (g (t)) remainsbounded.)

STEP 2. In Section 9, we obtain uniform bounds for the metric, thescalar curvature, the gradient of the scalar curvature, and the diameter of asolution on a surface of positive Euler characteristic. In particular, we showthat if I RI < t on a time interval [to, to + 1/ (4tc)], then IVRI < 4K3I2 on thesame interval. Together with the entropy result obtained in Step 1, we usethis to argue that RI is uniformly bounded independently of time, and thatthe diameter of the solution is bounded. In this step, the positivity of the

curvature allows us to apply Klingenberg's Theorem to obtain an injectivityradius bound.

STEP 3. In Section 10, we derive a differential Harnack estimate of Li -Yau-Hamilton type for the normalized Ricci flow. This is a lower bound forthe time derivative of the curvature

at log R - IV log RI 2.

(Again, there is a modified formula for the case that R (0) changes sign.)Integrating this along a space-time path from (xl, t1) to (x2, t2) such that 0 <ti < t2 yields a classical Harnack inequality, which is an a priori comparisonof the sort

R (x2, t2) > t1 eXp _ d (x1, x2, t1)2

R (xi,ti) t2 c(t2 - t1)

STEP 4. In Section 11, we combine the differential Harnack estimatewith the bounds on IRI and the diameter derived in Step 2 to obtain apositive lower bound for R. As outlined above, this allows us to prove thedesired convergence result for an initial metric of strictly positive curvature.

REMARK 5.36. In terms of getting a positive lower bound for R, theBBS estimates are useful locally, whereas the differential Harnack estimateis useful globally. Indeed, if

R(x,t)=f=Rma.(t)>0,

then the BBS estimates show that R (., t) > rc/2 in a g (t)-metric ball ofradius c/\, where c > 0 is a universal constant. On the other hand, ifthe diameter is bounded, then the Harnack estimate yields a uniform lowerbound for R.

7.2. The case that R 0) changes sign. To obtain convergence foran arbitrary initial metric having r > 0, we prove that the scalar curvatureof any such solution eventually becomes everywhere positive. Once thishappens, the argument outlined above goes through.

In order to obtain suitable bounds on the curvature, we need the mod-ified entropy formula and the modified Harnack estimate that apply whenthe curvature is of mixed sign. We also need an ad hoc injectivity radiusestimate, since Klingenberg's Theorem may not apply if the curvature isnegative somewhere. We obtain an adequate estimate for the injectivityradius in Section 12, and bound the curvature in Section 13.

In Sections 14 and 15, we illustrate alternative methods for boundingthe injectivity radius and curvature of solutions whose curvature is initiallyof mixed sign.

8. SURFACE ENTROPY 133

8. Surface entropy

Since we have not been able to employ the maximum principle to obtaina uniform upper bound for the curvature in the case that r > 0, we shallconsider integral quantities. Perhaps the most important such quantity forthe Ricci flow on surfaces is the surface entropy.

8.1. The case that R 0) > 0. We first consider the case that R > 0.The surface entropy N is defined for a metric of strictly positive curvatureon a closed manifold .M" by

N(g) _L.

R log R dµ.

The reason this quantity is called an entropy is because it formally resemblescertain classical entropies, each of which is the integral of a positive functiontimes its logarithm. Our sign convention is opposite the usual one, and so weshall show that the entropy is decreasing (instead of increasing) under thenormalized Ricci flow. We first compute the time derivative of the entropy.

LEMMA 5.37. If (M2, g (t)) is a solution of the normalized Ricci flowon a compact surface, then

(5.24) at (RdA) = ARdA.

PROOF. By (5.2) and (5.3), we have

(dA)at (RdA) = (R) dA + R

= [R + R (R - r)]] dA +R (r - R) dA.

LEMMA 5.38. If (M2, g (t)) is a solution of the normalized Ricci flowon a compact surface such that R 0) > 0, then the entropy evolves by

(5.25)N = -J 2 dA+J 2(R-r)2dAM M

PROOF. Using (5.24) and integrating by parts, we calculate

ddt =fM2 (log R I RdA + log R (RdA)

/=

fm[O R + R(R - r)] dA

f+

Jlog R A R dA

M2

IVRI dA.JM2IM22

11

PROPOSITION 5.39. If (M2, g (t)) is a solution of the normalized Ricciflow on a compact surface with R (-, 0) > 0, then

dN f FOR+RVf2 2

dt J 2 R dA - 2 Z IM1dA < 0,M M

where M is the trace-free part of the Hessian of f defined in (5.9). Inparticular, the entropy is strictly decreasing unless (JA42, g (t)) is a gradientRicci soliton.

PROOF. Recall that A f = R - r. Expanding the first term on theright-hand side and integrating by parts, we obtain(5.26)

f/M2 IVR RRVfI RI2 -2R(R-r)+RIVf12 dA.

On the other hand, commuting covariant derivatives and integrating by partsshows that

1M2fM2 (R - r)2 dA = (f)2 dA

=- (Vf,Vhf) dAJM2fM2 ((Vf,AVf) - Rc (Vf, Vf)) dA

= fM2(IVVf 12 + 2R IVf12) dA,

hence that

-2 fM2

IM12 dA = fM2 ((Af)2 - 2 IVVf 12) dA

= fM2 (RIvfl2 - IDVf12) dA

(5.27) = fM2 (R lVf 2 - (R - r)2) dA.

Subtracting (5.26) from (5.27) yields

-2IM2

IM12 dA - fM2 IoR Rof12

dA = rM2 ((R (_r)2 - 117R12dA,

R

whence the result follows by Lemma 5.38.

COROLLARY 5.40. Let (M2, g (t)) be a solution of the normalized Ricciflow on a closed surface with R (-, 0) > 0. Then the entropy is a strictly-decreasing function of time unless R (-, 0) - r, in which case it is constantin time.

8. SURFACE ENTROPY 135

PROOF. If dN/dt = 0 at some time to E [0, oo), then M (-,to) - 0.By equation (5.7), g (to) is a gradient Ricci soliton. Hence R (-,to) - r isconstant by Proposition 5.21, and thus g (t) = g (to).

8.2. The case that R 0) changes sign. We want to extend ourprevious entropy estimates to initial metrics on S2 or 11P2 whose curvaturechanges sign. We first need to define a suitable modification of the notion ofentropy. By passing to the double cover if necessary, we may assume X12 isdiffeomorphic to S2. Recall that the ODE corresponding to the PDE satisfiedby R is

dis = s(s - r).

Recall that r > 0 on .M2 ~ S2, and let s(t) be the ODE solution with initialcondition s(0) = so < Rmin(0) < 0, namely

(5.28) s(t) =r

(So < Rmin(0) < 0)1-(1-SO) ert

vThen the difference of R and s e olves by

at (R-s)=A (R-s)+(R-r+s)(R-s).

Since Rmin (0) - so > 0, the maximum principle implies that R- s > 0 for aslong as the solution of the normalized Ricci flow exists. This suggests thatwe define a modified entropy by

(5.29) 1V (g (t) , s (t)) -J (R - s) log (R - s) dA.

M2

LEMMA 5.41. Let (M2, g (t)) be a solution of the normalized Ricci flowon a closed surface with an arbitrary initial metric satisfying r > 0. Then

2

d 1V=f 2 -RRI +(R-s)(R-r+s+slog (R-s)) dA.M

PROOF. Put P - R - s. Noting that

at (P dA) = (AP + (R - r +s) P) dA+ P (r - R) dA = (AP + sP) dA,

we integrate by parts to obtain

dN= I [(A P + (R - r + s) P) + (logP) (AP + sP)] dA

= rM2 [-PhOPj2+(R-r+s+slogP)PJ dA.

When the initial metric has strictly positive curvature, Corollary 5.40shows that the entropy is decreasing. Not surprisingly, one can no longerprove this result for initial metrics whose curvature changes sign. However,one can show that the entropy is uniformly bounded from above, which iswhat we need for our applications.

PROPOSITION 5.42. Let (M2, g (t)) be a solution of the normalized Ricciflow corresponding to an arbitrary initial metric with r > 0. Then themodified entropy evolves by

dtN - JMZ11R + (R ss) Vf 12 dA - 2 IIM12M2 dA

(5.30) -8 fM2(Vf2 + s - r - (R- s) log (R- s)) dA,

where M is the trace-free part of V V f .

PROOF. Expand the first term on the right-hand side and integrate byparts to get

JVR+(R-s)VfI2dAX12 R-s

JVRI2

- 2R (R - r) + (R - s) IVf12 dA.M2 R - sBut as we observed in (5.27),

-2L IMI2 dA = fMs (RIVf12-R(R-r)) dA.

Subtracting these identities yields

-2 fM2IMI2 dA-fM2 IVR+(R s)Vf12 dA

2

JM2 RRIs dA,

whence the proposition follows by Lemma 5.38. 11

Since s < 0, we expect the last line of equation (5.30) to be positive. Themost difficult term to control is fM2 I V f 12 dA. To obtain an upper boundfor the modified entropy, therefore, we shall estimate its integral with respectto time.

LEMMA 5.43. There exists a constant C depending only on g (0) suchthat if the solution (M2, g (t)) of the normalized Ricci flow with r > 0 existsfor 0 < t < T, then

pTe-rt L lV f I2 dAdt < C.

Jo

9. UNIFORM UPPER BOUNDS FOR R AND IVRI 137

PROOF. By (5.10) and (5.2),

dtJM2f f((f+rf)+f(r_R)) dA

= 1M2 ((Of + rf) - fAf) dA

=JM2

(rf + Vf I2) dA.

Hence

d

dt(c_ri

fM2dA) = e-rt fM2 IV f 2 dA.

/J

But by Corollary 5.14, we have fM2 If I dA < Cert for some C < oc, so thatintegrating the equation above with respect to time yields

T e-rt Im I Vf 12 dAdt = [etf f dAlt T

< 2C.0 22 JJ t-0

PROPOSITION 5.44. Let (M2, g (t)) be a solution of the normalized Ricciflow corresponding to an arbitrary initial metric with r > 0. Then there isa constant C depending only on g (0) such that

N (g (t) , s (t)) < C.

PROOF. By (5.28), there is C > 0 depending only on g (0) such that-Ce-'t < s < 0. Hence (5.30) implies that

1V<-s1M2(Ipf12 +s-r-(R-s)log(R-s)) dA

< Ce-rt f I V f12 dA + Ce-rt 1V .

Nl

The proposition follows by integrating this inequality with respect to timeand applying Lemma 5.43.

9. Uniform upper bounds for R and IVRI

In this section, we derive further estimates for a solution g (t) on a surfaceof positive Euler characteristic. In the first part, we obtain a key doubling-time estimate, and derive upper and lower bounds for g (t). In the secondpart, we derive a BBS estimate for the gradient of the scalar curvature. Inthe final part, we apply the entropy estimate to obtain uniform bounds forthe scalar curvature itself under the additional hypothesis that R 0) > 0.

9.1. Bounds for the metric on a surface with X (M2) > 0. DefineRmin (t) - minXEM2 R (x, t) and Rmax (t) - maxXEM2 R (x, t).

LEMMA 5.45 (Doubling-time estimate). Let (M2,g(t)) be any solutionof the normalized Ricci flow on a closed surface with r > 0. Then for anyto E [0, oo), the estimate

Rma (t) < 2Rmax (t0)

holds for all x E M2 and

t E to, to +1

2Rmax (to)

PROOF. By (5.3), the scalar curvature evolves by

atR=OR+R2-rR.

Since r > 0, we have Rm (t) > 0 for all time. So at a maximum in space,

atR<AR+R2.

The solution of the initial value problem

dt= P2

dp (to) = Rm. (to)

is

1

P (t) -Rmax (to) + to - t

Hence the maximum principle implies that for to < t < to + 1/2Rmm (to),

Rmax (t) < 2Rmax (to).

REMARx 5.46. This is a prototype of the general Doubling-time es-timate derived in Corollary 7.5.

The lemma implies that the metrics are uniformly equivalent in the sametime interval. In particular, a lower bound for the metric is readily obtained.

COROLLARY 5.47. Let (.M2, g (t)) be any solution of the normalizedRicci flow on a closed surface with r > 0. Given any to E [0, oo), theestimate

g (x, t) > eg (x, to)

holds for all x E M2 and t E [to, to + 2Rmax to

PROOF. At any x E M2, we may write the metric as

(5.31) g (x, t) = e'fto (r-R(x,T )) dTg (x, to)

9. UNIFORM UPPER BOUNDS FOR R AND JVRI

If to < t < to + 1/2Rmax (to), then Lemma 5.45 implies thatt

tot Rdo(r -R(x,T)) dT > f

> -2t01- 2Rmaz (to

oitRmax (to) dT = -1.

139

It takes only slightly more work to derive an upper bound for the metric.

LEMMA 5.48. Let (M2, g (t)) be any solution of the normalized Ricciflow on a closed surface with r > 0.

If R 0) > 0, then for any times 0 < to < t < oo,

g (x, t)er(t-to)g (x, to)

If R 0) changes sign, then for any times 0 < to < t < oo,

g (x, t) < er(t-to)

e-rtg (x, to) .

\1 - ? -e-rto

so)

PROOF. Recall from Section 2 that Rmin (t) is bounded below by thesolution of the ODE

ds=s(s-r), s(0)= 0 ifRmin(0) - 0dt Rmin (0) if Rmin (0) < 0,

namely

s(t) =r

1- 1-r/Rm;n 0 )el

Hence for any x E M2, we havet

L

if Rmin (0) < 0-

t(r-R(x,T)) dT < Jo (r-s(T)) dT.t

In the case that Rmin (0) > 0, it follows therefore from (5.31) that

g (x, t) < of o (r-s(r)) d rg (x, to) < er(t-to)g (x, to)

In the case that R 0) changes sign, we computet t -re-rT [e_rr_ (1 7.-[ s(T)dr=J dT=logJ

to to a-rr (1 - ol so r=to

in order to conclude that

(-Lso /

- e-rt)

g (x, t) < _ft 0t o(r-s(r)) dTg (x, to) CLer(t-to) g (x, to)

C1 _ so_ e-rto

When the initial scalar curvature is nonnegative, these two lemmas leadto the following observation.

COROLLARY 5.49. Let (M2, g (t)) be a solution of the normalized Ricciflow on a closed surface with R (-, 0) > 0. Then for any to E [0, oo), theestimate

(5.32) eg (x, to) < g (x, t) C Veg (x, to)

holds for all x E M2 and t C[to, to + 2Rajax1 (to) ]

9.2. Estimating the gradient of the scalar curvature. If the scalarcurvature R is bounded above by some constant ic on a time interval [0, T),then applying the maximum principle to (5.20) proves that

IVRI2 < e4kt-3 fo r(t dt Sup IVRI2 (x, 0) .

xEM2

However, we prefer a bound for VRI2 which depends only on the initialscalar curvature R 0) and not on its derivatives. Of course, any suchbound must blow up as t \ 0.

PROPOSITION 5.50. There exists a universal constant C < oo such thatfor any solution (M2, g (t)) of the normalized Ricci flow on a compact sur-face that satisfies r > 0 and IR 0) 1 < ic for some c > 0, the estimate

SUPxE

cC

z

holds for all times 0 < t < 1/ (Ct).

PROOF. The idea is to compute the evolution equation of

G = t IVRI2 + R2,

and apply the maximum principle. Notice that G2 < R2 < K2 at t = 0. Itfollows from (5.20) that

at (tJVR12) =A (tIVRI2) -2tIVVRI2+[t(4R-3r)+1]IyRI2.

Adding this equation to

at(R2) = A (R2) - 2 I VRI2 + 2R2(R - r)

shows that

(5.33) atG < OG + (4tR - 1) IVRI2 + 2R3.

In order to apply the maximum principle to (5.33), we need to estimate R1on a suitable time interval. Since

a R> R(R-r)

9. UNIFORM UPPER BOUNDS FOR R AND IVRI 141

at a minimum of R, we have Rmin (t) > min {0, Rmin (0)} > -rc. And sincer > 0, we know Rmax > 0, which implies that

(R<R2

at a maximum of R. Combining these observations provesIR <

1-rctfor 0 < t < 1/rc. This implies that SRI < 2rc if 0 < t < 1/ (2rc), and hencethat 4t IRI < 1 if 0 < t < 1/ (8rc). Thus equation (5.33) yields the estimate

at G < AG + 16K3

for times 0 < t < 1/ (8rc), whence it follows from the maximum principlethat on that interval,

3

G < m2 + 16K3t < K2 +186

= 3rc2.

9.3. Estimates for solutions with R 0) > 0. In the remainder ofthis section, we assume that R 0) > 0. By the strong maximum principle,one then has Rmin (t) > 0 for any t > 0 unless R - 0 everywhere, which ispossible only if the initial manifold was a flat torus. Hence (by restartingthe flow after some fixed time e > 0 has elapsed) we may assume thatR 0) > 0. Then it follows from Lemma 5.9 and Proposition 5.18 that

ce-rt < R (x, t) < Ce't

for some constants c, C E (0, oc) depending only on go. We can now obtaina uniform upper bound for R by using the entropy estimate.

PROPOSITION 5.51. Let (M2, g (t)) be a solution of the normalized Ricciflow on a closed surface of positive Euler characteristic. If R 0) > 0, thenthere exists a constant C E [1, oo) depending only on go such that

sup R < C.M2 x [0,00)

PROOF. Since we already have time-dependent bounds for R, we mayset

rcl max R..M2X[o,1]

Given any time T E [1, oc), define

rc (T) max R > k1.M 2 x [0,T]

We want to show that tc is bounded independently of T. Assume thatrc (T) > max {rc1,1/4}, so that T > 1. Let (xi, t1) E M2 X (1, T] be a pointsuch that

R(x1,t1)= max R=rc.M2 x [0,T]

Let to t1 - -L > 0. Examining the proof of Proposition 5.50, we see thatif IRI t on a time interval [to, to + 1/ (4K)], then

IVR (x, t) I <2K

t - to

for all x E )42 and t E (to, to + 1/ (4K)]. In particular, we have

IVR(x,tl)I < 4tc3/2for all x E M2. Denote by Bg(t1) (x, p) the ball with center x E M2 andradius p, measured with respect to the metric g (t1). Let

y E Bg(t1) (x1,1/ 64K) ,

and choose a unit speed minimal geodesic ry (s) joining y to x1 at time t1.Then we have

R(xi,tl) -R(y,t1) = fas

[R('y(s),t1)] ds

Hence we getJ I VR (y (s) , ti) I ds < 4K3/2<

7 8r,1/2- 2

R (y, ti) >2

for all y E Bg(t1) (x1,1/ 64r,).Now by Klingenberg's Theorem, the injectivity radius of any closed ori-

entable surface whose sectional curvatures k are bounded by 0 < k < k*satisfies inj (M2, g) > 7r/ k*. (See Theorem 5.9 of [27].) Hence

inj (M2, g (tl )) >F_ >--.max (t1) /LR

2

By the maximum principle, we have R t1) > 0. So the entropy N (g (t1))is well defined. Then since R 0) > 0, we can let so / 0 in Proposition 5.44to obtain C (go) such that N (g (ti)) < C. Since infy>o (y logy) _ -1/e, wehave

C> N(g(tl)) =J

R(logR) dAMZ

> R (log R) dA -e

Area (.M2, g (ti)) .

Bs(t1)(x1,1/ 64,c)

But the area comparison theorem (§3.4 of [25]) implies that there is a uni-versal constant c > 0 such that

R (log R) dA > _(log-) Area (B9(1) (1, 1/64n) )Bs(t1) (x1 ,1/

clog -2

We conclude that x (T) has a uniform upper bound.

10. DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE 143

The uniform upper bound for the scalar curvature implies a uniformupper bound for the diameter.

COROLLARY 5.52. For any solution (.M2, g (t)) of the normalized Ricciflow on a closed surface with R 0) > 0, there exists a constant C > 0depending only on go such that

diam (M2, g (t)) < C.

PROOF. Suppose there are N points PI, ... , pN E M2 such that2rr

dist g(t) (pi pj) >Rm

(t)

for all 1 Rmax (t)

/2'

hence that the balls B9(t) (pi, 7r/ Rmax (t) /2) are embedded and pairwisedisjoint. The area comparison estimate then implies that there exists e > 0such that

NArea (M2, g (t)) > Area B9(t) (pi, (t)12) > N e

i=1 Rmax Rmax (t)

Hence by Proposition 5.51, there exists C > 0 such that

N < Rmax (t) Area (M2,g (t)) < C Area (M2, go) .E

10. Differential Harnack estimates of LYH type

In the context of parabolic differential equations, a classical Harnackinequality is an a priori lower bound for a positive solution of a parabolicequation at some point and time in terms of that solution at another pointand an earlier time. In their important paper [92], Li and Yau obtained newspace-time gradient estimates for solutions of parabolic equations. Their es-timates are differential inequalities which substantially generalize classicalHarnack estimates. Moreover, their method relies primarily on the maxi-mum principle, hence is robust enough to work in the context of Riemanniangeometry. Because of later advances made by Hamilton in the ideas Li andYau pioneered, such estimates are today called differential Harnack es-timates of LYH type.

We plan to discuss differential Harnack estimates for the Ricci flow inn dimensions in a chapter of the successor to this volume. Here, we de-rive certain differential Harnack inequalities that let us estimate the scalarcurvature function on a surface evolving by the normalized Ricci flow.


10.1. The case that R 0) > 0. Recall that the gradient Ricci solitonequation (5.7) implies that

(5.34) OR+RVf =0.Let L - log R and define

(5.35) Q+OL+R-r.Q is known as the differential Harnack quantity for a surface of posi-tive curvature. On a gradient soliton of positive curvature, we can divideequation (5.34) by R and take the divergence to find that Q - 0. As wasmentioned in Section 3, quantities which are constant in space on soliton so-lutions often yield useful estimates on general solutions. For such solutions,we shall obtain a lower bound for Q depending only on the initial metric goby applying the maximum principle to its evolution equation.

LEMMA 5.53. Let (M2, g (t)) be a solution of the normalized Ricci flowon a compact surface with strictly positive scalar curvature. Then

(5.36) 0tL=AL+IVL12+R-r.

PROOF. By (5.3), we have

8tL RatR= R(OR+R(R-r))=OL+IOLI2+R-r.

tityCOROLLARY 5.54. The differential Harnack quantity Q satisfies the iden-

Q 8tL -IVLI2

.

LEMMA 5.55. On any solution of the normalized Ricci flow on a compactsurface with strictly positive scalar curvature, one has

Q=OQ+2(VQ,VL)+2(5.37)at

VVL+ 2(R-r)g2

+ rQ.

PROOF. Using (5.5) with (5.36) and recalling the Bochner identity, wecalculate

\ / \

OQ + 2 ((VAL, VL) + IVVLI2)

+2RJVL12+ (2R-r)OL+R(R-r)=OQ+2(VQ,VL)+2IVVL12+2(R-r)AL+(R-r)2+rQ.

The result follows.

10. DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE 145

COROLLARY 5.56. Q satisfies the evolutionary inequality

(5.38) atQ > AQ + 2 (VQ, VL) + Q2 + rQ.

PROOF. Use (5.37) and the standard fact that n IS12 > (trg S)2 for anysymmetric 2-tensor S on an n-dimensional Riemannian manifold.

To obtain an estimate for Q from the maximum principle, we need toconsider the ODE corresponding to (5.38), namely

d _dtq =

q2 + rq.

The solution with initial data q(O) = qo < -r < 0 is

rgoe't Crertq goert - qo - r Cent - 1'

where C - qo/ (qo + r) > 1. Therefore, applying the maximum principle toequation (5.38) implies the following.

PROPOSITION 5.57. On a complete solution of the normalized Ricci flowwith bounded positive scalar curvature, there exists a constant C > 1 de-pending only on go such that

Tt

at log R - IV log R 12 = Q >_ -Ce re1.Tt -

This estimate for Q is known as a differential Harnack inequality.Integrating it along paths in space-time yields a classical Harnack inequality:a lower bound for the curvature at some point and time (x2i t2) in termsof the curvature at some (x1, ti), where 0 < t1 < t2. In particular, letry : [ti, t2l -> M2 be a C1-path joining x1 and x2. Then by the fundamentaltheorem of calculus,

f/'

log R (x1 t1) - Jt12 d (log R) (7 (t) , t) dtR

=112

at (log R) (Y (t) t) + V log R , ddtrl

(v'ogRI2>

Jtz C reTtl - JVlogRI dt dt

tl

t2 Crert - 1 d7 2 dtft1 Cert - 1 4 dt

4 Jt12dt

2

dt - log (Cert2 - 1) + log (Certl - 1) .

Now definerte I dy 2

dt,A = A(xl, tl e x2, t2) _ inft dt

1

where the infimum is taken over all C'-paths -y : [t1, t2] --> M2 joining x1and x2. Then exponentiation of equation (5.39) yields a classical Harnackinequality.

PROPOSITION 5.58. Let (M2, g (t)) be a complete solution of the nor-malized Ricci flow with bounded positive scalar curvature., Then there ex-ist constants Cl > 1 and C > 0 depending only on go such that for allx1i X2 E M2 and 0 < t1 < t2,

R (x2, t2) A Ciertl - 1 _A_ -> e 4 (t2 tl)(5.40) > e- 4 cR (xl ti) C1ert2 - 1 -

10.2. The case that R 0) changes sign. We now extend the dif-ferential Harnack inequality to solutions of the normalized Ricci flow whosecurvature changes sign. Assume that r > 0. As in Section 8.2, we considerthe solution

s(t) _

of the ODE

with initial condition

We define

s(0) = so < Rmin(0) < 0.

r1-(1-9) eTt

0

dts=s(s-r)

L = L (g, s) _ log (R - s).

A computation similar to that in Lemma 5.53 shows that L evolves by

(5.41)

Then we define

at L = AL +2VL +R-r+s.

Q =Q(g,s) +OL+R-r=at

VL2-8.

LEMMA 5.59. Let (M2, g (t)) be a solution of the normalized Ricci flowon a compact surface with any initial metric such that r > 0. Then

2

8tQ LQ+2(vQ,VL)+2 VVL+2(R-r)g1 2

(5.42) +s VL + (r - s) Q+ s(R - r).

10. DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE

PROOF. As in Lemma 5.55, we compute

IJL+ L +R-r+sat

2 \+(R-s) OL+ DL +R-r+sl+s(s-r)

= OQ + 2 KVQ, OL) + 2 VVL

+5 VL2

2

+ 2 (R - r) OL + (R - r)2

+(r-s) (AL+R-r) +s(R-r).

147

Our aim is to obtain a lower bound for Q. This is accomplished in thefollowing estimate.

PROPOSITION 5.60. Let (M2, g (t)) be a solution of the normalized Ricciflow on a closed surface. Let go be any initial metric such that r > 0. Thenthere exists a constant C > 0 depending only on go such that

Q > -C.

PROOF. By (5.28) and Proposition 5.18, there is C > 0 depending onlyon go such that -Ce-rt < s < 0 and R < Cert. Hence the only possi-bly troublesome negative term on the right-hand side of (5.42) is sI VLl2,over which we have no control. However, we can avoid this bad term byconsidering the quantity

P=Q+sL.Indeed, it follows from (5.4) and (5.41) that

at(sL) = A (sL) + s VL

2

+ s (R - r + s) + (s - r) (sL) .

Rewriting the gradient term and using the fact that L = log (R - s) >-c (1 + t) for some c > 0, we see that there is C,1 > 0 such that

at(sL) > (sL) + 2(D (sL) ,

VL)- s ILL

2

- C1.

By adding the evolution equations for Q and A and recalling the standardfact that n IS12 > (try S)2 for any symmetric 2-tensor S, we obtain

YtP>OP+2(VP,VL)+Q2+(r-s)Q-C2

for some C2 > 0. Since sL = P - Q is bounded, it follows that there existsa constant C > 0 such that

aApplying

the maximum principle, we( get

Q+ sL = P > min{ min P(x,0),-Cl xEM2 J

and hence reach the conclusion.

This is our differential Harnack inequality in the case that the curvaturechanges sign. As in the case of strictly positive curvature, we can integrateto obtain a classical Harnack inequality.

PROPOSITION 5.61. Let (M2,g (t)) be a solution of the normalized Ricciflow on a compact surface. Let go be any initial metric such that r > 0.Then there exists a constant C > 0 depending only on go such that for anyx1, x2 E M 2and 0 < t1 < t2i

R (x2, t2) - s (t2) > e_4_C(t2-t,)(5.43) R (xi,ti) - s(tl) -

where A--' inf ftt2 dt 2 dt.

11. Convergence when R (-, 0) > 0

We are now ready to prove Theorem 5.1 in the case the R 0) > 0.The first step is to apply the LYH differential Harnack inequality to obtaina uniform positive lower bound for R.

PROPOSITION 5.62. Let (.M2, g (t)) be a solution of the normalized Ricciflow on a compact surface such that R [go] > 0. Then there exists a constantc > 0 depending only on go such that

R(x, t) > c > 0

for all x E M2 and t E [0, oc).

PROOF. Observe that the differential Harnack inequality implies thatfor any points X1, x2 E .M2 and times 0 < t1 < t2, one has

R (x2, t2) > e-4-C(t2-t1)R (XI,ti)

whereft2 dy

A = A (xi,ti,x2,t2) _ in7 t

"

f

J dt

2

dt.

Notice that R > ce-r for times 0 < t < 1, and consider any point andtime (x, t) with t > I. Choose any x1 E M2 such that r < R (xi, t - 1) <Rn,ax (t - 1). Then we have

R (x, t) > e- 4 -CR (xi, t - 1) > re-C e-A/4

Thus to obtain a uniform lower bound for R, it will suffice to obtain auniform upper bound for A (x1, t - 1, x, t).

It follows from Proposition 5.51 and formula (5.31) that

e-Cg (t - 1) <g(r) < erg(t - 1).

12. AN INJECTIVITY RADIUS BOUND 149

for r E [t - 1, t]. Let 7 be a geodesic joining x and x1 of constant speedwith respect to g (t), so that

(r) 9(t) = distg(t) (x, x1) .

Then we have

A < 7 dTt-1

t

< er+C

-1 I(T) I s(t) d7- = eT+c [dist 9(t) (x, x1)] 2 .

Hence the desired uniform bound for A is implied by the diameter bound ofCorollary 5.52.

Applying Corollary 5.35, we conclude that there are constants 0 < c <C < oc such that

EMI < Ce-d

Then arguing as we did for the curvature in Sections 5 and 6, it is straight-forward to obtain estimates for all derivatives of M on a solution of themodified Ricci flow (5.23).

COROLLARY 5.63. If (M2, g (t)) is a solution of the modified Ricci flow(5.23) on a surface M2 of positive Euler characteristic, there exist for everyk E N constants Ck and Ck depending only on go such that

VkM < Cke-Ckt

Following the argument detailed in Section 7, we thus obtain the follow-ing result.

THEOREM 5.64. Let (M2, go) be a closed Riemannian surface with av-erage scalar curvature r > 0. If R (go) > 0, then the unique solution g (t)of the normalized Ricci flow with g (0) = go converges exponentially in anyCk-norm to a smooth constant-curvature metric g,,, as t --> oo.

12. A lower bound for the injectivity radius

In the next section, we shall derive uniform upper and lower bounds forthe scalar curvature of a solution (M2, g (t)) to the normalized Ricci flowon a closed Riemannian surface M2 of positive Euler characteristic whosecurvature is initially of mixed sign. In order to do this, we will require alower bound for the injectivity radius of such a solution. This section isdevoted to obtaining a suitable estimate.

PROPOSITION 5.65. Let (M2, g (t)) be a solution of the normalized Ricciflow on a closed surface with Euler characteristic X (M2) > 0, and letKmax (t) denote its maximum Gaussian curvature. Then for all 0 < t < oc,

inj (M2, g (t)) > min inj (M2, go) , miT t] Kmax

(T)

We begin with a collection of standard technical tools that are valid inany dimension. We denote the open ball of radius p centered 1; E TXM7z by

(l;, p) and the length of a path y by L (y).

LEMMA 5.66. Let (M", g) be a complete Riemannian manifold with sec-tional curvatures bounded above by Kmax > 0. Let 0 be a geodesic pathjoining points p and q in M' with L (,3) < 7r/ Kmax. Then for any pointsp* and q* sufficiently near p and q respectively, there is a unique geodesicpath /3* joining p* and q* which is close to ,3.

PROOF. By the Rauch comparison theorem (Theorem B.20 of AppendixB), the map

expp I Bp (0, 7r/ Kax) : Bp (0, 7r/ Kmax) - A4-

is a local diffeomorphism. We may assume that the geodesic 0 : [0, 1] -> A4'is parameterized so that R (s) = expp (sV) for some V E Bp (0, 7r/ Kmax)with expp (V) = q. Since expp I Bp (0, 7r/ Kmax) is a local diffeomorphism,there is for each p* and q* sufficiently close to p and q respectively a uniquevector V* E Tp*M' close to V E TpM" such that expp. (V*) = q*. (Here wehave identified TTM' ^-' ][8n = Tp*MI.) The path j3* : [0, 1] --* M' definedby,3* (s) - expp* (sV*) is the unique geodesic near /3 joining p* and q*.

We now anticipate Definitions B.66 and B.67 (found in Appendix B): ifk E N, a proper geodesic k-gon is a collection

r={yi:of unit-speed geodesic paths between k pairwise-distinct vertices pi E Nl'such that pi = yi (0) = yi-i (2i_1) for each i, where all indices are interpretedmodulo k. We say r is a nondegenerate proper geodesic k-gon ifZp, 1'i) 0 for each i = 1, ... , k; if k = 1, we interpret this to meanL (F) > 0. A (nondegenerate) geodesic k-gon is a (nondegenerate)proper geodesic j-gon for some j = 1, ... , k.

LEMMA 5.67. Let (Ms, g) be a closed Riemannian manifold with sec-tional curvatures bounded above by Kmax > 0. Let 0 be a nondegenerategeodesic 2-gon between p q E M' of edge lengths L ()3i) < 7r/ Kmax fori = 1, 2. Then there is a smooth nontrivial geodesic loop y on Mn with

L(y) <L(,31)+L(,32)

12. AN INJECTIVITY RADIUS BOUND 151

PROOF. Choose d < rr/ Kmax so that max {L (/3j)} < d. Let r bethe space of all nondegenerate geodesic 2-gons y in Mn with L (yi) < dfor i = 1, 2. Note that r E /3 is a nonempty open 2n-manifold locallyparameterized by points in A4' x Mn. Indeed, by Lemma 5.66 there isfor every pair of points p* and q* sufficiently close to p and q a uniquenondegenerate geodesic 2-gon /3* with L (/3z) < d for i = 1, 2. Define

m r inf {L(ryl) + L(y2)} .-YEr

Note that m > inj .M' > 0, since the map exp., I B. (0, p) : B. (0, p) --> Mnis an embedding at any x E Mn whenever p < inj Mn. Let /3 (i) E r be asequence of nondegenerate geodesic 2-gons with

Jim {L(/3 (i)1) + L(/3 (i)2)} = m.z-00

Let p (i) q (i) denote the endpoints of the geodesic paths /3 (i)1 and 3(')2.By compactness of M', we may pass to a subsequence such that the limits

limb,,,, p (j) and q,,) = limb + q (j) exist and satisfy dist (p00, q00) <d < it/. Hence geodesics

/3 (oo)1 (j)1 and 0 (00) 2 j1 /3 (j)2

exist and satisfy L (0 (oo)1) + L (/3 (00)2) = m by continuity.There are now two cases. If p..) = q,,, then some path, say /3 (00)1, is

a nondegenerate geodesic 1-gon at p,,,, satisfying L (/3 (00)1) = m. The loop/3 (o0)1 must be smooth at per, or else the first variation formula (see 1.3 of[27]) would let us shorten /3 (00)1 inside F.

In the second case, p,,,, # q,,,,. If /3 (oo) _ {/3 (oc)1, 0 (00)2} is degener-ate, then /3 (00)2 must be the same path as /3 (00)1 but with the oppositeorientation. By the uniqueness statement in Lemma 5.66, this is possibleonly if /3 (i)1 = /3 (i)2 for sufficiently large i, which contradicts the nonde-generacy of /3 (i). Hence /3 (o0) is nondegenerate. As above, /3 (oo) must besmooth, or else the first variation formula would let us shorten /3 (oo) insider.

DEFINITION 5.68. We say a geodesic loop y is stable if every nearbyloop y* satisfies L (y*) > L (-y). We say a geodesic loop -y is weakly stableif the second variation of arc length of y is nonnegative.

Clearly, stable implies weakly stable. The following result is well known.

LEMMA 5.69. Let (Mn, g) be a closed Riemannian manifold with sec-tional curvatures bounded above by Kmax > 0. If a is the shortest geodesicloop in Mn and L (a) < 2ir/ Km , then a is stable.

PROOF. If not, there exists a loop a* near a with L (a*) < L (a). Pickdistinct points p and q on a that divide a into two geodesic paths a1 anda2 of equal length. It is always possible to choose distinct points p* andq* on a* near p and q respectively that divide a* into paths cai and a2 of

length less than 7t/. Then since L (al) = L (a2) < ir/ Kmax, thereare by Lemma 5.66 unique geodesic paths ,Ql and 03 between p* and q* thatare close to al and a2 respectively, hence close to a* and a2. Now as aconsequence of the Gauss Lemma (Lemma B.1), we have L (0z) < L (a!)for i = 1, 2. But then by Lemma 5.67, there is a smooth geodesic loop ywith

L (y) < L (,Of) + L (/32) < L (a*) < L (oz),

which is a contradiction.

LEMMA 5.70. If -y is a weakly stable geodesic loop in an orientable Rie-mannian surface (M2, g), then

fRds <0 .

PROOF. Let T denote the unit tangent vector to y. Since M2 is ori-entable, there is a well-defined unit normal N along y. Since y is geodesicand n = 2, we have VTN = 0, because

(VTN,T) = T (N, T) = 0 and (VTN, N) = 2T (N, N) = 0.

Let -yo be a 1-parameter family of loops with yo = y and aye/aOle=0N. Then by the second variation formula (see 1.14 of [27])

2

aL(ye) = It (R(N,T)N,T) ds=- I Rds.L902 Lo ti 7 2

Since y is weakly stable, the result follows.

Now let yt be a smooth 1-parameter family of loops in a time-dependentRiemannian manifold (M',g (t)). That is, each yt is a smooth loop in Mndepending smoothly on t in some open interval Z. If Lt (-y) denotes thelength of a loop y with respect to the metric g (t), then by Lemma 3.11, wehave

(5.44) dLt (yt) = 2 f ag (T, T) ds - f (VTT, V) ds,7t 7t

where T is the unit tangent to yt, V is the variation vector field y* (a/at),and ds is the element of arc length. This implies the following observation.

LEMMA 5.71. If (M2, g (t)) is a solution of the normalized Ricci flowand -It is a smooth 1-parameter family of geodesic loops, then

dtLt (yt)1 (r - R) ds.

t=T 2 fTCombining this result with Lemma 5.70 yields the following result.

13. THE CASE THAT R 0) CHANGES SIGN 153

COROLLARY 5.72. Let yt be a smooth 1-parameter family of loops in asolution (.M2, g (t)) of the normalized Ricci flow. If yr is a weakly stablegeodesic loop, then

dLt (-It)

2LT('yr)

t=T

We are now prepared to show that the length of the shortest closedgeodesic is strictly increasing whenever it is small enough.

LEMMA 5.73. Let (M2, g (t)) be a solution of the normalized Ricci flowon a compact surface of positive Euler characteristic, and let Kmax (t) > 0denote its maximum Gaussian curvature. Suppose yr is the shortest closedg (T)-geodesic at some time r > 0. If LT (yr) < 27r/ Kmax (T), then thereis E > 0 small enough so that for all t E (T - e, T), there is a g (t) -geodesicyt such that

Lt (yt)<LT(yr)Note, however, that we do not claim the geodesics yt depend smoothly ont c (,r - E, T).

PROOF. By Corollary 5.72, there is Eo > 0 sufficiently small so thatLt (y,) < Lr (yr) fort E (T - Eo, r). Choose points p and q on yr that divideyT into two segments yT and yT of equal length with respect to the metricg (t). These may not beg (t)-geodesic segments, but for some E E (0, Eo),we have Lt (yr) < 7r/ Kmax (t) for i = 1, 2 and all t E (T - e,T). Thus byLemma 5.66, there exist unique g (t)-geodesics 01 and X32 between p and qwhich are near yT and y? respectively. Then by the Gauss lemma, we have

Lt (01) + Lt (02) :5 Lt (yT)+Lt ('T) <Lr(yr)

Hence by Lemma 5.67, there is a smooth g (t)-geodesic loop y with

Lt (-y) SLt(i31)+Lt(02) <LT(y7)

We are now ready to prove the main result of this section.

PROOF OF PROPOSITION 5.65. Lemma B.34 in Appendix B (Klingen-berg's Lemma) says that inj (M2, g (t)) is no smaller than the minimum of7r/ Kmax (t) and half the length of the shortest closed g (t)-geodesic. But byLemma 5.73, the length of the shortest closed geodesic is strictly increasingas long as it is less than 27r/ Km (t).

13. The case that R 0) changes sign

In this section, we complete our proof of Theorem 5.1 for the caseof a compact Riemannian surface (M2, go) such that x (M2) > 0 andRmin (go) < 0. The road map for obtaining convergence in this case isas follows. Armed with the entropy estimate of Proposition 5.44 and the

injectivity radius estimate of Proposition 5.65, we proceed as in Proposi-tion 5.51 to obtain a uniform upper bound for R. As in Corollary 5.52,this implies an upper bound for the diameter. Then we use the differentialHarnack estimate of Proposition 5.61 to argue that R eventually becomesstrictly positive within finite time. Once this happens, the proof of Theorem5.64 goes through exactly as written.

Our first result is an analog of Proposition 5.51.

LEMMA 5.74. If (M2, g (t)) is a solution of the normalized Ricci flow ona closed surface of positive Euler characteristic, then there exists a constantC > 0 depending only on go such that

sup R < C.M2 x [0,00)

PROOF. By Proposition 5.18, there exists C1 > 0 depending only on gosuch that SUPM2 x [O,I] R < r + C1 e'. Given T > 1, define

rc (T) max R.M2x[0,T]

We intend to prove that K is bounded independently of T. We may assumethat c (T) > max {ici,1/4}, so that T > 1. Choose (xi, ti) E M2 X (1, T]such that R (x1i ti) = maxM2x[o,T] R = n (T). Following the proof of Propo-sition 5.51, one shows that

(5.45) R (y,t1) > 2

for all y E By(tl) (x1,1/063-4-n) - Define p _ inj (M2, go). Then by Proposi-tion 5.65, one has

inj (M2,g (t)) > min f VIrrr-, Ip, .

As long as p:5 7r//, we have the uniform upper bound n < (7r/p)2. So wemay assume p > 7r/J. By Proposition 5.44, there exists C2 > 0 such that

C2 > N(g(ti))Im 2

(R - s) log (R - s) dA-

J (R - s) log (R - s) dABs(t1) (xl' 764w)

- 1 Area (M2, g (t1))e

where s = s (t1) _< 0 is given by formula (5.28). By the area comparisontheorem (§3.4 of [25]) and estimate (5.45), there exists E > 0 such that

fBg (t 1) (X 1,

1 (R - s) log (R - s) dA > (2 - s) log (2 - s)

> 2 log 2.

13. THE CASE THAT R(,0) CHANGES SIGN 155

Since Area (M2,g (t1)) = Area (M2,go), this implies the desired uniformupper bound.

We next bound the diameter of the solution as we did in Corollary 5.52.

COROLLARY 5.75. If (M2, g (t)) is a solution of the normalized Ricciflow on a closed surface of positive Euler characteristic, then there exists aconstant C > 0 depending only on go such that

diam (M2,g (t)) < C.

PROOF. Suppose there exist points pt,.. . , pN E M2 such that

dist 9(t) (PiPj)

_2

27r

(t)

for all 1 < i j < N. Define

S (t) - min inj (M2, go)Rmax (t)J 2

By Proposition 5.65, the balls B9(t) (pi, b (t)) are embedded and pairwisedisjoint. Arguing as we did in Corollary 5.52, one estimates that

N< Area (M2, go)

b (t)

But by Lemma 5.74, 5 is bounded from below by a constant depending onlyon go. The result follows.

We can now show that R becomes strictly positive within finite time.

LEMMA 5.76. If (M2,g (t)) is a solution of the normalized Ricci flowon a closed surface of positive Euler characteristic, then there exists T < 00such that

inf R > 0.M2x[T,oo)

PROOF. We begin by deriving a positive lower bound for R - s, where sis given by formula (5.28). Given any (x, t) with t > 1, chose xl E M2 suchthat

0 < r < R (xl, t - 1) <Rmax(t-1).Then the differential Harnack estimate (5.43) derived in Proposition 5.61implies that

(5.46) R (x, t) - s (t) > e-C [R (xi, t - 1) - s (t - 1)] e-A14 > e-Cre-A/4

where C is a constant depending only on the initial metric, and A is definedby

rtA (xl, t - 1, x, t) inf J 1f12 dr.

7 t-1

As in Proposition .5.62, the desired positive lower bound for R-s will followfrom (5.46) once we bound A from above. Recall the integral formula (5.31)for the metric, which implies that for any T > t - 1, one can write

itg (x, T) = g (x, t - 1) exp (r - R) dT.

1

Using the consequence of Lemma 5.9 that R > Rmin (0) and the uniformupper bound for R obtained in Lemma 5.74, it follows from this formula thatthere exist cl < C1 independent of x E M2 such that for all T E [t - 1, t],one has

<g(x,T)As in the proof of Proposition .5.62, this estimate implies that there existsC2 > 0 such that

A < C2 [dist 9(t) (x, x1)]2,

whence the desired upper bound for A follows from the diameter bound ofLemma 5.75.

Now by substituting the uniform upper bound for A into estimate (5.46)and using the fact that x E M2 is arbitrary, one concludes that there existse > 0 such that

Rmin (t) = inf R (x, t) _> e + s (t) .xEM2

Since s (t) / 0 as t -> oc, this proves the lemma. 0

Once the curvature becomes strictly positive, we may proceed exactlyas in Section 11. This completes our proof of the following result.

THEOREM 5.77. Let (.M2, go) be a closed Riemannian surface with aver-age scalar curvature r > 0. The unique solution g (t) of the normalized Ricciflow with g (0) = go converges exponentially in any Ck-norm to a smoothconstant-curvature metric g,,,) as t -> oo.

14. Monotonicity of the isoperimetric constant

In the remainder of this chapter, we illustrate an alternative strategyfor proving convergence of the Ricci flow starting from an arbitrary initialmetric on a surface of positive Euler characteristic. The basic idea is asfollows. A solution of the unnormalized Ricci flow with r > 0 will becomesingular in finite time. If one can estimate the injectivity radius of sucha solution on a scale appropriate to the curvature and if one can rule outslowly-forming singularities, one can then show that a sequence of parabolicdilations formed at the developing singularity will converge to a round spheremetric. This is called a blow-up strategy, because one `blows up' the solutionat the developing singularity. Blow-up strategies are extremely important inunderstanding singularity formation of the Ricci flow in higher dimensions.

14. MONOTONICITY OF THE ISOPERIMETRIC CONSTANT 157

REMARK 5.78. The result that a sequence of blow-ups (M2, gj (t)) pulledback by diffeomorphisms converges smoothly to a shrinking round spheremetric (S2, g,,,, (t)) is weaker than the pointwise C°° convergence obtainedabove. Moreover, if one did not already know the Uniformization Theorem,one could not conclude that g,, is in the same conformal class as the origi-nal solution. An analogous problem occurs when studying the Kahler-Ricciflow in complex dimensions greater than one: one does not know in gen-eral whether or not a limit g, is Kahler with respect to the same complexstructure as the original solution.

The strategy of proving sequential convergence gj (t) -, g,,,, (t) uses sev-eral techniques that will be developed later in this volume. We provide thedetails in Section 15. In this section, we develop the background necessaryto obtain a suitable injectivity radius estimate.

DEFINITION 5.79. Let (Mn, g) be a closed orientable Riemannian man-ifold. We say a smooth embedded closed (but possibly disconnected) hy-persurface En-1 C Mn separates Mn if Mn\En-1 has two connectedcomponents M1 and M' such that

i = aM2 = En-1.3M

DEFINITION 5.80. If En-1 separates M', the isoperimetric ratio ofEn-1 in (Mn, g) is

CS (En-1) - (Area (En-1))n 1 /1

1

n-1

Vol (Mn) + Vol (M2) /The isoperimetric constant of (Mn, g) is

CS (Mn, g) _ infl CS (En-1) ,

where the infimum is taken over all smooth embedded En-1 that separateMn.

REMARK 5.81. Cs (En-1) is invariant under homothetic rescaling of themetric g el`g and is equivalent to the standard isoperimetric ratio

(Area (En-1))n( n-1 )ECI

Indeed, one has

min {Vol (Mr) , Vol (M2)}n-1

CI (En-1):5 Cs (En-1) < 2n-1CI (En-1).

Now we restrict ourselves to the case of a closed orientable Riemanniansurface (M2, g). If ry separates M2 into two open surfaces M2 and M2, we

have

Cs (Y)-L9(7)2A9(Mi)+As(M2)

(5.47) = Lg ('Y)2Ag

(M2)As(Mi)-As(M2)

where L9 denotes length and Ag denotes area measured with respect to g.Moreover, the infimum in Definition 5.80 is attained by a smooth embeddedcurve.

LEMMA 5.82. If (M2, g) is a closed orientable Riemannian surface, thereexists a smooth embedded (possibly disconnected) closed curve -y such that

Cs ('Y) = Cs (M2, g) .

For a proof, see Theorem 3.4 of [69]. Analogous results in general di-mensions can be found in Theorems 5-5 and 8-6 of [99].

REMARK 5.83. The minimizing curve -y might be disconnected, as caneasily be seen by considering a rotationally symmetric 2-dimensional toruswith two very thin necks. In this case, -y is the union of two disjoint embed-ded loops. More generally, no connected embedded homotopically nontrivialcurve will separate a torus.

Now we restrict ourselves further to the case that (M2, g) is diffeomor-phic to S2. Define a loop in M2 to be a connected closed curve. It followsfrom the Schoenflies theorem that any smooth embedded loop 'y separatesM2 into smooth discs Mi and M. We denote the isoperimetric ratio of asmooth loop on a topological sphere by

CH ('y) - Cs ('y) -

Then we define

CH (M2, g) _ inf CH (-y),7

where the infimum is taken over all smooth embedded loops -y. Clearly,

Cs (.M2, g) CH (M2, 9) .

If (S2, gcan) is a constant-curvature 2-sphere, then CH (S2, gcan) = 47r.Moreover, this is an upper bound.

LEMMA 5.84. If (M2, g) is a closed orientable Riemannian surface, then

CH (M2, g) < 4nr.

PROOF. Let X E M2 be given and define

yE {yEM2:dist9(x,y)=E}.

Fore > 0 small enough, 'YE is a smooth embedded loop. Since g is Euclideanup to first order at x, we see from (5.47) that (as E \ 0,

CH(7) - [27r(E+o(E))]2O E

A9

A

2)

-[n(E + ( ))2] [ 9 ( 2)(62)]

11

We now sketch a proof of the following existence result; further detailsmay be found in §3 of [62].

LEMMA 5.85. If (M2, g) is a Riemannian surface diffeomorphic to S2such that CH (M2, g) < 47r, then there exists a smooth embedded loopsuch that

CH (l3)=CH (M2,g).

Our proof of Lemma 5.85 will use the following algebraic fact.

LEMMA 5.86. If L1, L2 and A1, A2, A3 are any positive numbers, then

1(L1 + L2)2

1

A1 + A2 + A3

> min { Li (Al + A2 + A3) ' L2 A2 Al + A3

PROOF. To obtaina contradiction, suppose the result is false. Thenthere exist positive numbers L1i L2 and A1, A2, A3 such that

(L1+L2)2(A1+A2+A3) <Ll(Al+A2+A3/

and

(L1+L2)2(Al+A2+)_<L(+AJA3).A3Werewrite these inequalities as

485Al (A2 + A3) < Ll

)( .

A3 (A1 + A2) - (Li + L2)2

and

L2A2 (A1 + A3)(5.49) <

A3 (Al + A2) (LI +L2)2.

Summing (5.48) and (5.49) yields

2A1A2 < 2L1L2< 0

A3 (AI + A2) (L1 + L2)2

which is impossible.

SKETCH OF PROOF OF LEMMA 5.85. Let L and A denote length andarea, respectively, measured with respect to the metric g. Define i > 0 by

CH (M2,9) = 41r - 277.

There exists co > 0 depending only on g and 77 such that L (y) > co whenevery is a smooth embedded loop such that CH ('y) < 41r -'q. By (5.47), we have

2

L (y)2 A(M2) A (M2)

CH (y) < 4ir,

whence it follows that2

A (M2)> 47r

for a = 1, 2. We also have

A M2) A M2(L (y)2 < 47r A

M2)2) < 27rA (M2).

This gives us some control on the curves -y. Now let ji be a sequence ofsmooth embedded loops such that

lim CH (=yi) = CH (M2, 9) .t-+oo

By applying the curve shortening flow (in particular, Grayson's theorem [49]that any embedded loop either converges to a geodesic loop or shrinks to around point) one may show that there exist a new sequence yi of smoothembedded loops such that

Jim CH ('yi) = CH (M2, 9)

and

JkZ ds < C,

yz

where ki denotes the Gaussian curvature of the curve yi, and the constantC < oc depends only on g and 77. From this, one shows that the yi arelocally uniformly bounded in L1,2 and C1,112. One then concludes that forany p < 2 and a < 1/2, there exists a subsequence that converges in L1,P

and Cl,0 to an immersed curve yam. Since -yam minimizes length among allnearby curves bounding regions of fixed areas, the formula for first variationof arc length shows that y,, has constant curvature. Hence 'y is smooth.

We now show that 'y is actually embedded. Since it is a smooth limitof smooth embedded curves, it can at worst have points of self tangency.If this is so, one can regard ryoo as the union of two curves y1 and y2 ofpositive lengths L1 and L2, respectively, bounding positive areas Al andA2, respectively. Then the `inside' of yoo has area Al + A2. Let A3 be the

area `outside' of -yam. We then have

CH (boo) _ (L1 + L2)2 ( A3 + Al -+A2

2 1 1CH L1 Al + A2 + A3

1 1

CH (72) = L2 A2 + Al + A3

But Lemma 5.86 implies thatlCH (`Yoo) > min {CH ('Yl) , CH (72)} ,

which contradicts the minimality of -y,,. El

It is interesting to note that CH ('y) may be represented in another way.

LEMMA 5.87. Let (M2, g) be diffeomorphic to S2. Let ry be a smoothloop in (M2, g), and let (S2, g) be a 2-sphere of constant curvature chosenso that

A (S2) = Ag (A42).

Let ry be a shortest loop (namely, a round circle) that separates (S2, g) intotwo discs Di and D2 with Ag (D2) = A9 (M2) and Ay (D2) = A9 (M2) .Then

CH ('Y) = 47rLs ('Y)2

Lg ('r)2

PROOF. Identify (S2, g) with the 2-sphere of radius p = Ag (.M2) /47rcentered at 0 E R3. We may assume without loss of generality that

A A < A2 = A9 (M2) .

Choose h E [0, p) such that

A9{(x,y,z) ES2:z>h} =A1Ag{(x,y,z) ES2:z<h}=A2.

Thenry={(x,y,z)ES2:z=h}

is a round circle that divides (S2, g) into two discs with areas Al and A2.Because

fhAg {(x, y, z) ES2 : z < hh 27rpdz = 27rp,P

we have A2 = 27rp (p + h) and Al = 27rp (p - h). Hence

47rA1A2 = [27rp (p - h)J [2irp (p + h)] = 47x2 (p2 - h2) = (Lg (7))2 .

Al -+A2 p2

The lemma follows.

We are now ready to state the main assertion of this section. Notice thatit does not matter whether we consider the normalized or unnormalized Ricciflow, because the isoperimetric constant is scale invariant:

Cs (M2, g) = Cs (M2,eAg).

THEOREM 5.88. Let (M2, g (t)) be a solution of the Ricci flow on atopological 2-sphere. At any time t such that CH (M2, g (t)) < 47r, one has

dtCH (M2,g(t)) > 0

in the sense of the lim inf of forward difference quotients.

REMARK 5.89. Because CH (M2, g (t)) is a continuous function of timesatisfying the upper bound CH (M2, g (t)) < 47r, the theorem implies inparticular that CH (M2, g (t)) is nondecreasing in time.

Before proving the theorem, we introduce some notation. Assume in theremainder of this section that (M2, g (t)) is a solution of the Ricci flow ona topological 2-sphere for t E [0, w). Let

A = A (t) - A9(t)(M2)

.

Observe that it follows from Lemma 6.5 and the Gauss-Bonnet Theoremfor a closed surface, that

(5.50)d

A = - fm RdA = -47rX (M2) = -8ir.Z

Given any to E [0, w) and any smooth embedded loop yo separating M2into Mo and Mo , let -yp denote the parallel curve of signed distance p from-yo measured with respect to the metric g (to), where p < 0 for curves inMo and p > 0 for curves in Mo M. For Ii sufficiently small, -y, is a smoothembedded loop that separates M2 into MP and MP M. We shall thus considerthe following functions of both p and t:

L = L (p, t) _ Lg(t) (7(P)

At = At (p, t) = A9(t) (MP )

CH = CH (p, t) - (L (p, t))2 A (t)A+(p,t)-A-(p,t)Notice that by (5.47), CH is the isoperimetric ratio of the curve 'yP takenwith respect to the metric g (t). The next three lemmas accomplish thecalculations necessary to prove the theorem.

LEMMA 5.90. Let (M2, g (t)) be a solution of the Ricci flow on a topo-logical 2-sphere. Then

L= fY,kds L=-fy,Kds

a A.=fL atA±=-47r±2fy,kds,


where K = R/2 is the Gaussian curvature and k = (T, VTN) is the geodesiccurvature of the curve 7p with unit tangent vector T and unit normal vectorN oriented outward to MP .

PROOF. The derivatives with respect to p are easily verified. By Lemma3.11,

aL=-2J Rds=-IK ds.Yp 7r

By the Gauss-Bonnet formula for a disc with outward unit normal N, onehas

27r = 27rx (MP) = J K dA +J

(VTT, N) dsiP raNtP

(5.51) _ f K dA f J (T, VTN) ds,A/I P -

and henceaAt=-J R dA=-47r±2f k ds.at MP 7p

LEMMA 5.91. Under the same hypothesis, L and At evolve by2

Lat L = apt

and

at At = - At -47r±J kds,7p

respectively.

Notice that these resemble heat equations.

PROOF. Differentiating (5.51) with respect to p yields

kds.0=J Kds+ aIrp7p ap

r

2L.a-L =

aJ

kds =02

p yp p

The formulas for A± follow from the observation that

ape At=±apL=±Jpkds=atAf+47rTpkds.

LEMMA 5.92. If (M2, g (t)) is a solution of the Ricci flow on a topological2-sphere, the isoperimetric ratios CH (p, t) of the parallel loops -y, measuredwith respect to the metric g (t) satisfy the heat-type equation

a 2 F 8 47r - CH A+ A_at

(log CH) = V (log CH) + L-p (log

CH) + A (A- + A+where

I,J lk ds7v

PROOF. Notice that

log CH = 2 log L - log A+ - log A_ + log A.

By Lemmas 5.90 and 5.91, we have

2 2

- (log L) =L-1Azpz

L = pz (log L) + I a- (log L)

2

= Apt (log L) + L ap (log L) .

)

2

Similarly, we obtain

[A±_4r](log At) = A1

= A2 (log A±) + 15p (log A±))2

- A At

, (_) --+-r. a(log A) + A+ L a (log At) ,

because L (aAtlap). Finally. we recall (5.50) to get

d dt(logA)=-A.

The lemma follows when we combine these results with the identities

47r A+ A- _ 47r (A+ + A_)z - 2A+A_ _ 47r 47r 87r

A (A- + A+) A++ A_ A+A_ A+ + A_ A

andCH A_) L2 (A+ A-) L2 L2(L+

A-+ / A+A- (A- + A+

A-)A? + A+

If CH < 47r, then Lemma 5.92 shows that log CH is a supersolutionof a type of heat equation. This leads one to expect that it should benondecreasing. We now prove this rigorously.

15. AN ALTERNATIVE STRATEGY FOR THE CASE X (M2 > 0) 165

PROOF OF THEOREM 5.88. We shall show that for any E > 0, we have

CH (M2, g (t)) > -Et=tp

at all times to such that CH (M2, g (to)) < 47r. Given such a time to, thereis by Lemma 5.85 a smooth embedded loop -yo whose isoperimetric ratiomeasured with respect to g (to) satisfies the identity

CH (0, to) = CH (M2, g (t)) .

Since 'yo is a minimizer of CH at time to, we have the relations

(9 log CH (0, to) = 0

a2 /

(logcH) (0, to) > 0.

Using these and the inequality 47r - CH > 0, we conclude from Lemma 5.92that

(at log CH) (0, to) > 0.

This is possible only if for each e > 0, there exists 8 > 0 depending on Esuch that for all t E (to - 8, to), one has

log CH (0, t) < log CH (0, to) + E (to - t) = log CH (M2, g (to)) + E (to - t) .

Hence for all t E (to - 8, to), we have

log CH (M2, g (t)) + Et < log CH (M2, g (to)) + Eto.

15. An alternative strategy for the case x (M2 > 0)

We shall now complete the alternative strategy for proving convergenceof the Ricci flow on a surface of positive Euler characteristic such that R 0)might change sign. This approach uses the the isoperimetric estimates de-rived in the previous section. It also uses the entropy estimate of Section 8and the LYH differential Harnack estimate of Section 10 for metrics of pos-itive curvature, but (perhaps surprisingly) not their extensions to the caseof variable curvature. It also relies on other important methods that will bedeveloped later in this volume. In particular, the approach uses techniquesof dilating about a singularity. (We will study such techniques in Chapter8.)

REMARK 5.93. The reader will find related results in Section 6 of Chap-ter 9.

For the remainder of this section, we shall adopt notation different fromthat used in the rest of this chapter. Namely, we will denote by (M2, g (t))the solution of the Ricci flow

a

atg__ -Rg,

9 (0) = go

and by (M2, g (t)) the corresponding solution of the normalized Ricci flow

a__atg -Rg, g (0) = go,

which exists for all t E [0, oo) by Proposition 5.19. As we shall see in Section9 of Chapter 6, these are related by rescaling time

1 rt 1 1t=r (1-e- t=rlogl -rtand dilating space

(5.52) g (t) = e-rt g O _ (1 - ft) 9 (i)Here r (t) - r (0) denotes the average scalar curvature of the metric go.Notice that g (t) exists for 0 < t < T, where

(5.53) T = 1.rIf A (t) _ Area (M2, g (t)), the Gauss-Bonnet formula

87r = 47rX (M2) - fa

R [g (t) ] dµ [g (t)]M

implies in particular that dA/dt = -87r and fA (0) = 87r. It follows that

(5.54) A (t) = A (0) - 87rt = A (0) (1 - ft) ,

hence that A (t) \ 0 and Rmax (t) / oo as t / T, where Rmax (t) denotesthe maximum scalar curvature at time t.

The first step in the current strategy is to show that one can bound theinjectivity radius by the isoperimetric constant. This fact is a consequenceof Klingenberg's result (Lemma B.34) that inj (Mn, g) is no smaller than theminimum of 7r/ KmaX and half the length of the shortest closed geodesic,where Kma., denotes the maximum sectional curvature.

LEMMA 5.94. If (M2, g) is a topological 2-sphere with isoperimetric con-stant CH (M2, g) and maximum Gaussian curvature Kma,,, then

C.(M2, g) .[inj (M2, g)] 2 > 4K..

PROOF. Let ry be a geodesic loop of length L (-y) dividing M2 into twotopological discs Mi and 1V12 with 5Mi = aM2 = ry. For i = 1, 2, let Ai

denote the area of M?. Since -y is a geodesic, the Gauss-Bonnet formula(5.51) reduces to

27r=X(MZ) =J KdAM?

for i = 1, 2. This implies that Kmax min {A1, A2} > 27r, equivalently that

1

,

I1

<

Kmaxmax f

Al A2 27r-Thus by (5.47), we get the estimate

rCFI (M 2, g) C CH (`Y) = L (7)2 l Al + 2 < Kmax L(1')2 ,

which we write in the form \L ('Y)2 > K CH (M2, g) .

max

If Lmin denotes the length of the shortest geodesic loop in (M2,g), thenKlingenberg's result says

2 2 l.[inj (M2, 9)] 2 > min Kmax, Lmin

Since we have CH (M2, g) < 47r by Lemmama 5.84, we conclude that

[inj (M2, g) ] 2 > min7r2 CH (M2,

g)'

7<

CH (M2, g) .

Kmax 47r 4Kmax

Combining this result with Theorem 5.88, one obtains the following:

COROLLARY 5.95. If (M2, g (t)) is a solution of the Ricci flow on atopological S2 with Gaussian curvature bounded above by Kmax (t), then

[inj (M2, g (t)) ] 2 > 4Kax

(t) CH (M2, g (0)) .

Our next task is to extract a limit of dilations about the singularity.Since the singularity time is T < oo, the solution (M2, g (t)) exhibits eithera Type I singularity

supM2 x [0,T)

or Type IIa singularity

sup IR t) l (T - t) = oo.M2 x [O,T)

(See Section 1 of Chapter 8.) In the first case, we take a sequence of pointsXj E M2 and times tj / T chosen to give a Type I limit, as in Section 4.1of Chapter 8; in the second case, we chose xj E M2 and tj / T to give aType II limit, as in Section 4.2 of Chapter 8. In either case, we ensure that

I R (xi, ti) I = maxM2 I R ti) I and consider the sequence of parabolicallydilated solutions

(tj(5.55) gj (t) _ R (xj, tj) g +t

R (xj, tj)

defined for -tjR (xj, tj) < t < (T - tj) R (xj, t.). The dilated solutions(M2, gj (t)) satisfy a curvature bound which depends on t but is uniform inj. Indeed, if we are dilating about a Type I singularity, there exists

SZ _ lim sup (T - t) Rmax (t) < 00t/'T

such thatSZ

R [g; (t)] <SZ - t'

whereas if the original singularity is Type IIa, there is a sequence f l / oosuch that

R [gj (t)] < 11- tlni'Since by Corollary 5.95, there is a uniform positive lower bound for allinj (M2, gj (0)), we can apply the Compactness Theorem from Section 3 ofChapter 7 to obtain a pointed subsequence (M2, gj (t) , xj) that convergesuniformly on any compact time interval and in every Ck-norm to a completepointed solution

(Moo, goo (t) , x.)of the Ricci flow with scalar curvature R. In the case of a Type I limit,g0C (t) is an ancient solution that exists for -oo < t < SZ and satisfies thecurvature bound

S2R°°<S2-t

In the case of a Type IIa limit, go,, (t) is an eternal solution that exists for-oc < t < oo and satisfies

The key step toward our goal will be to show that a Type IIa limitcannot occur. This is a consequence of the fact that an eternal solutionwith strictly positive curvature that attains the space-time maximum of itsscalar curvature must be a steady Ricci soliton. In the particular case of asurface, the result is the following result.

LEMMA 5.96. The only eternal solution (.N2, g (t)) of the Ricci flow ona surface of strictly positive curvature that attains its maximum curvaturein space and time is the cigar (R2, gE (t)).

PROOF. We claim that (N2, g (t)) is a gradient Ricci soliton. The resultfollows from this claim and Lemma 2.7, which proves that the only gradientsoliton of positive curvature in dimension n = 2 is the cigar.

To prove the claim we consider the trace Harnack quantity for the un-normalized Ricci flow on a surface with bounded positive scalar curvature,namely

Qat

log R - IV log R 12 =A log R + R.

This is the quantity defined in (5.35) without the -r term. Following theproof of Lemma 5.55, one computes that it evolves by

(5.56) aQ=OQ+2(VlogR,VQ)+2 VVlogR+ IRg7

2

hence estimates that

(5.57)at Q > OQ + 2 (V log R, VQ) + Q2.

(The reader should compare the equality and inequality above to (5.37) and(5.38), respectively.) If Q ever becomes nonnegative, then it remains so bythe maximum principle. On the other hand, if Qmin (t1) < 0 for some t1,then Qmin (to) < 0 for all to < tl, and one has

Qmin (t) >1 1

(Q (to)) - (t - to) t - to

for all t > to. Since (N2, g (t)) is an eternal solution, one may let to \ -oo,obtaining

1Qmin (t) > lim = 0.to -oo to - t

Hence on (N2, g (t)), we have

AlogR+R=Q? 0.This is the ancient version of Hamilton's trace Harnack inequality. (Notethe similarity between this argument and the proof of Lemma 9.15.)

Now by hypothesis, there exists (po, to) E N2 x Ilk such that

R (po, to) = sup R..I2XR

Thus we have aR/at = 0 and I VRI = 0 at (po, to), and hence Q (po, to) = 0.Since Q > 0, applying the strong maximum principle to equation (5.56) thenshows that Q must vanish identically on N2 x R. By examining its evolutionequation (5.56), we see that this is possible only if the tensor

VVlogR+ IRg

vanishes on N2 x R. In other wordsaatg = -Rg = C(V log R)d g.

Hence g(t) is a gradient Ricci soliton flowing along grad (log R).

PROPOSITION 5.97. Let (M2, g (t)) be a solution of the Ricci flow ona topological 2-sphere with g (0) = go. Then g (t) exists for 0 < t < T1/r, where r denotes the average scalar curvature of the metric go, and thesingularity at time T is of Type I.

PROOF. Suppose the singularity is of

r

Type IIa. Since

A(t)=A(0) -87rt=87r I 1 -t) =8rr(T -t)\by (5.53) and (5.54), this happens if and only if

sup Rma, (t) A (t) = oo.tE [0,T)

Since

Area (M2, gj (0)) = R (xj, tj) A (t;) = 87rR (xj, tj) IT - t3) -> oo

by (5.55), the limit ( ' A 429,, (0)) has infinite area. In particular, M2 is00 00noncompact, and g,, (t) has infinite area for all times t c (-oo, oc). Sincewe have

lim min 0,t--*oo M2

hence

lim min R (., t) / 0t- T M2

by Lemma 5.9, the scalar curvature of (M , goo (0)) is nonnegative, hence00strictly positive by the strong maximum principle. Moreover, R' E (0, 1]attains its maximum at (xc, 0). By Lemma 5.96, (M2, g (t)) is the cigarsoliton.

As we observed in Section 2 of Chapter 2, the cigar is the metric givenin polar coordinates (p, 9) on R2 by

19E = T+

p2dp2 + 1 +

P2d62.

As we saw in Section 2 of Chapter 2, it attains its maximum curvature atthe origin and is asymptotic to a cylinder as p - oo. In particular, for anye > 0 there is C >> 1 large enough so that the circle p = C has length27r - e < L < 27r and the open disc p < C has area A > 1/e. Henceif (M2, gj (t) , xj) is a sequence of closed pointed manifolds converging tothe cigar (]182, gr, 0), it is easy to see that there is j E N so large thatCH (M2, gj) < E. Since this contradicts Theorem 5.88, we conclude thatthe Type IIa singularity is impossible.

COROLLARY 5.98. The scalar curvature of any solution (M2, g (t)) ofthe normalized Ricci flow on a topological 2-sphere is uniformly bounded.

PROOF. Since

g(t)=(1-ft)g(0=f(T-t) 9 (t)

by (5.52) and (5.53), we have

But since g (t) has a Type I singularity, the right-hand side is uniformlybounded:

limsup (T-t)Rm.,. (t) =SZ<oo.t/'T

Now we finish the argument.

PROPOSITION 5.99. If (M2, go) is a closed Riemannian surface withaverage scalar curvature r > 0, then the unique solution g (t) of the unnor-malized Ricci flow with g (0) = go encounters a singularity at some finitetime T. A sequence (M2, gj (t)) of parabolic dilations formed as tj / Tconverges locally smoothly to a shrinking round sphere (S2, g. (t)) .

PROOF. We know that (M2, gj (t), xj) converges locally smoothly inthe pointed category to a limit (.M', g,, (t) , x,,,,) that must be an ancientsolution with bounded curvature. By Lemma 9.15, its scalar curvature ispositive. Let r = -t. Then by Proposition 5.39, the entropy N (g (T)) isincreasing in r and is bounded above by the entropy of a soliton. Thus(going backwards in time) we can take a limit as r --+ no to get a solu-tion (.M2., 92. (t) , x2.) which has constant entropy, hence must be a soli-ton. By Proposition 5.21, (M20, 92. (t) , x200) is a shrinking round sphere.But its entropy is minimal. Since N (g (r)) cannot decrease as T increases,this implies that N (g (r)) must be constant, hence that the original limit(M00, g. (t) , x,,,) was itself the shrinking round sphere.

REMARK 5.100. The assertion that the limit (.M2, g. (t), x.) must bethe shrinking round 2-sphere also follows from Proposition 9.23.


The main reference for this chapter is Hamilton's paper on the Ricciflow on surfaces [60]. Unless otherwise noted, the results in this chaptermay found there. In particular, the methods of entropy and Harnack esti-mates, Ricci solitons, and potential of the curvature were first introduced tothe Ricci flow in [60]. Curvature derivative estimates were established byShi in [117, 118]. Originally, curvature derivative estimates were obtainedin Corollary 8.2 of [60] by bounding the L2-norms of all derivatives and ap-plying the Sobolev inequality. (See also Theorem 13.4 in [58].) The methodof proof we give for the upper bound of R in Proposition 5.30 follows [30].The direct proof of the entropy estimate of Proposition 5.39 was given in[29]; the results in subsection 8.1 are also from there. The method of prov-ing the differential Harnack inequality in section 10 was first used by Li andYau [92] for solutions of the heat equation. (See Section 6 of [60].) Theextension to the case where R changes sign in Section 10.2 was established


in [28]. A new proof without the use of the potential function was givenby Hamilton and Yau [68]. The lower bound for the injectivity radius onS2 was proved by Hamilton. (See Section 5 of [28] or Section 12 of [63].)The isoperimetric estimate in Section 14 was given by Hamilton [62]; theapplication of this to give a new proof that R is uniformly bounded usesthe Gromov-type compactness theorem established in [64] and reviewed inSection 3 of Chapter 7.

Results on the Ricci flow on noncompact surfaces are established in[128], [129], [72], [73], and [35]. Convergence theorems on 2-orbifolds aregiven in [127] and [34]. A new proof of the convergence on S2 using theAleksandrov method was obtained by Bartz, Struwe, and Ye [10]. A relatedflow on S2 was studied by Leviton and Rubinstein [90].

The methods established for the Ricci flow are related to methods forother geometric evolution equations. Both the entropy and differential Har-nack estimates were established by Hamilton for the curve shortening flow[44]. The entropy estimate was extended to higher dimensional hypersur-face flows in [31] and [3]. References for differential Harnack inequalities ofLYH type in higher dimensions and for other geometric evolution equationswill be given in a chapter of the planned successor to this volume.

CHAPTER 6

Three-manifolds of positive Ricci curvature

The topic of this chapter is Hamilton's application of the Ricci flow to theclassification of closed 3-manifolds of positive Ricci curvature, in particularhis landmark result that any such manifold is diffeomorphic to a sphericalspace form. Let (Mn, g) be a closed Riemannian n-manifold with positiveRicci curvature. By Myers' Theorem, Mn is compact. Since its universalcover Mn also has positive Ricci curvature, it too is compact; and hence thefundamental group of ,Mn is finite. When n = 3, we have the well-known

CONJECTURE 6.1 (Poincare Conjecture). Any simply connected closedsmooth 3-manifold is diffeomorphic to S3.

A successful resolution of the Poincare Conjecture implies that . 43 isdiffeomorphic to S3. Furthermore, there is also the following

CONJECTURE 6.2 (Spherical Space Form Conjecture). Any finite groupof diffeomorphisms acting freely on S3 is conjugate to a group of isometricsof the standard sphere.

In the presence of a complete proof of the Spherical Space Form Conjec-ture, the diffeomorphism M3 = S3 would imply that .M3 is diffeomorphic toS3/I', where r is a finite subgroup of 0 (4). In other words, one could con-clude that M3 is diffeomorphic to a space form, all of which are classified.[126] As a consequence, M3 would admit a metric of constant positive sec-tional curvature. It is this last statement - the existence of a constant pos-itive sectional curvature metric on any closed 3-manifold with positive Riccicurvature - that Hamilton proved in 1982 using the then newly-introducedRicci flow. Our goal in this chapter is to present his result:

THEOREM 6.3 (Hamilton). Let (M', go) be a closed Riemannian 3-manifold of positive Ricci curvature. Then a unique solution g (t) of thenormalized Ricci flow with g (0) = go exists for all positive time; and ast --> oo, the metrics g(t) converge exponentially fast in every C' -norm to ametric gw of constant positive sectional curvature.

Although the result is stated for the normalized flow, it will be moreconvenient to work with the unnormalized flow for as long as possible. Thetwo equations differ only by rescaling space and time. (See Subsection 9.1.)In fact, the theorem may be interpreted as a statement of the convergence

173

174 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

of the unnormalized flow modulo resealing; but the advantage of finallyconverting to the normalized flow is that it will allow us more naturally todemonstrate exponential convergence. Our strategy for proving Theorem 6.3will be to learn enough about how curvature evolves under the unnormalizedflow to obtain certain a priori estimates that will suffice to prove long-timeexistence and convergence of its normalized cousin.

1. The evolution of curvature under the Ricci flow

In this section, we compute the evolution equations for the Levi-Civitaconnection and the Riemann, Ricci, and scalar curvatures of a solution tothe Ricci flow. We start by recalling certain general evolution equationsfor a one-parameter family of metrics which were derived in Section 1 ofChapter 3.

REMARK 6.4. Unless explicitly stated otherwise, all results in this sectionhold for any dimension n > 2.

LEMMA 6.5. Suppose that g (t) is a smooth one-parameter family of met-rics on a manifold M' such that

aatg=h.

(1) The Levi-Civita connection r of g evolves by

at r =19ke

(Dihje + Vj hie - Vehij)

(2) The (3, 1) -Riemann curvature tensor Rm of g evolves by

_ f ViVjhkp + ViVkhjp - ViVphjk

at iik 2" -V jVihkp - V jVkhip + V jVphik J

V hV V+ V h - ° V h V h- °i k jjp P ik i P j k ipjk1 9ep

2 -R khep - R phks(3) The Ricci tensor Re of g evolves by

atRjk =1gPg(VgVjhkp+VgVkhjp-VgVphjk-VjVkhgp)

}

2[Lhk + V j Vk (tr 9h) + V j (Sh)k + Vk (6h) j] ,

where AL denotes the Lichnerowicz Laplacian defined in formula(3.6).

(4) The scalar curvature R of g evolves by

a ij ke

atR = 9 9 (-V iV j hkl + V jV khje - hikRje)

= -0 (tr 9h) +div (div h) - (h, Re)

1. THE EVOLUTION OF CURVATURE 175

(5) The volume form du of g evolves by

atdµ = 2 (tr q h) dµ.

Substituting h = -2 Re into the lemma yields the following result.

COROLLARY 6.6. Suppose g (t) is a solution of the Ricci flow.


(6.1) ar -gke (V Rje +VjRi,e - VeRj)

(2) The (3, 1)-Riemann curvature tensor Rm of g evolves by

-V iV j Rkp - V iV kRjp + V iV pRjk(6.2) R2k = glp

+V jV iRkp + V jVkRip - V3V pRik

(3) The Ricci tensor Re of g evolves by

(6.3)aa Rik = ARik +VJVkR -gPq (VgVjRkp+VgVkRjp)


(6.4) aR = 2AR - 2gjkgpgVgV jRkp + 2 IRcl2

(5) The volume form dµ of g evolves by

(6.5)at

dµ = -R dp.

We shall soon obtain more useful forms of many of the equations aboveby applying various curvature identities. To motivate the type of equation weare looking for, recall formula (5.3) for the evolution of the scalar curvatureof a solution of the normalized Ricci flow on a surface:

aR=AR+R(R-r).This is an example of a reaction-diffusion equation with a quadraticnonlinearity involving the scalar curvature. Equations of this form are alsocalled heat-type equations. We shall see below that in any dimensionn > 2, the Riemann curvature Rm, the Ricci curvature Rc, and the scalarcurvature R all satisfy heat-type equations whose nonlinear terms are variousquadratic contractions of the Riemann curvature.

1.1. The scalar curvature function. We first consider the scalarcurvature function, since its evolution equation is the simplest. After twocontractions of the second Bianchi identity

VmRijkl + V kRijlm + VeRijmk = 0,

one gets gjkVjRkl = 20eR. Applying this to (6.4) gives a nicer equation.


LEMMA 6.7. The scalar curvature of a solution to the Ricci flow evolvesby

(6.6) (R = AR+ 2 IRcI2 .

This is the general (n-dimensional) version of the evolution equation forthe curvature of a surface derived in Chapter 5.

Although the evolution equation (6.6) for the scalar curvature R involvesthe Ricci tensor, the latter quantity enters only in a nonnegative term. Thusthe weak maximum principle for scalar equations implies that positive scalarcurvature is preserved under the Ricci flow. More generally, we have thefollowing result.

LEMMA 6.8. Let g (t) be a solution of the Ricci flow with g (0) = go. Ifthe scalar curvature of go is bounded below by some constant p, then g (t)has scalar curvature R > p for as long as the solution exists.

1.2. The Ricci curvature tensor. Applying the contracted secondBianchi identity to the second equation for the evolution of Rc in Lemma6.5 puts equation (6.3) in a nicer form.

LEMMA 6.9. Under the Ricci flow, the Ricci tensor evolves by

(6.7)aat Rjk = OLRjk = ARjk + 2g11grs p jkrRgs - 2gr9RjpRgk.

PROOF. We have "P

atRjk = OLRjk + VjVkR - gpq (VjVpRqk + VkVpRjq)

= OLRjk + VjVkR - 2 (VjVkR+ VkVjR).

0

The presence of the Riemann tensor in the evolution equation for theRicci tensor is an obstacle to showing that nonnegative Ricci curvature ispreserved in arbitrary dimensions. (In fact, there exist examples [101] wherepositive Ricci is not preserved.) One wants, therefore, to attain a better un-derstanding of the contribution of the Riemann tensor to (6.7). Recall thatin any dimension, the Riemann tensor admits the orthogonal decomposition

(6.8) Rm.= 2n(R-1) (g0g)+n1 2 (cog) +W,

where O denotes the Kulkarni-Nomizu product of symmetric tensors,

(P Q)ijke T PieQjk + PjkQie - PikQje - PjeQik,0

Rc denotes the trace-free part of the Ricci tensor, and W is the Weyl tensor.Because the Weyl tensor vanishes in dimension n = 3, equation (6.8) reducesto

R(6.9) Rijke = Riegjk + Rjk9ie - Rik9je - Rjegik -

2(9iegii - 9ikgje)


This identity shows in particular that the Ricci tensor determines the Rie-mann tensor when n = 3. Because of this, the reaction terms in (6.7) dependonly on the Ricci tensor.

LEMMA 6.10. In dimension n = 3, the Ricci tensor of a solution to theRicci flow evolves by

(6.10)at

Rjk = ARjk + 3RRjk - 6gp4RjpRgk + (2 IRcl2 - R2) gjk .

PROOF. Substituting (6.9) into the second term on the right-hand sideof (6.7) yields

2gpggrsRpjkrRgs = (2 jRc12 - R2) gjk -4g,qRjpRqk + 3RRik.

The weak maximum principle for tensors (discussed in Chapter 4) im-plies the following important consequence of formula (6.10).

COROLLARY 6.11. Let g (t) be a solution of the Ricci flow with on a3-manifold with g (0) = go. If go has positive (nonnegative) Ricci curva-ture, then g (t) has positive (nonnegative) Ricci curvature for as long as thesolution exists.

PROOF. At any point and time where the Ricci tensor has a null eigen-vector, it has at most two nonzero eigenvalues. Then one has IRc12 > R2/2,whence the result follows immediately from Theorem 4.6.

REMARK 6.12. More generally, in any dimension n > 3, the decomposi-tion (6.8) lets one write the evolution equation (6.7) satisfied by the Riccitensor in the form

a 2n 2n

at Rjk = ORjk - n - 2 Rek + (n - 1) (n - 2)RRjk

+ n 2 2 (Rc12_ n 1 1 R2 I gjk + 2RpgWjprk.

Since I Rc 1 2 > R2/ (n - 1) whenever Rc has a zero eigenvalue, one concludesin particular that the Weyl tensor is the sole obstacle to preserving thecondition Rc > 0.

1.3. The Riemann curvature tensor. The observations made in theprevious two subsections suggest that the Riemann curvature tensor alsosatisfies a heat-type (reaction-diffusion) equation.

LEMMA 6.13. Under the Ricci flow, the (3,1)-Riemann curvature tensorevolves by

l(6.11a)at

k + gpg (RjpRqk - 2R 2Rp. Rjgk)

(6.11b) - R'RZpk - Rkk. + RpRZk.


PROOF. By applying the second Bianchi identity and commuting covari-ant derivatives, we compute

gp4VpVgR k = 9 p9Vp (_ViRqk - VjRk)ViVpRgk +R ijRgk+ R

gpg

+VjVpRgik - R;jiRrgk - RPjgRirk - RpjkRzgr + RpjrRQk

and then apply the second Bianchi identity again to obtain

trngpqV pRjgk =

gpg9 (-V kRjgmp - V mRjgpk) = VkR - VtRjk.

This lets us rewrite the formula for ORijk given above as

-ViVkRj + ViVtRk + VjVkRZ - VjVPRik

R r RP + R r Rt + Rr RQ - Rt RTpij rqk piq irk pi,k jqr pir jqk+ gp4

- Rr Rt Rr Rp Rr Re +R1. R7 krqk pjq irk pjk iqr pjr aqk

Now the first Bianchi identity shows that

r P r P r eRpijRrgk - RpjiRrgk __ -RijpRrgk,

which implies that AM. may be written as

(6.12a)

(6.12b)

ORf -k = -ViVkR + V2VERjk + V jVkR2

- VjV'Rik - Ra Rjrk + RRark

(6.12c) + gpq

RpjrRrgk

R' T T Q-Rpit jqk - RpjkRigr}

On the other hand, rewriting the commutator term gEP (VjVi - ViVj) Rkpin formula (6.2) yields

(6.13a)BtRjk = -ViVkR3 +ViVtRjk + VjVkRi -VjVtRik

(6.13b) + gRP (RjkRqp + RPRkq) .

The reaction-diffusion equation for the Riemann curvature tensor now fol-lows from comparing terms in formulas (6.12) and (6.13).

We define the (4, 0)-Riemann curvature tensor by Rijk2 - gLmRmk, sothat R1221 > 0 on the round sphere. The following observation then followsimmediately from the lemma above.


COROLLARY 6.14. Under the Ricci flow, the (4, 0)-Riemann curvaturetensor satisfies the following reaction-diffusion equation:

(6.14a)at

Rijke = ARijke + gpq (R pRrgke - 2RpirtRgk)

(6.14b) - (RPRpjke + RPRipke + RpRijpe + RPRljkp) .

There is a nice way to rewrite the evolution equation (6.14) in terms ofa (4, 0)-tensor B defined by

(6.15) Bijke - -gprggsRpj,Rkres = -R jR9ek.

(The minus sign here, which does not appear in Hamilton's original paper[58], occurs because we are using the opposite sign convention for Rijke.)Note that B is quadratic in the Riemann curvature tensor and satisfies thefollowing algebraic identity

(6.16) Bijke = Bjiek = Bkeij

LEMMA 6.15. Under the Ricci flow, the (4, 0)-Riemann curvature tensorevolves by

(6.17a)aat Rijke = ARijke + 2 (Bijke - Bijkk + Bike - Bitik)

(6.17b) - (R'Rpjke + RRRipke + RkRijpe + RRljkp)

PROOF. First observe that the last two terms on the right-hand side of(6.14a) may be directly expressed in terms of B:

-2gpgRprikR.7gre = -2g gRipkrRj4er = 2Bikje,29P RpireRjgkr

= 2g RiperRjgkr = -2 2ejk .

This leaves us with gpgR U-pRgkt. Applying the first Bianchi identity gives

9pgK- Rrgke = gpggrsRpjiRgke

= gpggrs (_Rjip - Rripj) (-Rskeq - Rsigk)

= -Bjike + Bjilk + Bijke - Bijkk,

whence identity (6.16) implies that

9pgRijpR,.gke = 2 (Bijke - Bijkk

REMARK 6.16. Observe that although B does not in general satisfy thefirst Bianchi identity, the tensor C does, where

Cijke = Bijke - Bijkk + Bikke - Bitik-

Similarly, although Bjikt -Bijke in general, one has Cjike = -Cijke


2. Uhlenbeck's trick

Let (M", g (t)) be a solution of the Ricci flow, and let {ea}Q_1 be alocal frame field defined in an open set U C Mn. Suppose that {ea} isorthonormal with respect to the initial metric go. We would like to evolvethis frame field so that it stays orthonormal with respect to g (t). This mayeasily be done by considering the following ODE system defined in TXM1' foreach x EU:

(6.18a)

(6.18b)

dea (x, t= Re (ea (x, t)t

,

ea (x, 0) = ea (x) .

Here we regard the Ricci tensor Re - Rc [g (t)] as a (1,1)-tensor, that is,an endomorphism Re : TM' -> TMn. Since (6.18) is a linear system ofn ordinary differential equations, a unique solution exists as long as thesolution g (t) of the Ricci flow exists.

LEMMA 6.17. The inner product g (ea, eb) is constant for any solutiong (t) of the Ricci flow and any frame field {ea (t)} satisfying (6.18).

PROOF. Noting that g, Re, and each ea all depend on time, we computethat

aat [g(ea,eb)] =

(a19

9) (ea,eb)+g(ata

ea,eb I + g I ea, aeb)

_ -2 Re (ea, eb) + g (Re (ea) , eb) + g (ea, Re (eb))

= 0.

COROLLARY 6.18. If {ea} is orthonormal with respect to go, then {ea (t)}remains orthonormal with respect to g (t).

REMARK 6.19. If.Mn is parallelizable (in other words, if TM' is a trivialbundle) there exists a global frame field that is orthonormal with respect toany given Riemannian metric g. Indeed, one may take any global frame fieldand apply the Gram-Schmidt process (which preserves smoothness of theframe field).

REMARK 6.20. Every closed 3-manifold is parallelizable, hence may beassigned a global orthonormal frame field.

The idea of evolving a frame field to compensate for the evolution of aRiemannian metric has a more abstract formulation, due to Karen Uhlen-beck, which we now present. The Uhlenbeck trick allows us to put theevolution equation (6.17) satisfied by Rm into a particularly nice form. Webegin as follows. Let (Ma, g (t) : t E [0, T)) be a solution to the Ricci flowwith g (0) = go. Let V be a vector bundle over Mn isomorphic to T.Mn,and let to : V - TM' be a bundle isomorphism. (In other words, the

2. UHLENBECK'S TRICK 181

restrictions (to),, : Vx -+ TXM' are vector space isomorphisms dependingsmoothly on x E Ma.) Then if we define a metric ho on V by

ho - Lo (90) ,

we automatically obtain a bundle isometry

to : (V, ho) - (T-A4', go) .

Corresponding to the evolution of the metric g (t) by the Ricci flow, weevolve the isometry t (t) by

(6.19a) at t =Re oL,

(6.19b) L (0) = Lo .

Here as in (6.18), we regard the Ricci tensor Re = Re [g (t)] as a (1, 1)-tensor. For each x e M't, equation (6.19) represents a system of linearordinary differential equations. Hence a unique solution exists for t E [0, T)(namely, for as long as the solution g (t) of the Ricci flow exists). Clearly,t (t) remains a smooth bundle isomorphism for all t E [0, T). But more istrue.

CLAIM 6.21. Define h (t) _ (t (t))* (g (t)). Then the bundle maps

t (t) : (V, h (t)) -> (T.A4 , 9 (t))

remain isometries.

PROOF. Let x E Nl' and X, Y E VV be arbitrary. Then recalling thatg, Re, and t all depend on time, we compute that

at h (X,Y)

at [(c*9) (X, Y)]

= at [g(t(X),t(Y))I-2 Re (t (X) , t. (Y))

+ g (Re (t (X)) , t (Y)) + g (t (X) , Rc (t (Y)))

= 0.

11

The Levi-Civita connections

V (t) : C°° (T.Mn) x C°° (TM's) -} C°° (TM's)

of the metrics g (t) pull back to connections D (t) on V defined by

D (t) - L (t)* V (t) : CO° (TM') x C°° (V) - C°° (V) ,

where for X E C°° (TM 1) and E CO' (V), we have

(L*V) (X, ) - (t*V)x Vx (t (k))


Using the usual product rule, V (t) and D (t) define connections on tensorproduct bundles of TMn, V, and their dual bundles T*Mn and V*. Wedenote these connections by 17D.

The next step in Uhlenbeck's trick is to pull back the Riemann curvaturetensor to V. For x E .Mn and X, Y, Z, W E Vr, the tensor

t* Rm E C°° (A2V (gs A2V)

is defined by

(6.20) (t*Rm)(X,Y,Z,W) =Rm(t(X),t(Y),t(Z),L(W))

-Let {xk}k=1 be local coordinates defined on an open set U C Mn, and let{ea}a-1 be a basis of sections of V restricted to U. The components (tQ) ofthe bundle isomorphism t : (V, h) - (TMn, g (t)) are defined by

nkt (ea) to

k=1

axk .

Then the components (Rabcd) of t* Rm are defined by

n

Rabcd = (t* Rm) (ea, eb, ec, ed) _ E tatbtctdR'ijki, j,k,t=1

How is the evolution equation for t* (Rm) related to that for Rm? Inorder to answer this question, we define the Laplacian acting on tensorbundles of T.Mn and V by

n

OD - try (VD 0 VD) = E 923 (VD)i (VD)ji,j=1

where (VD)j (6) = Vi (t (c)). We then get the following formula.

LEMMA 6.22. If g (t) is a solution of the Ricci flow and t (t) is a solutionof (6.19), then the curvature t* Rm defined in (6.20) evolves by

(6.21)

where

Ct Rabcd = AD Rabcd + 2 (Babcd - Babdc + Bacbd - Badbc) ,

Babcd = heghfhRaebfRcgdh

PROOF. Observing that

3. THE STRUCTURE OF CURVATURE EVOLUTION 183

we computen

a P

at Rabcd = at (ti

tjit k

`d j kEi j,k,P=1

=(a i1 jkP i (a b) kP ij kl tataJ LcGdRijkP + acb atG J GdRijkP

+ Latbtc (:) +- (.kt)at= Rm,ta tbtctdRijkt + LaR

GaLbRTn czLdRijkP+ GaGbGcRmLdRijkP

ARijkP + 2 (BijkP - Bijlk + BikjP - Bi1jk)+

1 -Ri RpjkP - RRRipkt - RkRijpt - Rj'Rjkp

= Latbtctd [ORijki + 2 (BikjP - Bij k + Bikje - Bitjk)]

Since VD (t) = 0, we have [t* (ORm)]abd = ADRabcd Also, one may easilycheck that Babcd = (t*B) (ea, eb, e,, ed). The lemma follows.

3. The structure of the curvature evolution equation

By looking more closely at the structure of the Riemann curvature op-erator, we can obtain valuable insight into the structure of its evolutionequation (6.21).

We begin with the observation that although B is quadratic in Rm, it isnot the square of Rm. A natural way of defining the square is first to regardRm as the operator on 2-forms

Rm : n2T*Mn -r A2T*Mn

defined for all U E A2T*Mn by

(Rm (U))ij = -9kp9Pg1ijk1Upq.

The operator Rm is then self adjoint with respect to the inner productdefined for all U, V E n2T*Mn by

(6.22) (U, V) _ 9zk9j'UijVkP1

indeed, it follows easily from symmetries of the Riemann curvature tensorthat

(Rm(U),V) = (U,Rm(V)).Note that

Rm = 2r, id n2T.Mn

in case (Ma, g) has constant sectional curvature i, because then

RijkP = /c (giigjk - 9ik9j),


whence(Rm (U))ij = ,c (Uij - Uji) = 2icUij.

Now we can square the operator Rm to obtain the operator

Rm 2 = Rm o Rm : A2T*M' --> A2T*Mn

given in local coordinates by

(6.23) (Rm2) ijkP= gpg9rsRi7psR4r kP-

REMARK 6.23. An equivalent theory results if one regards Rm andRm2 as symmetric bilinear forms on A2TMn, hence as smooth sectionsof A2T*.Mn ®s A2T*.Mn. This point of view will be useful below.

Although Rm2 is the most natural definition of the square of the Rie-mann curvature operator, there is another concept of square which will beuseful. This can be defined whenever one has a Lie algebra g endowed withan inner product (, ). Choose a basis {co } of g and let C.y3 denote thestructure constants defined by

rya, I Ery

where is the Lie bracket of g. Let {cpa} denote the basis algebraicallydual to {cpa}, so that cpa (WO) = 6a. Given a symmetric bilinear form L ong*, we may regard L as the element of g ®s g whose components are givenby

Then there is a commutative bilinear operation # defined for all L, M Eg®sgby

(L#M)aQ =Abusing notation slightly, we define the Lie algebra square L# E g ®s gof L by

(6.24) (L#)ap - (L#L)al =The following property will be needed in showing that positivity of theRiemann curvature operator is preserved by the Ricci flow.

LEMMA 6.24. If L > 0, then L# > 0.

PROOF. Without loss of generality, we may choose a basis {cpa} thatdiagonalizes L, so that Lad = Sa$Laa. Then for any v = vacpa E g*, wehave

(vc6)L# (v, v) =(vC) LryeLa = (vaC)2 LryryLba

REMARK 6.25. The inner product of g defines an metric isomor-phism g --> g* by v (v, .). This isomorphism allows us to regard L : g -* gas a self-adjoint endomorphism.

3. THE STRUCTURE OF CURVATURE EVOLUTION 185

We now adopt this general definition to introduce another square of theoperator Rm. Observe that for each x E .M', the vector space A2TX.M canbe given the structure of a Lie algebra g isomorphic to so (n). Indeed, givenU, V E A2T*Mn, we define their Lie bracket by

(6.25) [U,V]ij _ gkP(UikVPj - VikUPj)

In a local orthonormal frame field {ei}, any 2-form U may be naturally iden-tified with an antisymmetric matrix (Uij) so that formula (6.25) correspondsto

[U, V]ij = (UV - VU)ij.This yields an evident Lie algebra isomorphism between g =and so (n) for each x E Mn'. We endow g with the inner product defined by(6.22). Then the construction outlined above applies. In particular, in localcoordinates {xi}, there is a basis {cp(ui) : 1 < i < j < n} for g defined by

cp('j) = dxti A dxp.

The corresponding structure constants are defined by

[dxp A dxq, dxr A dxs] C(pg)'(rs)dxi A dxj.

(ij)

Since

dxp n dxq = 2 (dxp ® dxq - dxq ® dxp)

=2

(Spde - 5g 5p\ dxk ®dxi,

formula (6.25) implies that

C(pq),(rs) = [dxp A dxq, dxr A dxs]ijij)

= 1 gkP

4

bPbk - I b"Ji'

A sk - bi bk) (jpq3- bQ b3

J

9graPsjs - ggsbPsjr-"

gprbgsjs +gpss9bjr

4gSP J

q +

9sgbz by

+grpb2

Sq - grga2

by

gqr Si SP) + gqs ((SZsP - PS;

(jiqjr+gpr+ 9ps

biSPI

Then formula (6.24) yields

(6.26) (Rm #ijkP

= RpquvRrs.,C(i)),(rs)C(ek)),(w)


Having completed these constructions, we can now write the evolution ofthe Riemann curvature operator in a way conducive to applying the tensormaximum principle.

THEOREM 6.26. If g (t) is a solution of the Ricci flow, the curvatureC* Rm defined in (6.20) evolves by

(6.27) at (t*Rm) =ODRm+Rm2+Rm#.

PROOF. By exploiting appropriate antisymmetries, we get

p (pq),(rs) = 1 ( qr (oo; ps (oo;RpquvC- Rpquv - bi bj J + g - jand

RpquvRrswxC(p)),(rs)Rrswx = RpquvRrswx9'gr 1 o s. - bibp) .

So by (6.26), we have

Rm # = RpquvRrsw.Tijkt

- RpquvRrswxggr (o; - S2 9vw (SQ Sk - bt ak )

= RuvpRsgx (Spa, - b'-"5p) (6, bk - St bk )

= Rg Rj Rk Rj Rg R" + Rg Rztvi qk - vi 7qt tvj iqk kvj aqt-

Now recalling that Bijkt = -RpijRgek by definition (6.15), we invoke (6.16)in order to write Rm# as

(Rm #ijkt

_ -Btikj + Bkitj + Btjki - Bkjti = 2 (Bikjt - Bitik)

On the other hand, applying the first Bianchi identity to formula (6.23) gives

(Rm 2 )ijkt ^ gpg9"R jpsRgrtk = gpggrs (R_,psj + Risjp) (Rkrgt + Rkgtr)r r- (& - Rpji

Rprkt - Rprtk = -Bijtk + Bijki + Bjitk - Bjikt= 2 (Bijkt - Bijtk)

So pulling back by the bundle isomorphism t defined in (6.19), we have(Rm 2)

abcd = 2 (Babdd - Babdc)

and

(Rm#)abcd 2 (Bacdd - Badbc)

Hence by Lemma (6.22), we conclude that

C a - OD) Rabcd = 2 (Ba6cd - Babdc - Badbc + Bac6d)at

= (Rm 2) abcd + (Rm #)abcd

4. REDUCTION TO THE ASSOCIATED ODE SYSTEM 187

Having written the evolution of the curvature operator in this form, wecan immediately apply the tensor maximum principle for systems (discussedin Chapter 4) to get the following important result.

COROLLARY 6.27. If (.M",g (t)) is a solution of the Ricci flow whosecurvature operator is positive (negative) initially, then that condition is pre-served for as long as the solution exists.

4. Reduction to the associated ODE system

The maximum principle for systems introduced in Section 3 of Chapter4 allows us to obtain qualitative information about the evolution of theRiemann curvature operator under the PDE (6.27) by studying associatedsystems (6.31) and (6.32) of ODE. This point of view originated in [59] andwill be useful below in proving pinching estimates for the Ricci curvatureof a solution to the Ricci flow on a closed 3-manifold with positive Riccicurvature.

Let (M', g) be a Riemannian manifold, and let {ei} be an orthonormalmoving frame defined on an open set U C M'. The frame {ei} induces anorthonormal basis f Bk = B J- ei A e j } of A2TU. Hence for any x E U, there

is a well defined Lie algebra isomorphism cpx : A2Tx.Mn ---> A2 R' taking theordered basis (61, ... , BIN to an ordered basis p = (,Ou, ...,ON) of A2Rn,where N = (2).

Let A denote the space of those self-adjoint linear transformations M ESym2 (A2W') that obey the first Bianchi identity. The ODE correspondingto the reaction-diffusion PDE (6.27) satisfied by the curvature operator of asolution to the Ricci flow may then be written as

(6.28)dt

M2 + 1M[#,

where M (0) = M0 E . and M# is the Lie algebra square introduced inSection 3. The basis Q allows one to represent any M E A by an N x Nsymmetric matrix M,3 defined by

M (/3) = f3 Q)ij,

with the obvious summation in effect. The structure constants Cp forA2TA2R'n ^_' so (n) with respect to the basis 0 are defined by

/3k] (CC)ijk .

In terms of the basis Q, the transformation M H Q _ M2 + M# is thengiven by

(QQ)ij = (Mp)ik (MO)kj + (C,3)ip9 (C13)jrs (M13), (MR),,

In what follows, we shall suppress dependence on the basis ,O.Let us now specialize to the case that M3 is a closed 3-manifold. Since

M3 is parallelizable (as we noted in Remark 6.20) we may assume there


exists a globally defined orthonormal moving frame {ei}. We fix an or-thonormal basis {ok = 0 ei A ej } of A2TM3. For example, such a basis isgiven by taking

(el) (e2 A e3)01 = 1_V2 vf2-

02 = * (e2) _ - (e3 A el)

0 0 0

0 0 1/v' ,

0 -1/\ 0

0 0 -1r2-0 0 0

1// 0 0

0 1/v 0-1// 0 0

0 0 0

where * : A1TM3 - A2TM3 corresponds to the Hodge star operator. Indimension n = 3, the Lie algebra square is easily computed. Indeed, it isreadily verified that ([ei, &i , 0') is fully alternating in (i, j, k), hence that

fa b c # df - e2 ce - b f be - cd(6.29) b d e = ce - bf of - c2 be - ae

c e f be-cd bc-ae ad-b2Now we identify Rm with the quadratic form M defined on A2T.M3 by

M(eiAej,ej Aek)_ (R(ei,63)ek,ee)

Using the basis {01, 02,031 of A2TU, we further identify M with the matrix(MP.) defined on each fiber n2TX.M3 of the bundle A2T.M3 by

(6.30) (R (ez, ea) ek a ee) = Mpg0p0q

If {ei} evolves to remain orthonormal, the PDE (6.27) governing the behaviorof Rm corresponds to the ODE

(6.31) dtM=M2+M#

satisfied by M in each fiber. If {ei} is chosen so that Mo is diagonal atx E M3 with eigenvectors A (0) > p (0) > v (0), then (6.29) and (6.31)combine to yield the system

(6.32) µ = µ2 + AVdt v2 A/-t

posed on 1[83. In particular, M (t) remains diagonal (which is not in generalthe case in higher dimensions). Moreover, A (t) > ft (t) > v (t) for all t > 0

5. LOCAL PINCHING ESTIMATES 189

such that a solution exists, because

dt (A(A -' µ) (A + µ - v)

dt(µ - v) v) (-A + µ + v) .

By (6.30), the eigenvalues A, µ, v of M are twice the sectional curvatures.Hence at x e M3, we may regard the Ricci tensor as the matrix

µ+v1

(6.33) Rc=2

A+vA+µ

For later reference, we note that the trace-free parts of Rm and Re arerelated at the same point by

0 2A-µ-v(6.34) Rm= 3 -A+2µ-v J =-2Rc.

-.X-µ+2v

5. Local pinching estimates

Now we are ready to state the pinching results which are true for 3-manifolds with positive Ricci curvature. The first estimate proves that cur-vature pinching is preserved, whereas the second shows that it improves,hence that a solution to the Ricci flow on a 3-manifold with positive Riccicurvature is nearly Einstein at any point where its scalar curvature is large.These are pointwise estimates; below, we shall develop techniques to com-pare curvatures at different points of a solution.

As in Section 4, we let A (t) > µ (t) > v (t) denote the eigenvalues ofthe curvature operator Rm of a solution (M3, g (t)) to the Ricci flow on aclosed 3-manifold, recalling that these are twice the sectional curvatures.

LEMMA 6.28 (Ricci pinching is preserved.). Let (M3, g(t)) be a solutionof the Ricci flow on a closed 3-manifold such that the initial metric go hasstrictly positive Ricci curvature. If there exists a constant C < oo such that

(6.35) A < C (v + µ)

at t = 0, then this inequality persists as long as the solution exists.

PROOF. Recall that A > 2 (v + µ) > 0, so that we can take logarithms.Compute

d

d log (v+µ) = A(v+µ) C(v+µ)dA

-Adt (v+µ)

= 1A(v+µ) ((v+µ) (A2+µv) - A (v2+A +µ2+Au))

(6.36) = µ2(v - A) +v2(µ - A) <0.A(v+µ)

Let A (IID) > ,u (IID) > v (IP) denote the eigenvalues of

P E (n2T*.M3 ®s A2T*.M3)X .

Define /C C (A2T*M3 ®s n2T*M3)X by

JC - {P: A (IP)-C(v(P)+A(lP)) <<0}.

It is easy to see that /C is invariant under parallel translation. /C is convexin each fiber, because the function

A (P)-C (v (IP) + u (IP)) = max P (U, U)+C ax [-P (V, V) - P (WI W)]UI=1

Ivy=mIWI=1

(V,W)=0

is convex. Let M (t) E (n2T*M3 ®s n2T*M3)X be the quadratic formcorresponding to Rm. [g (t)], and choose C < oo so large that M (0) E IC forall x E M3. By (6.36), M remains in K.

This estimate implies a global bound which will be useful later.

COROLLARY 6.29. Let (.M3,g(t)) be a solution of the Ricci flow on aclosed 3-manifold of initially positive Ricci curvature, and define Rmin (t)inf.EM3 R (x, t). Then there exists R > 0 depending only on go such that atall points of M3,

Rc > 2/32Rg > 2132Rming

PROOF. By the lemma and formula (6.33), there exists C > 0 dependingonly on go such that the inequality

Rc> N'+v > A > A+µ+y _ R > Rmin

2g 2Cg 6C g 6Cg 6C

holds everywhere on M3.

Lemma 6.28 also helps us prove the following result, which shows thatthe metric is nearly Einstein at points where the scalar curvature is large.

THEOREM 6.30 (Ricci pinching is improved.). Let (.M3, g(t)) be a solu-tion of the Ricci flow on a closed 3-manifold such that go has strictly positiveRicci curvature. Then there exist positive constants 8 < 1 and C dependingonly on go such that

a<

Cv+µ (v+p)S.

In particular, the left-hand side is invariant under homotheties of the metric,while the right-hand side tends to 0 as v +,a --> oo.

REMARK 6.31. Theorem 6.30 is equivalent to the following statement,which was proved in Hamilton's original paper [58]: There exist constants3 > 0 and C < oc depending only on go such that

(6.37)

0 02 2

1RR2 = 4 R2J < C R-S,

0

where as in (6.34), Rm and Rc are the trace-free parts of the Riemanncurvature operator and Ricci tensor, respectively. Once again, the left-handside is scale invariant, while the right-hand side tends to 0 uniformly ifinf,,cm, R (x, t) -> 00.

We shall give two proofs of the theorem: the first uses the maximumprinciple for systems and is in the spirit of [59], whereas the second is closerin spirit to the original argument of [58].

FIRST PROOF OF THEOREM 6.30. Because

dt (A- v) _ (A- V) (A-µ+v)

we may assume that A (M) > v (M) for M (t) E (n2T*M3 (&S n2T*M3)We calculate

d dtlog(A-v)=A-µ+vand

Hence

dtlog(v+µ) _ A+ v2 µ2

v+ µ

_ 2 2

d1og L(v+1_8] v+1l2 2=6a+vV +y+µ

<8A- 2(1-6)(v+µ),

where we used v < µ and v2 + µ2 > 2 (v + 9)2 to obtain the last inequality.By Lemma 6.28, the right-hand side is nonpositive for 6 chosen small enough.Now define the convex set

1C= {'P: [A(P)-v(p)]-C[v(P)+µ(P)]1-b <0}

and continue as before.

SECOND PROOF OF THEOREM 6.30. By Lemma 6.34 (below), the func-tion

f=

0IRml2

R2-e

satisfies the evolution equation

(6.38) 8 <i f+2(1-e)(Vf,V(logR))+2Q.

Here / 0

Q= R3 (eIRm12IRm12-P ,

where P may be written in terms of the eigenvalues A, p, v of Rm as

(6.39) P A2(µ- v)2+µ2(A-v)2+v2(A-tL)2>0.By the maximum principle, we only need to show that Q < 0 to establish the

lemma. Notice that the good term is -P, while the bad term IRm12 IRml2 is

scaled by E. So to prove Q < 0, one first observes that p > 62 Rm 2IRm12

if Rc _> SR g for some constant S > 0. Then one uses Corollary 6.28 toconclude that the inequality Rc > is preserved if it holds at t = 0. Thuswe can take e = S2 for a sufficiently small positive constant S = S (go).

REMARK 6.32. The hypothesis of strictly positive Ricci curvature isnecessary for either proof to work, since Q > 0 if A > 0 and µ = v = 0 at apoint. This failure is related to the fact that SI X S2 accepts no metric ofpositive Ricci curvature. (Indeed, by Myers' Theorem, a complete manifoldMn admits such a metric only if its universal cover is compact.)

Equation (6.38) is derived in the following two lemmas:

LEMMA 6.33. If cp is a nonnegative function and 0 is a positive functionon space and time, then

a

- \ a l (at - ) - 0+1(A)

'

a(a-1)2IV

I2- +1)(a+2V 12

I.+2ao00+1 (VIP,VV))

PROOF. Straightforward calculation.

LEMMA 6.34. The function f _ RF-2IRm12 satisfies the differential in-equality

of A f +2( 1 - e)

(V f, VR) + 2Q.at- R

PROOF. Recall that in the time-dependent orthonormal moving frame{ei}, we have

at Rm = O Rm + Rm 2 + Rm # .

Hence

a

atRrn 2 = 2 (A Rm. + Rm 2 + Rm #, Rm)

=AIRmI2-2IVRmI2+2(Rm2+Rm#,Rm)

= AIRm12 - 2IVRm12+2 (A3+µ3+v3+3aµv) .


Since the scalar curvature is R = tr (Rm) = tr (Rc) = A + µ + v, it followsfrom (6.33) that

(6.40)at R = OR + 2 I Rc12 = OR + A2 + µ2 + v2 + aµ + Av + ftv,

hence that

8(R2) = 0 (R2) - 2 1 vRI2

3t+2(A+p+v) (A2+µ2+v2+Ay +AV +/.tv).

Now since RmI2 = IRm2 - R2/3, we compute that0

0

2, 0 = R, a = 1, and $ = 2 - E in Lemma 6.33, we get

+2(A3+/t3+v3+3Apv)

- 3 (A+µ+v) (A2+/2+v2+Ai +Av+µv)

=zIRm12-21VRm12

+ 3 (a3 + µ3 + v3 + 3.µv)

- 4 (A2I.L +A2v+p2A+/.t2v+v2A+v2µ)

0

Taking co = IRm0

219

atf Of-R2EIVRm1

4A3+µ3+v3+3Auv

RmI2 - 30 (R2) - 2 IV RmI2 + 3 IVR12

-(a µ+a v+µ A+/1 V A+v µ)2

- (2-E) IR31 (A2+µ2+v2+Aµ+AV+µv)

}

-(2-E)(3-E)(RmI2IVR2+2 (2-E)VIRm12,OR

R4-E: R3-E

Simplifying and combining terms, we obtain

f =A f

+ 2(1-6) V IRmI2oR + 2E IRmI2 IReI2

at R R2-E R3-E

o \ a JVR- R?E R//(VRm l - VR ® Rm2 - 6(1-,-) IRmR4-c 12

R4rA2µ2 + AY + µ2v2 - (A2µv + /t2Av + v2A.G

2 2 2 2 2 2

The result follows when we note that IRc12 < IRm12, discard the negativeterms on the second line, and compute

2 [A2µ2 + A2v2 + 127,,2 - (A 2µv + )U2Av + v2AIL) ]

6. The gradient estimate for the scalar curvatureIn this section we obtain a gradient estimate for the scalar curvature.

This estimate is important because it enables us to compare curvatures atdifferent points, whereas the pinching estimate of Section 5 is a pointwiseestimate which compares curvatures at the same point.

As motivation, let us recall Schur's Lemma. If g is an Einstein metricon a manifold M', then Rc = f g for some function f on Mn and one has

VkR = Vk (gz'Rig) = nVkf.

On the other hand, the contracted second Bianchi identity implies that

VkR = 2V'R3k = 2V3 (fgjk) = 2Vkf

So we have (n - 2) V f 0, which shows that f = R/n is constant indimensions n > 2.

Now suppose we have a solution (M3, g (t)) of the Ricci flow on a closed3-manifold. The pinching estimate (6.37) proved in Section 5 can be writtenin the form

IRc-3RgI26(6.41) R2 <CR- ,

where the left-hand side is a scale-invariant quantity which measures howfar the metric is from being Einstein, and the right-hand side is small whenthe scalar curvature is large. So if the scalar curvature becomes uniformlylarge, it is natural to expect its gradient to approach zero uniformly. It isalso natural to expect that the contracted second Bianchi identity will bekey to proving this result. Both expectations are in fact correct, and weshall now establish the following result.

THEOREM 6.35. Let (M3, g (t)) be a solution of the Ricci flow on aclosed 3-manifold with g (0) = go. If Rc (go) > 0, then there exist /3, d > 0depending only on go such that for any ,Q E [0, // , there exists C dependingonly on /3 and go such that

VR2R3I < /3R-b12 + C R-3.

Here, the left-hand side is a scale invariant quantity, while the right-handside is small when the scalar curvature is large.

6. THE GRADIENT ESTIMATE FOR THE SCALAR CURVATURE 195

To prove this estimate, we need to compute several evolution equations,the first of which is for the square of the norm of the gradient of the scalarcurvature.

LEMMA 6.36. If (.MT, g (t)) is a solution of the Ricci flow, then

(6.42)at

VRI2 = 0 IVRI2 - 2 IVVR12 +4 (VR, V IRcl2) .

PROOF. By (6.6), one has

at at(gz'VZRVjR)

= 2Rc(VR,VR) +2CVR,V (OR+2 IRcI2)>,

whence the lemma follows from the Bochner-Weitzenbock formula

0 IVR12 = 2IVVR12 + 2 (VR,OVR) +2Rc(VR,VR) .

Next we calculate the evolution of IVRI2 divided by R. This choice ofpower is not strongly intuitive, but yields a useful equation.

LEMMA 6.37. Let (M',g (t)) be a solution of the Ricci flow such thatR > 0 initially. Then for as long as the solution exists,

.43a)(VR2)

=A 1::12) -(6 2R V ()2

(6.43b) - 2 I

IRc12+ R (VR, V IRc12> .

PROOF. By Corollary 6.8, the inequality R > 0 is preserved. Using (6.6)and (6.42), one computes that

a IVRI2 _ 1 a 2IVR12 a

at R R at VR - R2 atR

= R (A VRI2 - 2IVVR12 +4 (VR, V IRc12))

- I VR12 (R+2Rc).2R2Since for any smooth functions u and v,

(u) vDu 2(Vu, Vv) + 2v3 IVv12

V V2

we have

a IVRI2 0)(VR)2 1VR14 (VVRI2,VR) +

at R R R3 R2 R

- 2

2

IVR2IRc12+R

(VR,V IRcI2).


The lemma follows by completing the square.

The evolution equation forVRI2

has only one bad (potentially positive)

term on the right-hand side, namelyR

(OR, V IRcl2). In dimension n = 3,

we can remedy this by adding the quantity I Rc 12 - 3 R2 to 'VR2 , therebyintroducing a good (negative) term that cancels out the bad term. Beforeproving this, we compute two more evolution equations valid for a generaln-dimensional solution.

LEMMA 6.38. If (MI, g (t)) is a solution of the Ricci flow, then

(6.44) at R2 = A (R2) - 2 IVRI2 + 4R jRc12

and

(6.45)at

IRc 2 = D IRcl2 - 2 IV RcI2 + 4RZjkjki Rjk.

PROOF. It follows from (6.6) that the square of the scalar curvatureevolves by

atR2=2R(AR+2IRc12)

= A (R2) -2IVR12+4RIRc12.

On the other hand, (6.7) implies that the square of the norm of the Riccitensor evolves by

a JRc12 =at

(giigkjR)= 4tr9 (Rc3) +2 (RC, AL Rc)

= A Rc12 - 2 IV Rc12 + 4RZjk$RifRjk.

COROLLARY 6.39. If (M3, g (t)) is a solution of the Ricci flow on a3-manifold, then

IRc12 - 3R2 J = A I IRc12 - 1R2 I - 2 l IV Rc12 - 3 IVRI2at 3

- 8tr9(Rc3) + 3R IRc12 - 2R3.

PROOF. Formula (6.9) implies that (6.45) can be written in dimensionn=3 as

8tIRc2 =AIRc12-210Rc12-2R3-8trg(Rc3)+1ORIRc2.

Alternatively, one may obtain this formula by using (6.10).

The useful term in the evolution equation for IRcI2 - s R2 is the quantity-2(VRc12 - s IVRI2). Our next task is to show that this term dominates

the bad term R(VR, V IRcI2) in the evolution equation (6.43) forIVI2.

Itis easy to see that IV RcI2 - 3 IVRI2 > 0, but we need a slightly betterestimate.

LEMMA 6.40. In dimension n = 3,

I V Rc12 - 3IDRI2

> 37 ID RcI2.

PROOF. Define a (3,0)-tensor X by Xijk - ViRjk - s ViRgjk and a(1, 0)-tensor Y by Yk - gi3Xijk. The contracted second Bianchi identityshows that

IYI2 = (vR.k3 VkR)

(viR3VkR) 36

IDR12.

But the standard estimateIZI2 > 1 (tr9Z)2

nfor any (2, 0)-tensor Z (not necessarily symmetric) implies in dimensionn = 3 that

3 IYIZ < IXI2 = IDRcI2 - 3 IDRI2.Hence we get

(6.46) IDRcI2 > 1 (1 +36

IDRI2 08 IDRI2,

from which the result follows easily. /REMARK 6.41. By decomposing the (3, 0)-tensor V Re into irreducible

components, Hamilton proved a stronger inequality

IV Rc12-3IDRI2>21]VRcI2

in Lemma 11.6 of [58]; but the estimate above suffices for the proof ofTheorem 6.35. Hamilton observes that VRc may be written as the sum oftwo irreducible components

DiRjk = Aijk + Bijk,where

Aijk = 20 [(VkR) gij + (V R) gik] + 10(ViR) gjk.

The constants 1/20 and 3/10 are determined by the condition that alltraces of Bijk are zero. The contracted second Bianchi identity implies thatgjkAijk = ViR and gij Aijk = gijAikj = 2VkR. It follows that (A, B) = 0.Calculating that

8 27

0+00+200)IVRI2=20

IDRI2,JAI'=(40

one concludes that

IVRc12 = IAI2 + IBI2 > 7 IVRI2,20

whence Hamilton's estimate follows.

Combining Corollary 6.39 with Lemma 6.40 gives the following differen-tial inequality.

COROLLARY 6.42. If (M3, g (t)) is a solution of the Ricci flow on a3-manifold, then

(RcI2\\(Ic821 < - 3R2 I 37 IV RcI2

- 8 tr 9 (Re 3) + 3 R IRcI2 -2R3.

Now we return to equation (6.43) for the evolution of IVRI2 /R. Onany manifold of positive Ricci curvature, one can estimate IRcI < R. Thenestimating

V IRcI2 < 2IVRcI IRcl

and recalling (6.46), we apply the Cauchy-Schwarz inequality to the badterm R (VR, V IRcI2) on the right-hand side of (6.43), obtaining

R\VR, V IRcI2

<IVRI V IRcI2

< 8IVRI IVRcIIRcI

(6.47) < 8' IV Rc12 .

This motivates us to consider the quantity

(6.48) V = ERR 2+ 2(8 + 1) (lRcI2_R2)

which gives an upper bound for IVRI2 /R. Combining Lemma 6.42 andCorollary 6.42 with estimate (6.47) shows that V satisfies the following dif-ferential inequality.

LEMMA 6.43. If (M3, g (t)) is a solution of the Ricci flow on a 3-manifold whose Ricci curvature is positive initially, then

atV <AV-2R V I RR 12-2IR2I2IVR2-I RcI2

.+32 (8,`+1) I 3RIRcI2-8tr9(Rc3)-2R3)

The only potentially positive term in the evolution equation for V is

W = 36R IRc12 - 8tr9 (Re3) - 2R3.

One expects this quantity to be small when the metric is close to Einstein,because the formula derived in Corollary 6.39 shows that W vanishes iden-tically on an Einstein 3-manifold. We now make this expectation precise.

LEMMA 6.44. On a 3-manifold of positive Ricci curvature, one has

W<50R(IRcl2-IR2).

PROOF. If we define

X-8`Rc-3Rg,Rc2),

then X = 8 R IRc12 - 8 tr 9 (Re 3), and we may write W as

W=X+6RIIRcj2-3R2).

Define a (2, 0)-tensor Y by Y + Re2 \-9R2g, observing that

Yj _ Ri Rkj - 9 R29Z; =9k (ik - R9k) (Rjt +

Then using Cauchy-Schwarz, we canestimate X in dimension n = 3 by

X=-8(Rc-1R9,Y>

< 8 Rc +3Rg Re -3Rg2

<33R Re-3I R92

=33R(RcI2 - R2).

We used the positivity of Re to get the last inequality. The lemma follows.

We are ready to prove the main result of this section.

PROOF OF THEOREM 6.35. If (M3, g (t)) is a solution of the Ricci flowon a closed 3-manifold whose Ricci curvature is positive initially, we maycombine the results of Lemmas 6.43 and 6.44 to obtain

aV < AV - JV Rc12 +7400v'3- + 925R (RcI2 - 1R2)

Applying the result of Theorem 6.30 in the form (6.41) lets us estimate thisfurther as

a V < AV_ I V RcI2 + CR3-try,

where C and 6/2 depend only on go. On the other hand, it follows fromformula (6.6) that

atR2--r = A (R2-,) - (2 - Y) (1 - 7)

R-7 IVRI2 + 2 (2 -'y) RI-ry Rc12.

Choose /3 depending only on go such that

0</3< 3(2--y)(1--y)'

and recall thatIVR12 < 3 IV Rcl2 .

(This is a consequence of (6.46), but is easy to prove directly.) Then for any/3 E I0, one has an estimate

at (V - OR2-7) < A (V - /3R2-ry)

+ [,3 (2 --y) (1-y)R_yIVR12 - IVRc12]

+ CR3-27 - 2/3 (2 - -y) R1-ry IRc12

<A(V-OR 2-ry)+CI,

where C1 depends only on /3 and go. By the maximum principle, we concludethat

V -,3R 2-,' < Clt.

But formula (6.6) implies that

at R>AR+2R2;

as we shall see in Lemma 6.53, this forces the solution to become singularwithin a finite time T depending only on go. Hence we conclude that

I VRI2< V <,3R 2-7

+ C1T.R

7. Higher derivative estimates and long-time existence

If go is a smooth metric on a compact manifold M7 , Theorem 3.13implies that a unique solution g (t) of the Ricci flow satisfying g (0) = goexists for a short time. It follows that there is a maximal time interval0 < t < T < oc on which the solution exists. If T < oo, it will be importantfor us to understand what goes wrong. The goal of this section is to provethe following result.

(Rmin (0))"

7. HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 201

THEOREM 6.45. If go is a smooth metric on a compact manifold M",the unnormalized Ricci flow with 9 (0) = go has a unique solution g (t) on amaximal time interval 0 < t < T < oo. Moreover, if T < oo, then

lim I sup I Rm (x, t) = 00.t/'T \zEMn

We shall in fact prove the contrapositive of the theorem. Namely, wewill show that if the maximum curvature were to remain bounded alonga sequence of times approaching T, the solution could be extended pastT. In order to make this argument rigorous, we must first discuss a prioriestimates for any solution whose curvatures remain suitably bounded.

7.1. Higher derivative estimates. In Section 5 of Chapter 3, we sawhow the Ricci flow may be regarded heuristically as a nonlinear heat equationfor a Riemannian metric. This viewpoint admits a rigorous interpretationin the sense that the curvatures of an evolving metric all satisfy parabolicequations. In particular, bounds on the curvature of an initial metric goautomatically induce a priori bounds on all derivatives of the curvature fora short time. We call these Bernstein-Bando-Shi (BBS) derivativeestimates, because they follow the strategy introduced by Bernstein [16,17, 18] for proving gradient bounds via the maximum principle, and werederived for the Ricci flow in [9] and [117, 118]. (See also [11] and Section7 of [63].)

Because the BBS estimates for the Ricci flow are not surprising in thatthey follow the natural parabolic scaling in which time scales like distancesquared, we shall postpone our technical discussion of them until Chapter 7.(Certain local analogues of the BBS estimates will also be discussed in theplanned successor to this volume.) It will be enough for our present needsto assume the following result from Chapter 7.

THEOREM 6.46 (Theorem 7.1). Let (.M', g (t)) be a solution of the Ricciflow for which the maximum principle holds. (This is true for instance ifM is compact.) Then for each a > 0 and every m E N, there exists aconstant C, depending only on m, and n, and max {a,1 }such that if

IRm (x, t) I9 < K for all x E Mn and t E [0, K

then

JVt Rm (x, t) lg < K for all x E Mn and t E (0, aC7n

As stated above, the BBS derivative estimates deteriorate as t 0. Thisis due to the fact that bounds on the Riemann tensor of an initial metricimply nothing about its derivatives. Furthermore, the BBS estimates aboveallow the factors C; to grow as the time interval over which we estimatebecomes larger. Because our ultimate goal is to establish long-time existenceof the flow, we want a practical criterion that will let us know when a solutionwhich exists up to an arbitrarily large time T can be extended past T. We

can accomplish this by the following strategy: when we have an upper boundon IRml which is uniform in time, we can apply Theorem 6.46 at large timesby taking a fixed small step backwards in time. This technique lets us provethe following consequence of the theorem.

COROLLARY 6.47. Let (.Mn, g (t)) be a solution of the Ricci flow forwhich the weak maximum principle holds. If there are 3 > 0 and K > 0such that

IRm (x, t)19 < K for all x c M and t E [0, T],

where T > f3/K, then there exists for each m E N a constant C,,,. dependingonly on m, n, and min {,3,1 } such that

IVm Rm (x, t) I9 < CmK' 2 for all x E Mn and t E [m1n{1}T]

PROOF. Let ,81 _ min {j3, 11. Fix to E [j31/K, T] and set To--* to -061 /K.Let t = t - To, and let g (t) solve the Cauchy problem

aatg = -2Rc

(0) =g(To)-

Then by uniqueness of solutions to the Ricci flow, g (t) = g (t + To) = g (t)fort E [0,131/K]. Thus by hypothesis on the solution g (t), we have

IRm(x,t)I9 <K

for all x E Mn and T E [0, 01 /K] . Applying Theorem 6.46 with a = i3 , weget a family of constants C,, depending only on m and n such that

J V-R.m (x, 011 < t KC

for all x E Mn and t E (0, X31 1K]. Now when t E [2K , x ] , we have

t ''/2 > )3-/22-m/2K-"'./2.

Taking t = 01 /K, we find in particular that

I

f9 < KI+'n./2 for all x E Mn.VmRm(x,to)(_n2/2)

"'/20m)31

Since to E [Q1 /K, T] was arbitrary, the result follows.

The main objective of this subsection is to obtain the following gradientestimates. They are stated with respect to a half-open time interval [0, T)because one of their applications (moreover, the one of current interest) isto help us study obstacles to long-time existence of the flow.

PROPOSITION 6.48. Let (M', g (t)) be a solution of the Ricci flow on acompact manifold with a fixed background metric g and connection V. Ifthere exists K > 0 such that

Rm (x, t) l9 < K for all x E Mn and t E [0, T),

then there exists for every m E N a constant C,,,, depending on m, n, K, T,go, and the pair (g, V) such that

7mg (x, t) I9 < C,,,, for all x E Mn and t E [0, T).

The first step in proving the proposition (and ultimately in establishinglong-time existence of the normalized flow) is to obtain a sufficient conditionfor the metrics composing a smooth one-parameter family to be uniformlyequivalent. Recall that one writes A 0for all vectors V.

LEMMA 6.49. Let Mn be a closed manifold. For 0 < t < T < oo, letg (t) be a one-parameter family of metrics on M' depending smoothly onboth space and time. If there exists a constant C < oo such that

f0T

for all x E M', then

19

9(x,t)g(t)

dt < C

e-cg (x, 0) < g (x, t) < ecg (x, 0)

for all x E M' and t E [0, T). Furthermore, as t / T, the metrics g (t)converge uniformly to a continuous metric g (T) such that for all x E Mn'

e-c9 (x, 0) < 9 (x, T) S eC9 (x, 0)

PROOF. Let x E Mn, to E [0, T), and V E TXM' be arbitrary. Thenusing the fact that I A (U, U) A 9 for any 2-tensor A and unit vector U,we obtain

log g(x,to) (V,V)

1C 9(x,o) (V, V) fto a

at[logg(x,t) (V, V)] dt

Lto

N9(x't) (V, V) dt9(x,t) (V, V)

oto

atg(x't) \ JV[ ' TV1/dtJO

ftoL3t g ( x't)

9(t)

dt < C.

The uniform bounds on g (t) follow from exponentiation.

Since Mn is compact, f T as (x, T) dT , 0 as t / T uniformly in9(t)

x E M'. Hence the function f : TM --j JR defined by

f (x, V) = lim T9(x,t) (V, V)

exists and is continuous in x c M and V E T1M'. By polarization, wecan define a (2, 0)-tensor g(x,T) on Mn by

9(.r.T) (V. W) = 4[.f (x, V + W) - f (x, V - W)1'

noting that 9(x,T) (V, W) = limt-T (V, W). The uniform bounds ong (t) ensure that

e-C9(x.o) (V, V) < f (x, V) < eC9(x,o) (V, V) ,

hence that 9(x,T) is a Riemannian metric.

In particular, the lemma implies that if (JVla, g (t)) is a solution of theRicci flow satisfying a uniform curvature bound on a finite time interval[0, T), then all metrics in the family {g (t) : 0 < t < T} are uniformly equiv-alent.

COROLLARY 6.50. Let (Mn, g (t)) be a solution of the Ricci flow. Ifthere exists a constant K such that IRcl < K on the time interval [0, T],then

e-21Tg(x, 0) g (x, t) < e2<T9 (x, 0)

for all x E .Mn and t E [0, T].

We are now ready to prove the proposition.

PROOF OF PROPOSITION 6.48. Since Mn is compact, it is covered bya finite atlas in which we have uniform estimates on the derivatives of thelocal charts. Henceforth, we fix such a chart : U -> lRn. Since g and theconnection V are fixed, it will suffice to prove that for each m E N, theordinary derivatives of g of order in satisfy an estimate

IBmg(x,t)I <Cm for all x E U and t E t E [0,T),

where the norm = I' I & is taken with respect to the Euclidean metric 5 in U,and Cm, depends on only on m, n, K, T, and go. Adopting this point of view,we shall in particular regard r as a tensor in U, namely as the difference ofthe Levi-Civita connection of g and the background (flat) connection in U.The proof will be by complete induction on m E N.

In the estimates which follow, C will denote a generic constant that maychange from one inequality to the next, but which depends only on m, n,K, T, and go. Choose 0 = 3 (K, T) so that 0 < Q < min {KT,1}. ByCorollary 6.50, we have uniform pointwise bounds from above and below for

the metrics g (t) on the time interval [0, T). To estimate the first derivativesof the metric, we begin by writing

a s 1 a s

gjk/

=-2 a Rat axigjk) = axi Cat ax' k

= -2 CviRjk + r Rek + rekRje)

Because jRm K by hypothesis, this implies that

(6.49)at ag

= 210 Re l < 2 IV RcI + CK Irl.

By equation (6.1), the tensor atr is given by

atr _whence we get the estimate

aat r

<CIVRcI .

Now by Corollary 6.47, there exists B = B (m, n, K, 0) such that the boundIV Rcj < B holds on the time interval (<3/K, T). Since [r[ is bounded on[0, 0/K] by some A = A (K, 3, go), we see that

(6.50) IF (x, t) I < A + BC (T - Q/K) < C

for all x E M' and t E [0, T). Since IV Rcl is bounded on [0, /3/K] by someD = D (K, Q, go), we conclude by (6.49) that

agI < jagol + CD

K

+ (2B + C) (T - /3/K) < C.

To do the general (inductive) step, let a = (al, ... , a,.) be any multi-index with I a I = m. Then since

a ( al-IY 5xa9ij = -2 a- Rij ,

it will suffice to bound 101* Rd. A moment's reflection reveals that we canestimate

m m-1(6.51) 10tm Rd <

E C Iri lDrn-i Rc

+ E ci lair) lam-1-i Rcli=0 i=1

where the constants ci, c2 depend only on m and n. For instance, the casem = 2 corresponds to the explicit formula

aiajRke = VivjRkP

+ (rvRk1 + rp v j Rpe + rp V j Rkp + rPkviRpt + rppviRkp)

+ (FpFlRkq + r prp R# + r Qr1kRgp + r kr1QRpq)

+ (airpkRP2 + airPQRkp ) .

Applying Corollary 6.47 and estimate (6.50), we see that

m

EIi=0

By the inductive hypothesis, we may assume that jatapg1, hence IOP Rcl, hasbeen estimated for all 0 < p < m - 1. By (6.51), this implies in particularthat I OF I has been bounded for all 1 < i < m - 2. So to finish the proof, itwill suffice to estimate Iam-lrl. We shall accomplish this by taking a timederivative and integrating. By equation (6.1), we have

m-1a a k

at axpl ... axPm_ rij

am-1 (a

atki

axpl ... axp--i at\

So by the inductive hypothesis and Corollary 6.50,

M-1<C > Iam-1-i (g-1) I IaivRcl

i=0

M-1< C E Iam-l-igl IaiVRcl

rn mvm-i

Rc I < C supo<t<p/x i-o

vm-i Rc + C L Cm,-iK1+"2 t < C.i=0I

a. .

M-1

axpm- 1[9k

axPl(-viRje - V R%P + VQRij)

i=0M-1

(6.52) <CE aiv Rcl .

i=0

Here we used the fact that bounds on g and its derivatives induce boundson the derivatives of g-1 via the identity

a gjk = -gipgkga

.axi axi gpq

Reasoning as in the derivation of (6.51) with the tensor Rc replaced by V Rc,we observe that

laivR

j=0

IVi+1-i-1

Rcl + E c'j I ajrl lai-j Rcl,j=1

where the constants cj, c depend only on p and n. Applying this estimateto (6.52) then shows that

amrat <C> r'Ioi+1-iRcl+lajrllai-jRc1

,

i=0 j=o j=1

where all of the terms on the right-hand side have already been bounded.Thus we have I gam-,rl < C and hence lam-lrl < C+CT. This completesthe proof.

An examination of the proof makes it evident that we have also estab-lished the following result.

COROLLARY 6.51. Let (Mrs, g (t)) be a solution of the Ricci flow on acompact manifold with a fixed background metric g and connection V. Ifthere exists K > 0 such that

IRm (x, t)19 < K for all x E .A4' and t E [0,T),

then there exists for every m E N a constant Cm depending on m, n, K, T,go, and the pair (g, V) such that

l Or'Rc (x, t)l9 < Cm' for all x E 'Mn and t E [0,T).

REMARx 6.52. As an alternative to the argument used to prove Propo-sition 6.48, one may begin by writing the identity

a0 = Digjk axi9jk - r j92k , ric9jP

in the forma

ax29jk = r jgek + rik9jt-

Then using the uniform bounds on g (t) given by Corollary 6.50, one getsthe estimate

aaxi 9j k < C IrI

and proceeds as before.

7.2. Long-time existence. Now we are ready to show that the cur-vature becoming unbounded is the only obstacle to long-time existence ofthe flow.

PROOF OF THEOREM 6.45. Define M (t) supXEMf IRm (x, t) I. Weshall first prove the claim that

lim sup (M (t)) = 00.t/'T

By Theorem 3.13, a unique solution 9 (t) of the Ricci flow satisfying g (0) _go exists for a short time. Suppose that the solution exists on a maximalfinite time interval [0, T). To obtain a contradiction, suppose further thatthere is a positive constant K such that

sup M (t) < K.o<t<T

Fix a local coordinate patch U around an arbitrary point x E .M', and letT E (0, T) be arbitrary as well. Then by Lemma 6.49, a continuous limitmetric g (T) exists and is given in component form by the integral formula

T9ij (x, T) = 9ij (x, T) - 2 j Rij (x, t) dt.

T

Let a = (a1,... , a,.) be any multi-index with Jal = m E N. By PropositionRij are uniformly bounded on6.48 and Corollary 6.51, both ax gij and TiF

U x [0, T). Thus we can write

l

am am am R_,,)

which shows that a'gI < C for some positive constant C, hence that g (T)is a C°° metric, and also

am am

(x'T) - (-9i) (x, T) < C (T

which shows that g (T) --+ g (T) uniformly in any Crn norm as t / T.Now since g (T) is smooth, Theorem 3.13 implies there is a solution of

the Ricci flow g (t) that satisfies g (0) = g (T) and exists for a short time0 < t < E. Since g (T) -* g (T) smoothly, if follows that

9(t)-{ g(t-T) T<t<T+Eis a solution of the Ricci flow that is smooth in space and time (in particularnear T) and satisfies g (0) = go. This contradicts the assumption that T ismaximal, and proves the claim.

Finally, we prove the theorem by replacing the lim sup of the claim witha bona fide limit. Suppose the theorem is false. Then there exists Ko < 00

8. FINITE-TIME BLOWUP 209

and a sequence of times t2 / T such that M (ti) < Ko. A consequence ofthe differential inequality

atIRm2 <AIRm2-21VRmI2+C,,,IRml3

we shall derive in Lemma 7.4 is the doubling-time estimate of Corollary 7.5,which implies that there exists c > 0 depending only on the dimension nsuch that

M (t) < 2M (ti) < 2Ko

for all times t satisfying

t2<t<min {T,ti+K0

Since t2 / T as i -+ oc, there exists io large enough that do + c/Ko > T.But then this implies that

sup M (t) < 2Ko,t=o <t<T

which contradicts the claim established above and hence proves the theorem.0

8. Finite-time blowup

In Section 9 below, we shall shift our attention to the normalized Ricciflow in order to complete our proof of Theorem 6.3. Along the way, we willneed to prove that the unique solution of the normalized Ricci flow startingon a closed 3-manifold (M3, go) of positive Ricci curvature exists for all pos-itive time. Paradoxically, the key to proving this fact is obtaining a betterunderstanding of exactly how the corresponding unnormalized solution be-comes singular. We begin with the observation that a finite-time singularityis inevitable.

LEMMA 6.53. Let (M', g (t) : 0 < t < T) be a solution of the unnormal-ized Ricci flow for which the weak maximum principle holds. If there areto > 0 and p > 0 such that

inf R (x, to) = p,ZEMT

then g (t) becomes singular in finite time.

PROOF. By equation (6.6), we have

atR=AR+2jRcI2>AR+nR2.

Consider the solution

r (t) _ 1

I - 2 (t _to)

to the initial value problem

dr/dt

=2r2/n

1 r(to) p

By the maximum principle, Rinf (t) _ infxEM. R (x, t) satisfies the inequal-ity Rinf (t) > r (t) for as long as both solutions exist.

Once we know a solution becomes singular in finite time, Theorem 6.45tells us that its Riemannian curvatures must blow up as we approach thesingularity time.

COROLLARY 6.54. Any solution (M3, g (t)) of the unnormalized Ricciflow on a compact manifold whose Ricci curvature is positive initially existson a maximal time interval 0 < t < T < oc and has the property that

t T C sup oo.xEM3

By combining this fact with results of Sections 5 and 6, we can obtainglobal pinching estimates for the curvature.

LEMMA 6.55. Let (.A43,9 (t)) be a solution of the unnormalized Ricciflow on a compact manifold whose Ricci curvature is positive initially. Thenthe solution becomes singular at some time T < oo. Moreover, it obeys thefollowing a priori estimates.

(1) There exist positive constants C and -y depending only on the initialdata such that

Rmin

Rmaxfor all times 0 < t < T. In particular, Rmin/Rma.x -3 1 as t / T.

(2) For x E M3 and t E [0, T), let A (x, t) > ,u (x, t) > v (x, t) denotethe eigenvalues of the curvature operator at (x, t). Then for anyE E (0, 1), there exists T6 E [0, T) such that for all times TE < t < T,one has

min v (x, t) > (1 - E) ax (y, t) > 0.

In particular, the solution eventually attains positive sectional cur-vature everywhere.

PROOF. By Lemma 6.53, we know the solution becomes singular at sometime T < oo. Since c IRcl < IRm1 < C IRcl in dimension n = 3, it followsfrom Theorem 6.45 that

-

(6.53) lim sup IRc (x, t) =Co.t/T xEM3 /

To prove claim (1), define

Rmin (t) _ inf R (x, t) and Rm. (t) _ sup R (x, t) .xEM3 xEM3

> 1 - CIt_

8. FINITE-TIME BLOWUP 211

By Theorem 6.35, there are positive constants A, B, and a such that

IVRI2< 2A2RmaXa + B2

Since IRc12 < R2 and R is positive, it follows from (6.53) that there existsr E [0, T) such that

IVRI < AR3/a2 a

at all times t E (T, T). Consider any time t E (r, T). Since M3 is compact,there exist x (t) E M3 such that Rmax (t) = R (x, t). Givens > 0, considerthe geodesic ball B (x, L), where

1

L(t)

E Rmax (t)

If ry is any minimizing geodesic from x to x E B (x, L), we can estimate that

Rmax - R (x) < IVRI7

Hence on B (x, L), we have the lower bound

(6.54) R > Rmax (l - AR-a \\\ E

< AR__a.E

It follows that there exists t E (-F, T) depending only on A, a, and E suchthat

(6.55) R > (1 - E) Rmax

on B (x, L) for all t E [t, T). Now by Corollary 6.29, there exists > 0depending only on go such that the inequality

Rc > 2/32Rg

holds at all points of M3. By Myers' Theorem, this implies that the minimiz-ing geodesic ry emanating from x must encounter a conjugate point withinthe distance

7r

0 in B(.t,L)R.

Now when E > 0 is sufficiently small, estimate (6.55) implies that

7r 7r <

Q infB(x L) R - Q (1 - E)Rm.

E Rm.hence that B (x, L) is all of M3.

To prove claim (2), recall that Theorem 6.30 implies there exist positiveconstants C and d < 1 depending only on go such that

v> A-C(µ+v)1-8>A-C(A+ii+v)1-a

at all points on the manifold. Thus we have the pointwise inequality

(6.56) > 1 - 3CR-a > 1 - 3CR-3min'

Let x, y E M3 and 77 > 0 be given. Then by (6.53), (6.55), and (6.56), thereexists T,, E [0, T) such that for all times Tv < t < T one has

v (x, t) > (1 - 71) A (x, t)

> t)1 77 R(3 x,

> (1 1) R( t)3

y,

1- 2l)(> ) A ( t)]t) + 2 (1 - 7[A (37 y,y,

> (1-71)3A(y,t)

The claim follows by taking the infimum over all x E M3 and the supremumover all y E .M3.

This result lets us conclude in particular that g (t) is approaching anEinstein metric uniformly as t / T.

COROLLARY 6.56. If (M3, g (t)) is a solution of the unnormalized Ricciflow on a compact manifold whose Ricci curvature is positive initially, then

02

tim sup I R2I = 0.

-EM3

PROOF. Applying the local pinching estimate of Theorem 6.30 in theform (6.37) shows that there are positive constants C and b such that

0

IRc < CR-a < R-SR2 - - min'

Since limt/T Rmax (t) by Theorem 6.54 and limt/T (Rmin/Rmax) = 1 byLemma 6.55, the result follows.

9. Properties of the normalized Ricci flow

We now have at our disposal all the facts about the unnormalized Ricciflow that we will need to prove Theorem 6.3. But since the theorem is statedfor the normalized flow, we must convert to that flow in order to completeits proof. In this section, we shall study how this is done and will then showthat the normalized flow exists for all time and asymptotically approachesan Einstein metric. In the final part of this chapter (Section 10, below) wewill prove that the convergence is in fact exponential.

9.1. Equivalence of the flows under resealing. We begin by show-ing that the unnormalized and normalized Ricci flows differ only by a rescal-ing of space and time. Specifically, suppose that (M', g (t)) is a solution ofthe unnormalized Ricci flow

tg = -2Rc

9. PROPERTIES OF THE NORMALIZED RICCI FLOW 213

on a manifold of finite volume. To convert to the normalized flow, definedilating factors ' (t) > 0 so that the metrics g (t) _' (t).g (t) have constantvolume

JMdi

1,

and put t = fo (rr) dr. Then dt/dt = (t), while the geometries of g andare related by the following lemma.

LEMMA 6.57. Let (Mn, g) be a Riemannian manifold. If g = ig forsome 0 > 0, then the following relations result.

(1) The Levi-Civita connections of g and g are related by(2) The (3,1)-Riemann curvature tensors of g and g are related by

e eRisk = Rjjk.(3) The (4, 0)-Riemann curvature tensors of g and g are related by

='bRzjk(.(4) The Ricci curvature tensors of g and g are related by R2j = RZ2.(5) The scalar curvatures of g and g are related by R = ,-1R.(6) The volume elements of g and g are related by dµ = ,m/2 du.

Now writing equation 6.5 in the form

aat

log det g = g2jat (gig) -2R,

we see that

R du,dt fMdu - f nM

hence that 0 is a smooth function of time. Thus we get the evolutionequation

(6.57)atg dt at (0g) = -2Rc +

(1d\

Denote the average scalar curvature of g (t) by

fMnRdµp (t) -

fMn dtc= fMn R dµ.

Then observing as above that

a log det (0g) = 1 gzi a (i,Og?j) = -2R +dd?p

at at 0 dtwe obtain

0 = dt J dµ = f t log det (fig) d1

= f (-Of?+2 at l dµ

n do_-P+20 dt.

This shows that 215 _ dt , hence by (6.57) that g evolves by the normalizedRicci flow

atg = -2Rc + 2n

9.2. Long-time existence of the normalized flow. Let (.M3, g (t))be the unique solution of the unnormalized Ricci flow starting on a 3-manifold of positive Ricci curvature. By Theorem 6.54, g (t) becomes sin-gular in the sense that it exists on a maximal time interval 0 < t < T < 00and satisfies limt/T Rmax (t) = oo, where Rmax (t) - SUPxEM3 R (x, t). Let(M3, g (t)) be the corresponding normalized solution constructed in Sub-section 9.1. Then g (t) exists on a maximal time interval 0 < f < T < oo.We shall now prove that gg enjoys long-time existence, namely that T = oo.

THEOREM 6.58. If (M3, g (t)) is a solution of the unnormalized Ricciflow starting on a compact manifold of positive Ricci curvature, then thecorresponding normalized solution (M3,. (0) exists for all positive time.

We begin with some preliminary results. Define

Rmin (t - inf R (x, t)xEM3

and

Rmax ( --.,sup R (x, t) .xEM3

LEMMA 6.59. If (M3, g (t)) is a solution of the Ricci flow with initiallypositive Ricci curvature, then

foRma (t) dt = oo.

T

PROOF. Since Rm,, is a continuous function of t E [0, T), there is aunique solution p to the initial value problem

5 dp/dt = 2Rma p1 P(0) = Rmax (0)

Since g (t) has positive Ricci curvature for as long as it exists, we haveRc 1 2 < R2 < R Rmax, which implies that

( R A

<0(R-p)+2Rmax(R-p).So by the maximum principle, we have R R,,,,,,, hence byTheorem 6.54 that liM,/T p (T) = oo. Since for 0 < T < T, we have

log p (T) = f d log p dt = 2 f Rmax (t) dt,P (0) o dt o

the integral on the right-hand side must diverge.

9. PROPERTIES OF THE NORMALIZED RICCI FLOW

LEMMA 6.60. There exists a positive constant C such that

Rmax < C

215

PROOF. Let L () and V (t-) denote the diameter and volume of g (t),respectively. Since Rc = Rc > 0, the Bishop-Guenther volume comparisontheorem implies that

(6.58) 1-V< 37rL3.

On the other hand, Corollary 6.29 shows there is a positive constantdepending only on go such that

2 2Rc = Rc > 2,3 Rming = 203 Rming

So by Myers' Theorem,

(6.59) L <

Since

7r

/3 v nmin

lim Rmin () _ lim Rmin (t)FIT Rmax (t) t/'T Rmax (t)

by Lemma 6.55, there exists a positive constant C such that

(6.60)Rmin > 1

Combining estimates (6.58), (6.59), and (6.60), we get the desired result:

,T

(0)2 (4)2/3

1 < CRmax < CRmin < C()2

Q

O

We are now ready to prove the main result of this section.

PROOF OF THEOREM 6.58. Let

and

(t) - fM3 R dµfM3 dµP

fM3 RdµP ( = = J dµ

fM3 dTt M3

denote the average scalar curvatures of the unnormalized and normalizedsolutions, respectively. Recall that dt = Odt. By Lemma 6.57, we have

R = RIO, and dµ = ,On/2dµ = 03/2dE.G. Thus for any T E [0, T), we have

/T /T RdµJ

p (t) dt = J dt0

0

fMad

T fM3 dµ 1-0

fM3 0_3/2 dµ Z/J dt

0

JTP(i) dt,

where T = fo 0 (t) dt. Now since

Rmin (t) 0 such that

J T p ( dt < f Rmax (t1 dt < C.

Hence T = oo.

Noting that the expression in Corollary 6.56 is invariant under dilationof the metric, we obtain the following convergence result for free.

COROLLARY 6.61. If (.M3, g (t)) is a solution of the unnormalized Ricciflow on a compact manifold whose Ricci curvature is positive initially, thenthe corresponding normalized solution g (t exists for all positive time andasymptotically approaches an Einstein metric uniformly in the sense that

O

2

tlsup I R2 = 0.

xENI3

In Section 10, we shall show that the convergence is exponential.

9.3. Evolution of curvature under the normalized flow. To aug-ment the scaling arguments employed above, it will be useful in the sequelto observe directly how various geometric quantities evolve under the nor-malized Ricci flow. To wit, suppose that we are given a solution of thenormalized Ricci flow

atg -2 Rc +rg

on a manifold M', where r is the function of time alone defined by

2P (t) = 2 fm R dp(6.61) r (t) n p fMn dµ

9. PROPERTIES OF THE NORMALIZED RICCI FLOW 217

(Note that we are now dropping the g () notation used above.) The easiestway to obtain the evolution equations for quantities depending on g is byfirst making a more general observation.

LEMMA 6.62. Suppose that g (t) is a smooth one-parameter family ofmetrics on a manifold Mn such that

a-9g 1P9,

where co is a function of time alone.(1) The Levi-Civita connection r of g is independent of time.(2) The (3, 1)-Riemann curvature tensor of g is independent of time.(3) The (4, 0)-Riemann curvature tensor Rm of g evolves by

aatRijk@ _ coRijke

(4) The Ricci tensor Rc of g is independent of time.(5) The scalar curvature R of g evolves by

at R = -cpR.

PROOF. All the formulas are immediate consequences of Lemma 6.5except for (3), which follows from (2) when we write

a _a(at (Rijke) = at (gtmRmk) (PRijkt

REMARK 6.63. The formulas above may also be deduced from Lemma6.57 by observing how the various geometric quantities under considerationrespond to scaling the metric. (Compare with Lemma 17.1 of [58].)

COROLLARY 6.64. Let (Mn, g (t)) be a solution of the normalized Ricciflow, and let r (t) be defined by (6.61).


atr = -ViR - V3R + V

(2) The (3,1)-Riemann curvature tensor evolves by

at RQ = AM' + rq (RjpRrqk 2RR+ 2Rp- Rj)g - pik jqr pxr qk

- RpRpjk - RpRepk - RkRf- + RPRJk.(3) The (4, 0) -Riemann curvature tensor Rm of g evolves by

aORijkt + 2 (Bijke - Bitjk +Bikjt)atRijke =

+ Rm Rmi - RkPjRmj + R rRijke,

where the tensor B is defined in (6.15).

(4) The Ricci tensor of g evolves by

Rjk = OLRjk = ORjk + 2RpjkgRP4 - 2MRtk.


atR = AR + 2 I Rc 12 - rR.

10. Exponential convergence

We may now assume we are given a solution (.M3, g (t)) of the normal-ized Ricci flow

19

at g -2 Rc +rg

starting on a closed 3-manifold of positive Ricci curvature, where r (t) isgiven by formula (6.61). By Theorem 6.58, we know that g (t) exists for allpositive time. By Lemma 6.8, Corollary 6.11, and Lemma 6.57, we knowthat the Ricci and scalar curvatures of g (t) remain strictly positive. ByLemma 6.60, we know that there exists a positive constant A such that

(6.62) 0<R<A,hence such that

(6.63) 0<r<AV,where V is the constant volume of (.A43, g (t)). And by Corollary 6.58, weknow g (t) becomes asymptotically Einstein in the sense that

O

2

lim sup I R2l = 0.

In order to show that this convergence is exponential, we shall followthe method of Section 4 (which uses techniques from the later paper [59]instead of the method originally employed in [58]). As in Section 2, letV be a vector bundle over M' isomorphic to T.Mn. Corresponding to theevolution of g (t) by the normalized Ricci flow, we construct a one-parameterfamily of bundle isometries t (t) : V -* TA4' evolving by

r

Then recalling Corollary 6.64, we compute that the analogue of formula(6.21) for the evolution of the components (Rabd) of t* Rm is

49

aRabcd = ADRabcd + 2 (Babcd - Babdc + Bacbd - Badbc) - rRabcd

It follows that the quadratic form M defined by the curvature operator oneach fiber of the bundle A2TM3 evolves by the ODE

dtM=M2+M#-rM.

10. EXPONENTIAL CONVERGENCE 219

In particular, the eigenvalues A > µ > v of M (which are twice the sectionalcurvatures) evolve by

dt A

A2 + µv - rA

d

d /.iA2 + Av - rµ

dtv=v2+A - rv.

It is easy to check that

log[A (µ + v)]d

= (µ + v) (A2 + µv) - rA (µ + v)dt + v

-A[µ2+v2+A(µ+v)] +rA(µ+v),hence that the proof of Lemma 6.28 goes through for the normalized flowexactly as written. Thus we can immediately state the following result.

LEMMA 6.65. Let (M3, g(t)) be a solution of the normalized Ricci flowon a closed 3-manifold of initially positive Ricci curvature. Then there existsa constant B such that

A<B(µ+v)for all positive time.

REMARK 6.66. One can also conclude directly that the estimate ofLemma 6.28 applies to the normalized Ricci flow by writing it in the scale-invariant form

+

We shall need one more simple observation.

LEMMA 6.67. Let (M3,g(t)) be a solution of the normalized Ricci flowon a closed 3-manifold of initially positive Ricci curvature. Then there existse > 0 depending only on go such that R > e for all positive time.

PROOF. By (6.6), R satisfies the differential inequality 8R/at > OR.Hence Rmin (t) > Rmin (0) > 0. 0

Now we are ready for the key estimate of this section.

PROPOSITION 6.68. Let (M3, g(t)) be a solution of the normalized Ricciflow on a closed 3-manifold of initially positive Ricci curvature. Then thereexist constants a E (0, 1), Q, and C depending only on go such that

A - v < C (µ + v)1-« e-pt

for all positive time.

PROOF. Following the proof of Theorem 6.30, we compute thatdtlog(A-v)=A-IL +v-r

and

The calculation

2 2

dtlog(I-i+v)=A+-f-v -r.+

d

dt(µ-v) (µ+v-A-r)(µ-v)

shows that µ - v stays positive. Thus we can compute that

ddtlogLept (µ

+u)I-"=a(A_r)+(µ-v)µµ+v2.

Recalling estimates (6.62) and (6.63), and applying Lemma 6.65, we estimatethis as

d A

dtlog

[eat (µ + v)I-a J< aB (µ + v) +

2 a+ v) .

If a > 0 is chosen smaller than 1/ [2 (1 + 2B)], then

dA - 1-a(l+2B)

(µ+v) µ+v(µ + v)'-'] 2 4

By Lemmas 6.65 and (6.67), there exists e > 0 such that

4e<A+µ+v<(1+B)(µ+v).Hence

d log eat A -v <Q- E

dt (µ+v)I-" 1+B*For sufficiently small 0 > 0 depending only on B and e, the right-hand sideis negative. So there exists a constant C = C (go) large enough that M (t)starts inside and hence remains in the convex set

>c={P:eat[A(?)-v(P)]-C[y (P)+v(P)]1-"<o}.

By Lemma 6.65 and estimate (6.62), the proposition implies that

A - v < CRl-"e-at < ACe--at

whence formula (6.34) and Lemma 6.67 imply that g (t) is exponentiallybecoming a metric of constant positive sectional curvature.

COROLLARY 6.69. If (M3, g(t)) is a solution of the normalized Ricciflow on a closed 3-manifold of initially positive Ricci curvature, then thereexist positive constants 0 and B such that

Rc -3

R g = Rcl < Be-at


for all positive time.

From here it is not hard to show that all derivatives of the curvaturedecay exponentially. This implies that g (t) converges exponentially fast inevery Cm norm to a smooth Einstein metric gam; for details, the reader isreferred to Section 17 of [58]. (Also see Section 5 of Chapter 5.)


The pinching estimate in Lemma 6.28 has an analogue for the meancurvature flow [74]. This is also true for Theorem 6.30; however, the proofof the analogous estimate for the mean curvature flow requires integral es-timates. In dimension n = 4, pinching estimates for the Ricci flow weobtained by Hamilton under the assumption of positive curvature opera-tor [59]. Higher dimensional estimates under stronger pinching assump-tions were obtained independently by Huisken [75], Margerin [95, 96], andNishikawa [102, 103].

CHAPTER 7

Derivative estimates

We saw in Chapter 3 that the Ricci flow is equivalent via DeTurck's trickto a quasilinear parabolic PDE, and may in fact be regarded heuristically asa nonlinear heat equation for a Riemannian metric. Furthermore, we sawin Chapter 6 that the intrinsically defined curvatures of a Riemannian met-ric evolving by the Ricci flow all obey parabolic equations with quadraticnonlinearities. Knowing this, anyone familiar with the smoothing proper-ties of parabolic equations would expect that appropriate bounds on thegeometry of a given Riemannian manifold (M', go) would induce a prioribounds on the geometry of the unique solution g (t) of the Ricci flow suchthat g (0) = go. Moreover, one would even expect the geometry to improve,at least for a short time.

In this chapter, we verify those expectations by proving global short-time derivative estimates for all derivatives of the curvature of a solutionto the Ricci flow. As we remarked in Chapter 6, we call these Bernstein-Bando-Shi estimates (briefly, BBS estimates) because they follow Bern-stein's technique of using the maximum principle to establish gradient esti-mates, and were applied to the Ricci flow in papers authored independentlyby Bando and Shi.

1. Global estimates and their consequences

Local derivative estimates are important for performing dimension re-duction and in other cases where one wants to take a 'semi-global limit' ofa sequence of geodesic balls whose radii go to infinity, or a `local limit' ofa sequence of geodesic balls of uniform size. Local derivative estimates willbe discussed in the successor to this volume. In the present chapter, we willprove those global derivative estimates that are important for establishinglong-time existence of the flow, as was discussed in Section 7 of Chapter 6.Our main goal for this chapter is to prove the following result.

THEOREM 7.1. Let (Mn,g (t)) be a solution of the Ricci flow for whichthe maximum principle holds. (This is true in particular if Mn is compact.)Then for each a > 0 and every m E N , there exists a constant Cm dependingonly on m, and n, and max {a, 1} such that if

Rm (x, t)I9(x,t) < K for all x E M' and t E [0, K],

223

224 7. DERIVATIVE ESTIMATES

then

117 ' Rm. (x, t)I g(x t) < C /K for all x E A4' and t E (0, K].

Note that the estimates in Theorem 7.1 follow the natural parabolicscaling in which time behaves like distance squared. Note too that we havestated the estimates in a form that deteriorates as t \ 0. This is the best onecan do without making further assumptions on the initial metric. Indeed,it is easy to construct examples of rotationally symmetric metrics on Snwith IRS < 1 and IVRI arbitrarily large by writing the metric as a warpedproduct. (Recall that we used warped product metrics on spheres in Section5 of Chapter 2.)

Theorem 7.1 has several important consequences, of which the followingtwo are particularly useful.

COROLLARY 7.2 (Long-time existence). If go is a smooth metric on acompact manifold M', the unique solution g (t) of the Ricci flow such thatg (0) = go exists on a maximal time interval 0 < t < T < oo. Moreover,T < no only if

sup I Rm (x, t) oo.lim LEM-t/TThis result equivalent to Theorem 6.45.

COROLLARY 7.3 (Uniform bounds for sequences). Let { (MZ , g9) : i E NJbe a sequence of compact Riemannian manifolds. If their curvatures areuniformly bounded in the sense that

Rm [g°] 1 go <_ K

for some constant K < no independent of i c- N, then there exists a constantc > 0 depending only on n such that for each i E N, the unique solutiongi (t) to the Ricci flow such that gti (0) = g° exists on Mi for the uniformtime interval t E [0, c/K]. Moreover, there is for each m E N a constantC, < no depending only on m and n (that is, independent of i E N) suchthat

sup IVm Rm[9i (t)]lgi(t) S C,,,KxEMra

a Wt-)In particular, for any time to E (0, c/K), there is for each m E N someC,',L = Cm (m, n, K, to) such that

sup I Vm Rm[9i] Igi < C,n

M' x [ta,c/K]

This fact is an immediate consequence of Theorem 7.1 and the minimalexistence time result (Corollary 7.7) we shall prove below. An importantapplication of Corollary 7.3 is to obtain convergence in each Cm norm whenone proves the compactness theorem for the Ricci flow, which we state inSection 3 of this chapter.

1. GLOBAL ESTIMATES 225

Before proving Theorem 7.1, we will calculate the evolution of the squareof the norm of the curvature tensor. Besides being a prototype for the sortof equations we will encounter in the proof of the theorem, this result hasuseful applications of its own.

LEMMA 7.4. If (Ma, g (t)) is a solution of the Ricci flow, then the squareof the norm of its curvature tensor evolves by

atIRm2 = OIRml2 -2IV Rm2

ri sj pk qt RI+ 49 9 9 9 rspq (BijkP -BijPk + Bikje -Biejk)

where

BijkP = -Rpij

In particular, one has

IRmI2 < A IRm12 - 2IVRm12 +CIRmI3,at

where C is a constant depending only on the dimension n.

PROOF. By formula (6.17), the (4, 0)-Riemann curvature tensor evolvesby

a(7.1a) atRijkt = ARijki + 2 (BijkP - BijPk + Bikje - Biejk)\

(7.1b) (R'RPIk-P + RRRipkt + RkRijpt + RpRijkp)

It is easy to check that

a IRmI2 = agri Sj pk9g9RrspgRjkPat at

= 2gragsj9 pkggPRspq [ARijkt + 2 (BijkP - BijPk + Bikje - Biejk)

indeed, the terms that arise when we differentiate g-1 exactly cancel theterms that appear in line (7.1b). Since

O lRml2 = 2grigsjgpkgq'RrpgORjke + 2 IV Rm12 ,

the result follows.

The following important consequence of Lemma 7.4 explains why theassumption in Theorem 7.1 that IRml is bounded for a short time is areasonable one.

COROLLARY 7.5 (Doubling-time estimate). There exists c > 0 dependingonly on the dimension n such that if (M', g (t) : 0 < t < -r) is a solution ofthe Ricci flow on a compact manifold and

M (t) - sup IRm (x, t)I9(,,t)seMn

then

M (t) < 2M (0) for all times 0 < t < min {-r,c

M (0)

PROOF. By Lemma 7.4, M (t) is a Lipschitz function of time whichsatisfies dMCM3C< - M2

dt - 2M 2in the sense of the lim sup of forward difference quotients, where C dependsonly on n. This implies that

M (t) <1

1

cM(o) - 2 t

as long as t E [0,,r] satisfies t < 2/ (CM (0)). Let c = 1/C. Then one hasM (t) < 2M (0) for all times t satisfying 0 < t < min {T, c/M (0)}.

REMARK 7.6. We encountered a special case of this result in Lemma5.45.

Combining Corollaries 7.2 and 7.5 yields the following useful observation.

COROLLARY 7.7 (Minimal existence-time). If (Ma, go) is a Riemannianmanifold such that IRm [go] I90 < K, then the unique solution g (t) of theRicci flow with g (0) = go exists at least for t E [0, c/K], where c > 0 is aconstant depending only on n.

2. Proving the global estimates

In proving Theorem 7.1, we shall have to estimate the time derivativeof the quantity I Vk Rm I2 on a solution (.Mn, g (t)) of the Ricci flow. This isthe generalization to n dimensions of the computations in Chapter 5 wherewe considered the evolution of IDkRI2 on a surface. To prepare for thecalculations at hand, let us consider a simpler but representative problem.If Q (t) is a 1-parameter family of (1, 0)-tensor fields on a solution (Mn, g (t))of the Ricci flow, then formula (6.1) implies that

aViQ;=5t-xi ; - rQk)

= ViaatQ)j + (viR +VjRk - VkRj) Qk.

Hence by Lemma 3.1, we have

at IVQ12 a (gikgivjQjVkQt)

= 2Vi j Via

Q (Q) + 2V'Qj (vR + V .3 2 - VkRi;)Qk

j+ 2RikVjQ'V kQj + 2R3'VzQjViQ,.

2. PROVING THE GLOBAL ESTIMATES 227

Simply put, in taking the time derivative of a quantity such as IVQI2, onemust take into account the evolution of the metric and its Levi-Civita con-nection as well as the evolution of the tensor itself.

To avoid a notational quagmire, we adopt the following convention inthe proof. If A and B are two tensors on a Riemannian manifold, we denoteby A * B any quantity obtained from A ® B by one or more of these oper-ations: (1) summation over pairs of matching upper and lower indices, (2)contraction on upper indices with respect to the metric, (3) contraction onlower indices with respect to the metric inverse, and (4) multiplication byconstants depending only on n and the ranks of A and B. We also denoteby A*k any k-fold product A * * A. For example, this convention lets uswrite the conclusion of Lemma 7.4 in the form

atIRmI2 = A IRmI2 - 2 Io RmI2 + (Rm)*s

PROOF OF THEOREM 7.1. The proof is by complete induction on m.We first consider the case m = 1. The evolution equation for IV RmI2 is ofthe form

IVRm12 =2(0 (-atRm) ,VRm)

+VRc*Rm*VRm+Rc*(VRm)*2The standard technique of commuting derivatives and using the secondBianchi identity shows that for any tensor A, the commutator [V, A] A isgiven by

[0,A]A = DOA-AVA=Rm*VA+VRc*A.By applying formula (6.17) and replacing instances of Re with Rm, we get

v=v(ARi+(Rm)*2) =AVRm+Rm*VRm.Because

AIAI2=z (A, A) =2(L (AA, A) + 2 (VA, V

for any tensor A, we conclude that

(7.2) atIVRmI2=AIVRmI2-2JV2Rml2+Rm*(VRm)*2

There are two obstacles to our deriving a satisfactory estimate for IV Rm12directly from equation (7.2). The first difficulty is the potentially bad termRm * (V Rm) *2 on the right-hand side. The second problem is that ourassumptions give us no control on IVRm12 at t = 0. To circumvent theseobstacles, we define

F=tIV RmI2+,3IRmI2,where Q is a constant to be chosen below. The strategy for this choiceis that when t is small, it will allow us to control the bad term we get

when we differentiate IVRm]2 by the good term -2/0 IV Rm12 we get whenwe differentiate IRm12. Moreover, we get an upper bound F < /3K2 att = 0 by hypothesis on without needing further assumptions on theinitial metric. Combining equation (7.2) with the result of Lemma 7.4 anddiscarding the term -2t I V2 Rml2, we observe that F satisfies the differentialinequality

a F < AF + (1 + clt IRmI - 2 / 3 ) 10 Rm12 + c20 IRml3

where the constants c1, c2 depend only on n. By hypothesis, IRm] < K forall t E [0, a/K]. So for such times,

atF< OF + (1 + cia - 2/3) IV Rm12 + c20K3.

Choose ,Q _> (1 + cla) /2, noting that 3 depends only on n and max {a,1}.Then for t E [0, a/K],

at F < OF + c2i3K3.

Thus by the parabolic maximum principle, we have

sup F (x, t) < /3K2 + c2/3K3t < (1 + c2a),3K2 < Cl K2xEMV

for 0 < t < a/K, where CI is a constant depending only on n and max {a,1}.Hence

IVRmIt1/

for 0<t<K.This completes the proof of the case m = 1.

Now by induction, we may assume that we have estimated I V RmI forall 1 < j < m. Let 1 < k < in. We begin by computing

(7.3)at I

VkRm2

2Cat(VkRm),VkR)+Rc* (VkRm)*2

To evaluate the time derivative on the right-hand side, we again recall for-mulas (6.1) and (6.17) and write

at(VkRm)

=v k (Rm) +

k-1T Vj (VRc*V'Rm)j=

k

_ Vk (Rm+(Rm)*2) + V Rm*Vk-j Rmj=1

k

_ VkARm+EVjRm*Vk-j Rmj=o

2. PROVING THE GLOBAL ESTIMATES 229

In order to put (7.3) into the form of a heat equation, we need AVk Rm.Since for any tensor A, the commutator [Vk, A] A is given by

k

[Vk, A] A = VkAA - AVkA = E Vi Rm *Vk-jA,j=o

we find thatk

atVkRm = AVkRm+1: VjRm*Vk-j Rm.j=o

Substituting this formula into (7.3) and recalling that

2 (AA, A) = A JAI2 - 2 IVAI2,

we obtain

(7.4a)a

atVk Rm

2

=A VkRm2

-2IVk+1Rm2

(7.4b) + E V3 Rm *Vk-j Rm *Vk Rm.j=0

Now applying identity (7.4) in the case k = m, we get a differential inequalitym

VmRm 2 < A VmRml2+Ecmj IVjRrI . IVm-j . IVmRmI,j=o

where the constants cmj depend only on j, m, and n. The inductive hy-pothesis then gives an estimate

at< A IVmRmI2 + (cmo +cmm) KIV-RmI2

m-1Cj Cm-j 2 m+ Grnj tj/2 t(m-j)/2 K V Rm

j=1

<AIV-Rml2+K (CjVmRmI2+t/2KIV'Rml

on the time interval 0 < t < a/K, where the constants C,n and C,, dependonly on m and n. Completing the square on the right-hand side and usingthe fact that (a + b)2 < 2 (a2 + b2), we obtain Cm depending only on m andn such that

z

(7.5) atlV"Rml2<AIV-Rm12+CmK(IVmRml2+ m).

As in the case m = 1, we shall not try to control I Vm Rml2 directly fromthis equation; instead, we define

G =tm l Vm Rm12

-+ 0M

in(m - 1)l t,,,,m-k

(m - k)1k=1

Vm-k Rm2

where /3m, is a constant to be chosen momentarily. We will explain the fullstrategy behind this choice after we estimate the evolution equation satisfiedby G. For now, we simply observe that our assumption on IRml implies thatG < 13m (m - 1)! K2 at t = 0. By (7.4) and the inductive hypothesis, thereare constants Ck depending only on k and n such that for all 1 < k < m wehave

(7.6)a

atVk Rm

2

<0 Vk Rm2

-2 Vk+1 Rm2CkK3+

tk

on the time interval 0 < t < a/K. It will be significant that we retain thegood term -2 I Vk+1 Rm I2 that appears in the inequality (7.6) for k < m,although we discarded it in estimate (7.5). Indeed, using formulas (7.5) and(7.6), we compute that G satisfies the differential inequality

09 -at G<OG+C,,,,KtmlVmRm12+CmK3+mt"ri-1 jVmRm12

+ pmm (m - 1)!

-2tm-k Ivm-k+l Rm12 + Cm-kK3

E (m --k)! + (m - k) tm-k-1 I Vm-k RmI2

This reveals why we defined G as we did: the good terms

-2 (m - 1)! tm-k vm-k+l Rm 2(m - k)!

that we get when we differentiate IVm-k Rml2 compensate for the bad terms

(rn - 1)! m-k(m-k+1)! (m-k+l)t vm-k+l Rm12

we got when we differentiated tm-(k-1). In this way, we get the estimate

atG < OG + (C nKt + m - 2 )3m) tm-1 I Vm RmI2 + (Cm + 00,'n) K3,

where

-. (rn1)!Cm L Om-k- k)!k=1

(m

Choose 0m > (Cm,a + m) /2, noting that /3m, only depends on m, n, andmax {a, 1}. Then for t E (0, a/K], we have

at G< OG + (Cm. + 00',) K3.

Since G < /3m, (m - 1)! K2 at t = 0, the maximum principle implies that

supxEM'ti

3. THE COMPACTNESS THEOREM 231

for 0 < t < a/K, where Cm _ V/3m, (m - 1)! + a (C., + iC',l) depends onlyon m, n, and max {a,1 }. Thus we have

072 Rm 1 < V m < /K for 0 < t < K .

This completes the inductive step, hence our proof of Theorem 7.1.

3. The Compactness Theorem

The Compactness Theorem for the Ricci flow [64] tells us that anysequence of complete solutions to the Ricci flow having curvatures uniformlybounded from above and below and injectivity radii uniformly bounded frombelow contains a convergent subsequence. This result has its roots in theconvergence theory developed by Cheeger and Gromov. In many contextswhere this theory is applied, regularity is a crucial issue. By contrast, theproof of the Compactness Theorem for the Ricci flow is greatly aided bythe fact that sequences of solutions to the Ricci flow enjoy excellent regu-larity properties. (See Corollary 7.3 above.) Indeed, it is precisely becausebounds on the curvature of a solution to the Ricci flow imply bounds onall derivatives of the curvature that the compactness theorem produces C°°convergence on compact sets. For this reason, we state it in this chapter.We plan to present a detailed proof in a successor to this volume.

The most important application of the compactness theorem is the for-mation of singularity models (final time limit flows). These are completenonflat solutions of the Ricci flow that occur as limits of sequences of di-lations about a singularity. Indeed, we already used it for this purpose inSection 15 of Chapter 5. We shall discuss singularity models in greatergenerality in Chapter 8.

THEOREM 7.8 (Compactness Theorem). Let

{Mi, gi (t) , Oil Fi : i E N}

be a sequence of complete solutions to the Ricci flow existing for t E (a, w),where -co < a < 0 < w _< oo. Each solution is marked by an originOi E J1i12 and a frame Fi = {ei, ... , 4} at Oi which is orthonormal withrespect to gi (0). Suppose that there exists K < oo such that the sectionalcurvatures of the sequence are uniformly bounded by K in the sense that

sup (sect [gi] I < KMn x (a,w)

for all i E N. Suppose further that there exists 8 > 0 such that the injectivityradii of the sequence are bounded at Oi E Mi and t = 0 in the sense that

inj gti(a) (Oi) > 5

for all i E N. Then there exists a subsequence which converges in the pointedcategory to a complete solution

{M 00, goo (t),0.,F.}

of the Ricci flow existing for t E (a, w) with the properties that

sup sect [g00] < KM n x (a,w)

and

injgoo(o) (O".) > J.The convergence is in the following specific sense: there exist a sequenceof open sets Ui C M' such that 0,,. E Ui for all i E N and such thatany compact subset of M' is contained in UZ for all i sufficiently large,a sequence of open sets Vi E Mi such that Oi E Vi for all i E N, anda sequence of diffeomorphisms oi : Ui -- Vi such that ci (Occ) = Oi and

(F.) = Fi for all i E N; these diffeomorphisms have the property thatthe pullbacks cpj (gi) converge uniformly to g, in every C'`r' norm on anycompact subset of Mn x (a,w).


Two good references for derivative estimates that apply to general par-abolic equations are [94] and [89]. The reader may also be interesting inconsulting some of Bernstein's original papers, notably [16, 17, 18]. Forthe Ricci flow in particular, the early Bemelmans-Min-Oo-Ruh paper [11],Bando's paper [9], and Shi's papers [117, 118] are all useful references. Ourapproach is a modification of the method outlined in Section 7 of [63].

The convergence theory of Cheeger and Gromov and in particular theGromov-Hausdorff convergence of Riemannian manifolds is currently an ac-tive and rich area of research. An interested reader is urged to consult theinfluential book [53] and the fine survey [109]. Good examples of conver-gence results in the literature can be found in [2], [26], [45], [50, 51], [108],and [130, 131].

CHAPTER 8

Singularities and the limits of their dilations

In this chapter, we shall survey the standard classification of maximalsolutions to the Ricci flow. From the perspective of this classification, oneregards a solution (Mn, g (t)) that exists up to a maximal time T < oo asbecoming singular at T, because it cannot be extended further in time. Soeach maximal solution may in this way be thought of as a singular solution.We shall then study how singularities may be removed by dilations. In manycases, one can pass to a limit flow, called a singularity model, whose proper-ties can yield information about the geometry of the original manifold nearthe singularity just prior to its formation. We discussed some examples ofsingularity models in Chapter 2. Singularity models always exist on infinitetime intervals. In dimension n = 3, singularity models have other specialproperties (in particular nonnegative sectional curvature) that imply thatthey are simple topologically, and thus make it reasonable to expect thatone could obtain a classification of such limit solutions that is adequate toderive geometric and topological conclusions about the underlying manifold.M3 of the original singular solution g (t). In Chapter 9, we shall see someof the reasons why singularity models in dimension three are so nice.

1. Classifying maximal solutions

Consider a solution (Ma, g (t)) of the Ricci flow which exists on a maxi-mal time interval [0, T), where T E (0, oo]. We call T the singularity time.This divides solutions into two categories: those where T < oo and thosewhere T = oc. In both cases, there are further subdivisions into two typesof singularities.

To motivate these subdivisions, we look to the simplest solutions, thosewhich are the fixed points of the normalized Ricci flow. So let us considerthe evolution (M', g (t)) of an Einstein metric by the Ricci flow. There arethree possibilities: the scalar curvature R may be positive, zero or negative.If R (x, 0) > 0, it follows from the evolution equation

aR=OR+2IRcI2=OR+2R2n

and the fact that R is constant in space that T < oo; in fact,

R (x t) =n

233

234 8. DILATIONS OF SINGULARITIES

for all x E 1Vl'. Then since Rm (t)I = CR, where C > 0 is a constantdepending only on g (0), we have

IRm(t)I(T-t)=Cfor another positive constant C.

Analogously, if R (x, 0) < 0, thenn

R (x, t)

so T = oo and

nR (0)-I - 2t'

IRm(t)I(Cl+t)=C2for some C1, C2 > 0. Finally if R (x, 0) = 0, then R = 0, T = no and

jRm(t)I =C>0.These considerations lead us to subdivide solutions of the Ricci flow into

two types: those whose curvatures are bounded above by a constant timesthe curvature of an Einstein solution with R > 0 or R < 0 (Type I or III)and those whose curvatures are not (Type Ha or IIb). More precisely, theclassification of maximal solutions (.Mn, g (t)) by their singularity type isas follows.

If the singularity time T is finite, Theorem 6.45 tells us that the curvaturebecomes unbounded,

sup IRmt = no,M" x [0,T)

and classify the solution by its curvature blow-up rate - the rate atwhich the curvature approaches infinity as time approaches T. (Curvatureblow-up rates were discussed briefly in Subsection 6.2 of Chapter 2.)

DEFINITION 8.1. Let (Ma, g (t)) be a solution of the Ricci flow thatexists up to a maximal time T < no.

One says (M7z, g (t)) encounters a Type I Singularity if T < noand

sup Rm t) (T -t) < oo.Mn x [0,T)

One says (M', g (t)) encounters a Type IIa Singularity if T < noand

sup Rm t) I (T - t) = oc.Mn x [0,T)

One says (Mn, g (t)) encounters a Type IIb Singularity if T = noand

supMn x [0,00)

One says (Mn' g (t)) encounters a Type III Singularity if T = noand

sup IRm ( , t) I t < oo.M'1 x [0,00)

2. SINGULARITY MODELS 235

An Einstein solution with R > 0 is Type I; an Einstein solution withR < 0 is Type III; and a nonflat Einstein solution with R = 0 is Type IIb.Although Type Ila singularities are strongly conjectured to exist, it is be-lieved (but not proved) that they will occur only for very special initial data.The degenerate neckpinch discussed heuristically in Section 6 of Chapter 2is simplest example in which a Type IIa singularity is expected to form.

REMARK 8.2. Perelman's recent work [105, 106, 107] does not explic-itly distinguish between Type I and Type IIa singularities. Nonetheless, theconcepts of rapidly-forming and slowly-forming singularities are importantin so many aspects of the analysis of geometric evolution equations that wewill endeavor to help the reader be aware of the distinctions between thesesingularity types for the Ricci flow.

2. Singularity models

Singularity models are complete nonflat solutions which occur as lim-its of dilations about singularities. Any such solution exists for an infinitetime interval. Each type of singular solution (I, IIa, IIb, and III) has its ownsingularity model.

For insight, we turn again to the Einstein metrics. Notice that an Ein-stein solution with R < 0 is an immortal solution: a solution that existson the future time interval [0, oo). Indeed, we may extend such a solutionto its maximal time interval of existence via the relation

9 (t) = Cl - 2R (0) t) g (0) tE (20)oc).An Einstein solution with R > 0 on a time interval [0, T) with T < oo maybe uniquely extended backward in time to an ancient solution: a solutionthat exists on the past time interval (-oo, T). Here

9 (t) = (l_)g(o), t E (-oo, T) .

Finally, an Einstein solution with R = 0 (namely, a Ricci-flat solution) maybe uniquely extended to an eternal solution: a solution that exists for alltime (-oo, oo). Clearly, a Ricci-flat metric is static:

g (t) = g (0) t E (-oo, +oo) .

Singularity models will turn out to be solutions of one of these three types- ancient, eternal, or immortal. (Recall that we saw other examples ofancient, eternal, and immortal solutions in Chapter 2.)

DEFINITION 8.3. Let ( 00, g,, (t)) be a limit solution of the Ricci flow.

One says (Mn, g,, (t)) is an ancient Type I singularity modelif it exists on a time interval (-oo, w) containing t = 0 and satisfies

the curvature bound

sup Rm,,, (', t) jtI < oc.Mn X(-00,0j00

One says (.M', g,,, (t)) is an ancient Type II singularity modelif it exists on a time interval (-oc, w) containing t = 0 and satisfiesthe curvature bound

sup Rm t) I ' ti = cc.x (-00,0]Mn00

One says (M', g,,, (t)) is an eternal Type II singularity modelif it exists for all t E (-cc, oo) and satisfies the curvature bound

sup IRmt)I < 00.M ,x(-00,00)

One says (M', g (t)) is an immortal Type III singularitymodel if it exists on a time interval (-a, oo) containing t = 0 andsatisfies the curvature bound

sup IRm,, t) t < oo.M1 X[0,00)

REMARK 8.4. As we shall see when we study dilation below, it is alwayspossible to apply a more careful `point picking argument' and thereby tochoose the sequence of points and times about which one dilates so that thesingularity model satisfies

(8.1) sup IRmoo (x,0)I = IRmoo (y,0)1xEMn

for some y E M'. If the curvature operator of the solution g (t)) isnonnegative, one can replace this with

(8.2) sup R. (x, 0) = R. (y, 0) .xEMn

The latter condition allows application of the strong maximum principleand a differential Harnack inequality. (We introduced Harnack estimates inSection 10 of Chapter 5 and will see them again in Section 6 of Chapter9. We will make a thorough study of differential Harnack estimates in thesuccessor to this volume.) Condition (8.2) is useful primarily for Type IIsingularities. (For example, see Section 6.)

REMARK 8.5. By Theorem 9.4, any limit (M00,g,,,, (t)) of a finite-timesingularity in dimension n = 3 has nonnegative curvature operator. So indimension 3, one can always satisfy (8.2) by making appropriate choices ofpoints and times about which to dilate.

3. PARABOLIC DILATIONS 237

REMARK 8.6. The distinction between ancient and eternal Type II mod-els should be carefully noted. It arises because any limit of a Type II sin-gularity is noncompact. In order to study its geometry at spatial infinity,one may take a second limit by a procedure called dimension reduction.(Dimension reduction will be discussed briefly in Section 4 and thoroughlyin a planned successor to this volume.) This technique produces ancientsolutions which are either Type I or Type II.

EXAMPLE 8.7. The Rosenau solution introduced in Section 3.3 is anexample of a Type II ancient solution that is not eternal but has a `backwardlimit' (as described in Section 6 below) that is an eternal solution. In fact,this limit is the cigar soliton studied in Section 2.1.

REMARK 8.8. It is finite-time singularities (Type I and Type IIa) thatare most important in applications of the Ricci flow program towards provingthe Geometrization Conjecture for closed 3-manifolds.

EXAMPLE 8.9. In dimension n = 3, the neckpinch (Section 5 of Chapter2) and the conjectured degenerate neckpinch (Section 6 of Chapter 2) arethe canonical examples of Type I and Type Ha singularities, respectively.

3. Parabolic dilationThe basic idea of dilating about a finite-time singularity is to choose

a sequence of points and times where the norm of the curvature tends toinfinity and is comparable to its maximum in sufficiently large spatial andtemporal neighborhoods of the chosen points and times. Heuristically, onetrains a microscope at those neighborhoods and magnifies so that the normof the curvature is uniformly bounded from above and is equal to 1 atthe chosen points. Because Perelman's No Local Collapsing theorem [105]provides an appropriate injectivity estimate in those neighborhoods, one canthen obtain a nontrivial limit solution of the Ricci flow. Complete solutionswhich arise as such limits have special properties which suggest that theycan be classified in a way that will yield information about the singularity.We shall make these ideas precise in what follows. The case of an infinitetime singularity is analogous, and will be considered in Section 5.

3.1. How to choose a sequence of points and times. If (M , g (t))is a solution of the Ricci flow on a maximal time interval 0 < t < T _< oo, wemay study its properties by dilating the solution about a sequence of pointsand times (xi, ti), where xi E .tit' and ti / T. Of course, there are manypossible sequences of points and times around which to dilate, and the limitswe get may depend on our choices. However, each sequence should at thevery least satisfy certain conditions which we outline below. The guidingprinciple is that if we want to get a complete smooth limit flow, then we wantthe norm of the curvature of each dilated solution to be suitably boundedin an appropriate parabolic neighborhood of each (xi, ti).

In order to develop intuition for the dilation criteria introduced below,recall that if the singularity time T is finite, then Theorem 6.54 tells us that

(8.3) lim sup IRm ti) I = 00t1/T Mn

regardless of whether the singularity is forming quickly (Type I) or slowly(Type IIa). In light of this fact, one may choose the sequence of timesti so that the maximum of the curvature at time ti is comparable to itsmaximum over sufficiently large previous time intervals. Similarly, one maychoose the sequence of points xi so that the curvature at (xi, ti) is com-parable to its global maximum at time ti. Occasionally (for example, indimension reduction) one chooses points xi so that the curvature at (xi, ti)is comparable to its local maximum over a sufficiently large ball. (In somecases, it may be possible to get a complete limit for any sequence (xi, ti)such that IRm (xi, ti) I -+ oo if one has good enough bounds on IV Rm weshall not discuss such limits in this volume, however.)

Motivated by these considerations, we adopt the following definitionsfor any solution (Mn, g (t)) of the Ricci flow on a maximal time interval0 < t < T < oo. Notice that they reflect the way parabolic PDE naturallyequate time with distance squared.

DEFINITION 8.10. We say a sequence (xi, ti) is globally curvatureessential if ti / T < oo and there exists a constant C > 1 and a sequenceof radii ri E (0, ti) such that

(8.4) sup { IRm (x, t)l : x E B9(t) (xi, ri), t E [ti - r?, ti] } < C IRm (xi, ti)j

for all i E N, where

(8.5) lim r? IRm (xi, ti) I = oc.

DEFINITION 8.11. We say a sequence (xi, ti) is locally curvature es-sential if ti / T < oc and there exists a constant C < oo and a sequenceof radii ri E (0, ti) such that for all i E N, one has

(8.6) sup { IRm (x, t) I : x E B9 (t) (xi, ri) , t E [ti - r?, ti] } < Cry 2.

In this volume, we shall deal mainly with globally essential dilation se-quences. In a later volume, we hope to discuss the relevance of locallyessential sequences to the surgery arguments in [106].

REMARK 8.12. In some cases, one works with a special case of condition(8.4) in which

(8.7) sup {IRm (x, t)l : x E Mn, t E [0, ti]} < C Rm. (xi, ti) I .

REMARK 8.13. By (8.4), any smooth limit of a globally curvature es-sential sequence formed by the parabolic dilations (8.8) has its curvaturebounded by some function K (t). Moreover, property (8.5) ensures that anysuch limit is complete. A smooth limit of a locally curvature essential se-quence will have bounded curvature on a metric ball of radius 1 for times-1<t<0.

3. PARABOLIC DILATIONS 239

Once we have chosen a sequence (xi, ti) of points and times such thatt2 / T E (0, oo], we consider a sequence of parabolic dilations: thesolutions (.Ms, gi(t)) defined by

(8.8) 9i (t) - IRm (xi, ti)g

t2+JRm(xi,ti)I

that exist for

-ti IRm (xi, ti) I < t < (T - ti) IRm (xi, ti) I .

(The right endpoint is oo if T = oo.) The translation in time ensures thatgi (0) is a homothetic multiple of g (ti), and the dilation in time ensures thatgi(t) is still a solution of the Ricci flow. The spatial dilation is chosen so thatthe curvature Rm [gi] of the new metric gi has norm 1 at the new `origin' xiand the new time 0, namely so that

(8.9) IRm[gi] 1.

This guarantees that the limit of the pointed solutions (.M',gi(t),xi), if itexists, will not be flat.

REMARK 8.14. In dimension 3, one can replace IRm by R in the discus-sion above. Indeed, the ODE estimate for the curvature obtained in Lemma9.10 implies that there exist constants C, C' > 0 such that

R>CIRmI - C'in dimension n = 3. But in any dimension n, there exists C,, depending onn such that

R<C,,,IRmI.Hence the scalar curvature R is equivalent to IRm whenever the latter islarge enough.

If a sequence (xi, ti) is globally curvature essential, then estimate (8.4)implies that IRm [gi] (x, t) I will be bounded on arbitrarily large balls as ioo. So in order for the limit solution to exist, it suffices to have

inj [9i] (xi, 0) > c > 0

for some constant c > 0 independent of i. A sufficient condition is given inthe following.

DEFINITION 8.15. A solution (.M'"2, g(t)) of the Ricci flow on a timeinterval [0, T) is said to satisfy a global injectivity radius estimate onthe scale of its maximum curvature if there exists a constant c > 0such that

inj (x, t)2 >SUPMn t)

C

for all (x, t) E Mn x [0, T).

In more advanced applications to be considered in a later volume, onemay replace this by a local condition.

DEFINITION 8.16. A solution (M", g(t)) of the Ricci flow on a timeinterval [0, T) is i-noncollapsed on the scale p if for every metric ballBg(t) (xo, r) of radius r ttr".

If (A4 , g(t)) obeys a suitable injectivity radius estimate, the Com-pactness Theorem from Section 3 of Chapter 7 shows that a subsequenceof the pointed sequence (M ,gi(t),xi) converges to a complete pointedsolution ge(t), xc) of the Ricci flow on an ancient time interval-oc < t < w < oc. This singularity model satisfies

IRm [g.] (x, t)Ig < C

x (-oo, 0].for all (x, t) E /W00If (.M", g(t)) becomes singular in finite time - in other words, whenever

(M", g(t)) encounters a Type I or Type IIa singularity - Perelman's NoLocal Collapsing Theorem [105] provides such an estimate. In particular,Perelman's result implies that there exists i > 0 such that the singularitymodel (M', ge(t), x,,,,) it k-noncollapsed at all scales.

THEOREM 8.17 (Perelman). Let (M",g(t)) be a solution of the Ricciflow that encounters a singularity at some time T < oc. Then there existsa constant c > 0 independent oft and a subsequence (xi, ti) such that

inj (xi, ti) >c

VmaxMn IRm (', ti) I

In forthcoming work, we plan to provide a detailed explanation of theproof.

REMARK 8.18. A global injectivity radius estimate on the scale of themaximum curvature was originally established for Type I singularities byHamilton's isoperimetric estimate. (See §23 of [63].)

4. Dilations of finite-time singularities

There is a lower bound for the curvature blow-up rate which appliesboth to Type I and Type IIa singularities.

LEMMA 8.19. Let M" be a compact manifold. If 0 < t < T < oo is themaximal interval of existence of (M", g (t)), there exists a constant co > 0depending only on n such that

(8.10) xsupn I (x, t) > ,co t.

4. DILATIONS OF FINITE-TIME SINGULARITIES 241

PROOF. Recall from Lemma 7.4 that

atIRm 22 < A IRmj2 +C IRmf3

for some C depending only on n. Define

K (t) _ sup IRm. (x, t) I2x E.M n

By the maximum principle, we have dK/dt < CK3/2, which implies that

dtK-1/2 > - 2 .

Integrating this inequality from t to r E (t, T) and using the fact that

lim inf K (T)-1/2 = 0,T-->T

we obtain

K- 1/2 (t) < C (T - t).

Hence SUR,EM,i IRm (x, t) > 2/ [C (T - t)].

4.1. Type I limits of Type I singularities. Given a Type I singularsolution (.Mn, g (t)) on [0, T), define S by

(8.11) sup IRm (x, t) I (T - t) - S < oc.Mn x [O,T)

Let (xi, ti) be any sequence that is globally curvature essential. Then ti / T,and the norm of the curvature at (xi, ti) is comparable to its maximumat time t in the sense of (8.4). Consider the dilated solutions (M',gi(t))of the Ricci flow defined by (8.8), and recall that Rmi - Rm [gi] satisfiesIRmi (xi, 0) 1 = 1 for all i.

For simplicity, we will assume the global bound (8.7); the general caseis essentially the same. By using definitions (8.8) and (8.11), estimate (8.4),and Lemma 8.19, we find that the solutions (Mn, gi(t)) obey the uniformbound

IRmi (x,t)I = IRm(xi,ti)IIRm

(try + IRm (xi,ti)I )S<C( -ti)

T - (ti IRm x. t,)I)

t=CS

C1- (T-ti)IRm(xi,ti)l)<CSI+S

for all x E M' and t E [-ai, 0], whereai = ti IRm (xi, ti)I. This implies theestimate

sup I tl .1 Rm i (x, t) I < CS2 < oo,Mn X [-caj,O)

which is the curvature bound we need for a subsequence to converge to aType I singularity model.

By classifying the limits of dilations of all sequences (xi, ti) chosen inthis manner, one can understand a Type I singularity at all fixed relativescales in the interval (0, 1], where the relative curvature scale of a pointx E .M' at time t E [0, T) is defined to be the ratio

IRm (x, t) IsupMn

There are certain properties common to any such limit.

PROPOSITION 8.20. Let (M', g (t)) exhibit a Type I singularity at T E(0, oo), and let (xi, ti) be any globally curvature essential sequence. Thenany limit (Mn , g,, (t)) is an ancient Type I singularity model that exhibitsa Type I singularity at some time w < no.

PROOF. By the Type I singularity condition and Lemma 8.19, thereexist positive constants co < C such that

co<supIRm(,t)I <

CT-t - Mn T-t'

Then by our choice of (xi, ti), there is c > 0 such that

c < IRm (xi, ti)I CT-ti - T-tiThe solutions (Mn, gi (t)) defined by (8.8) exist for -ai < t < wi, where

ai = ti IRm (xi, ti) I > 0,

wi = (T - ti) I Rm (xi, ti) I > 0.

For any t E (ai,wi), the curvature Rmi of gi obeys the estimate

(8.12)C wi

< I Rm i (x t) I < C WiCWi - t - cwi -t'

Indeed, one has

Rmi(x,t) = IRm(xi,ti)I Rmti +IRm(xi,ti)I/I<T - ti C

C T - ti - t IRm (xi, ti) I-'C wicwi -t'

with the other inequality being obtained similarly.Now since c < wi < C for all i, we can choose a subsequence (xi, ti) such

that w - limi-. wi exists. Since ai > T IRm (xi, ti) I - C, we have ai --> noby (8.3). So any limit (M ,goo (t)) is defined on (-oo,w). For each timet E (-oc,w), there is It so large that t E (-ai,wi) for all i > It, whence

it follows from (8.12) that (M', g,, (t)) exhibits a Type I singularity at00W E [c, C] .

REMARK 8.21. A consequence of the results in Chapter 6 is that anysolution starting at a compact 3-manifold of positive Ricci curvature en-counters a Type I singularity. Furthermore, any Type I limit formed at thatsingularity will be a shrinking spherical space form.

We can satisfy the stronger condition (8.1) by being more careful inpicking the sequence of points and times. In particular, a necessary but notsufficient condition is that the relative scales of the points (xi, ti) tend to 1in the sense that

(8.13) limIRm (xi, ti) - 1

i--;oo suPMn IRm (ti)I

Note that when M" is compact, we may choose xi so that

Rm (xi, ti) I = max IRm (., ti)

but this is not necessary (and not always possible when Mn is noncompact).Define

(8.14) w - lim sup IRm (-, t) I (T - t) .t/'T

By Lemma 8.19 and definition (8.11), we have w E (0, oo) . Take any se-quence of points and times (xi, ti) with ti J' T such that

(8.15) IRm (xi, ti) I (T - ti) wi -> w.

Assuming (8.13) holds, this condition is equivalent to the requirement thatthe times ti satisfy

sup IRm. (., ti) I (T - ti) -* w.Mn

In other words, we want the Type I rescaling factors to tend to their lim sup.Consider the dilated solutions gi(t) of the Ricci flow defined by (8.8).

By definition of w, there is for every e > 0 a time t, E [0, T) such that

IRm(x,t)I (T - t) < w + e

for all x E M and t E [ti, T). The curvature norm of gi then satisfies

IRm i (x, t) I =IRm (xi, ti) Rm Cx' ti + IRm (xi, ti)

Rm (x, ti + Rm(tti1t01)

I

(T - tz - Rm(xj,ti)I)

Rm. (xi, ti) I (T - ti) - tw + e

(8.16)<

if

wi - t'

tE C ti + t Rm. (xi, ti) 1 -1 < T,

hence ift E [- IRrn (xi, ti)I (t2 - t'), wi)

Note that wi --- w and that

lim Rm (xi, ti) I (ti - tE) = 00

for any E > 0. So for a pointed limit solution (M', g... (t) , x,,,.) of a sub-sequence of (M", gi (t) , xi), we can let i ---* oc and then E \ (l in estimate(8.16) to conclude that

Rm (x, t) I <w

w-tfor all x E Mx and t E (-oc, w). Because IRmi (xi, 0) = 1 for all i, thelimit satisfies IRm (x, 0) = 1.

4.2. Type II limits of Type Ha singularities. An important sourceof intuition into the expected behavior of Type Ha singularities is given bythe conjectured degenerate neckpinch described in Section 6 of Chapter 2.Recall that the Type Ha condition says that

sup IRm t)I (T - t) = oo.Mn x [0,T)

This will allow us to select a sequence of points and times so that the di-lated solutions have uniformly bounded curvatures on larger and larger timeintervals both forwards and backwards in time.

First choose any sequence of times Ti / T. Given any c E (0, 1), let(xi, ti) be a sequence of points and times such that ti -+ T and

(8.17) Rm (xi, ti) I (Ti - ti) > c sup IRm (x, t) I (Ti - t).Mn X [O,T2]

In other words, if we were doing Type I rescalings relative to the intervals[0, Ti], we would choose the curvatures at (xi, ti) to be comparable to theirmaxima on A4' x [0, Ti]. Since Ti < T, such points and times always exist.When MI is compact, they also exist for c = 1.

Consider the dilated solutions given by (8.8). Condition (8.17) impliesthat for

t E [-ti Rm (xi, ti) I , (Ti - ti) IRm (xi,ti)I )we have

IRmi(x,t) Ti - ti - tIRm (xi, ti)

IRm (x, ti + Rm x{,t2 ) (Ti - t2 Rm(k,tz) )

Rm(xi,ti)I (Ti -ti)(Ti - ti)

1 (Ti - ti)c

or equivalently,

IRmi (x,t)I <_ 1(T i -ti) IRm (xi,ti)I

c (Ti - ti) I Rm (xi, ti) I - tSince Ti -4T, the Type IIa condition implies that

lira sup IRm (x, 01 (Ti - t) = 00.i +oo Mrex[O,TT]

Thus, by our choices of xi and ti we have IRm (xi, ti) I (Ti - ti) , oo.Applying Theorem 8.17 and the Compactness Theorem, one obtains a

72 ,limit solution (A4,, g,, (t)) that exists for all t E (-oo, oo) with uniformlybounded curvature

(8.18) sup IRmo,,I <Mnx(-oooo) c

As in the Type I case, a more carefully chosen sequence (xi, ti) will guar-antee that the pointed limit solution (M', g. (t) , x.) attains its maximumcurvature at (x', 0). Again choose any sequence of times Ti / T. The basicidea is to choose points and times (xi, ti) such that

(8.19) lim IRm (xi, ti) I (Ti - ti) - 1.i oc supMl x [O,T] IRm (x, t) I (Ti - t)

We shall give the proof for the case that .M' is compact, leaving the generalcase to the reader. Then since Ti < T, there exist points and times (xi, ti)such that equality holds in (8.19). By repeating the argument of Section 4.2with c = 1, we obtain the estimate

IRmi (x,t)I <(Ti -ti) IRm(xi,ti)I

(Ti - ti) IRm (xi, ti) I - tfor all x E Mn and

where we have

t E [-ti I Rm (xi, ti) I, ( T i - ti) IRm (xi, ti) I) ,

(Ti - ti) IRm (xi, ti) I -? oo.Hence the pointed limit g (t) , x,,,,) is defined for all t E (_00, 00)and satisfies the uniform curvature bound

(8.20) sup IRmMl9. < 1.Mn X(-00,00)

Moreover, equality holds at (x, 0), because IRmi (xi, 0) I = 1.

REMARK 8.22. When n = 3, we have the estimate

CRmI-C'<R<C3IRmlof Lemma 9.10, which implies that

sup R (x, t) (T - t) = oc.M3 x [0,T)

Thus we may replace IRml by R in the discussion above, whence it followsthat

R. (x, t) < 1 = R... (x,,,,, 0)for a pointed Type II limit (M3, g,, (t) , x,,,). The advantage of the scalarcurvature R,, attaining its maximum in space and time is as follows. Astronger consequence of Theorem 9.4 is Corollary 9.7, which says that anytype of limit of dilations about a finite-time singularity has a non-negativecurvature operator. When R,,, attains its maximum, this enables the appli-cation of a differential Harnack estimate together with the strong maximumprinciple.

EXAMPLE 8.23. An example of a rotationally symmetric Type II limitis the Bryant soliton [21] on 1R3.

5. Dilations of infinite-time singularities

5.1. Type II limits of Type IIb singularities. This case is dual tothe Type Ha case. Recall that the Type IIb condition is

(8.21) sup t IRm (x, t) I = oo.Mn X[0,00)

Analogous to Type Ha, choose any sequence Tj / oo. If we follow what wedid before, it is natural to choose a sequence (xi, ti) such that

ti IRm (xi, ti) I _ ai --> 00

andti IRm (xi,ti)I >c>0 .

SUPMnX[0T] tIRm(x,t)I -This is always possible but may not be a good choice, because it does notguarantee that the curvatures of the dilated solutions can be uniformlybounded on finite time intervals. To see this, notice that the dilated so-lution gi (t) is defined for t E [-aj, 00) and satisfies

IRmi (x, t)I = IRm (xi,ti) I

IRm Cx, t2+ IRm (xi, ti) I

ti IRm (xi, ti) I IRm \x, t2 + IRm (xi, ti)Il

x(ti +

t ) ti lRm (xi, ti) IRm (xi, ti) I / t i I Rm (xi, ti) I + t

(8.22) < 1az

cai+tfor all x E M' and t such that

0 < ti +t IRm(xi,ti)1'I <Ti,hence for all t E [-ai, wiI, where

wi - (Ti - ti) IRm (xi,ti)I

5. DILATIONS OF INFINITE-TIME SINGULARITIES 247

Now ai --* no, but we do not know that wi also tends to infinity. Thismotivates finding a better sequence of points and times (xi, ti).

Refining our choice by the method of Section 4.2, we pick points (xi, ti)so that

ti (Ti - ti) IRm (xi, ti) I * 1- - Si --> 1.supMn x [O,TT] [t (Ti - t) IRm (x, t) I}

This guarantees that ai -4 no and wi ---> no. Indeed, it follows from (8.21)that

1 aiwi _ ti IRm (xi, ti)I (Ti - ti)ai 1 + wi 1 ai + wi Ti

= 1 sup t I Rm (x, t) I (Ti - t)Ti Mn X [O,Ti]

> 1 sup t IRm (x, t)I --> oo,2 Mn x [O,T,/2]

hence that ai 1 - 0 and wi 1 --* 0. Then as in (8.22), we get

IRmi (x,t)I

(ti +i ) (Ti - t2 Rm(xi,t,)I) Rm (x, tz + q,tti

ti (Ti - ti) IRm(xi,ti)Iti I Rm. (xi, ti) I (Ti - ti)

X

(ti IRm (xi, ti) + t) (Ti - ti - Rm(at,tz)I )< 1 x ai wi

ai+twi - tfor all x E Mn and t E [-ai, wi). Since Si -- 0, ai -4 oo, and wi -> oo,we conclude that the pointed limit solution (A1 , goo (t) , x.), if it exists,00is defined for all t E (-oo, oo) and satisfies the curvature bound

sup IRni I 1= IRmc (x,,,0)IM" X(-00,00)

5.2. Type III limits of Type III singularities. Recall that a solu-tion (Mn, g (t)) of the Ricci flow forms a Type III singularity at T = no ifthe solution exists for t E [0, oo) and satisfies

sup t I t) I < no.Mn x [0,00)

Define

(8.23) a - 1im sup I t sup IRm t) I ] E[0,00).t--roo \ Mn /

The nonnegative number a is analogous to w defined in equation (8.14). Weshall now demonstrate that one may assume a is strictly positive. We begin

by noting a consequence of the formula

(8.24) d-Lt (7t) =2f at

(S, S) ds - f (VS,S, V) ds7t t

derived in Lemma 3.11.

COROLLARY 8.24. Let ^/t be a time-dependent family of curves with fixedendpoints in a solution (.M1, g (t)) of the Ricci flow. If 'yt is a constant pathor if each -yt is geodesic with respect to g (t), then

tLt (7t) _ -f Re (S, S) ds.

PROOF. The last term in formula (8.24) vanishes if the paths 1't areindependent of time or variations through geodesics.

PROPOSITION 8.25. Let (Mn, g (t)) be any solution of the Ricci flow on[0, oo). There exists a constant e > 0 depending only on n such that a > eif Mn is not a nilmanifold.

PROOF. We claim there is e = E (n) > 0 small enough so that if

<E'(8.25) a = lim sup (tsuPIRm(.t))Mtt-00

then there exist C < oo, 5 > 0, and TE < oo depending only on n such that

(8.26) diam (M', g(t)) < CtI/2-a

for t > T. Setting

S sup Ct sup IRm t)I < oo,MnX

0,00)

Mn

the claim implies that for all t E (0, oo),

sup IRm (', t) I [diam (Mn, g(t))]2 <SC2t-21

Mn

hence that Mn is a nilmanifold, because SC2t-26 -i 0 as t ----> oo.To prove the claim, let 'y : [a, b] -f Mn be a fixed path and put

L (t) - length ry.g(t)

Then

by Corollary 8.24, and so

dL f Re (T, T) dsdt

dLdt

IRcIg(t) ds.

5. DILATIONS OF INFINITE-TIME SINGULARITIES 249

If (8.25) holds for some e > 0, there exists a time TE < oo such that for allt > TE we have t supMn IRm t) I < 2E. In particular,

EMn

for all t > TE, where C depends only on n. Hence

dt (T)

CE< L(T)

for T > T. Integrating this inequality from time TE to time t > TE implies

C-CE tCEL (t) < L (TE)

TE J

CE

=L (TE)TE

In particular, if we choose 0 < E\< 1/ (2C), then any two points in (M n' . (t))can be joined by a path of length

L (t) < [diam (.M", g(TE))] TECE . tl/2-a

where 8 - 1/2 - CE > 0. This implies (8.26) and proves the claim.

Because 3-dimensional nilmanifolds are known to be geometrizable, wenow describe how to obtain Type III limits under the assumption a > 0.By definition of a, there exist sequences (xi, ti) with ti / oo such that

ti IRm (xi, ti) I - ai - a.Choose any such sequence. Also by definition of a, there is for any e > 0 atime TE E [0, oo) such that

tIRm(x,t)I <a+Efor all x E Mn and t E [Ti, oc). The dilated solutions gi (t) exist on thetime intervals [-ai, oo) and satisfy

IRm i (x, t) I = IRm (xi,ti) I Rm x, ti + IRm (xi, ti) I

Rm I x, ti + Rmt

,ti)) (t2 + Rm xt,ti

IRm (xi, ti) l ti + t< a+E

ai+t'if

ti + t IRm (xi, ti) > T,that is if

t > I Rm (xi, ti) I (TE - ti) .

Note that for any fixed e > 0, we have

IRm (xi, ti) (TE - ti) -f-aand a+E a+E

ai+t a+t

for t > -a. Hence the pointed limit solution g00 (t) , x00), if it exists,is defined for t E (-a, oo) and satisfies

sup IRm 00I :5 = IRm (x00, 0) IaMnx(_0 oo) a + t

6. Taking limits backwards in time

Assume that we have taken the limit of a sequence of dilations andobtained a Type II ancient solution (Mncc, g,, (t)) for t E 00) w), where

(8.27) sup tI IRm oo (x, t) I = cc.Mn X(-0C'0]

Type II ancient limit solutions typically arise from taking a second limit,as in dimension reduction. However, such limit solutions might not attaintheir maximum curvatures, hence might not satisfy condition (8.1). Whenn = 3, these limit solutions have nonnegative sectional curvature, whichallows for the application of the Harnack estimate. But to combine theHarnack estimate with the strong maximum principle, it is necessary thatthe limit solution attain its maximum scalar curvature. This motivates usto take a another (third) limit to obtain a solution where this maximumis attained. We can do this by translating and dilating about a suitablesequence of times ti \ -oo and points xi E .M' where the curvaturesalmost attain their maximums.

So let (M.,g00(t)) be a complete Type II ancient solution defined fort E (-oo,w). Let ej < 1 be any sequence of constants such that Ei \ 0, andchoose any sequence of times Ti -> -oo and corresponding points and times(xi, ti) E M' x (Ti, 0) such that(8.28)

til(ti-TT)IRm0(xi,ti)I > (1-ei) sup I t I (t - TT)IRm0(x,t)IMoo x [Ti,O]

This is very similar to how we chose the points and times in the case of aType lib singularity. Set

ai (ti -Ti) IRm00 (xi,ti)I > 0,Wi = -ti I Rm 00 (xi, ti) I > 0,

and observe that limi.00 ai = limi.00 wi = oo. Indeed, by assumption(8.27) and estimate (8.28), we have

1 _ aiwi Iti l (ti - Ti) IRm 00 (xi, ti) l1/ai+1/wi ai+wi ITii

uItl ->oo> (1 - Ei)Mgx[TT,0 ITi I

asi ->oo, because T2 -+ -oo.Now consider the dilated solutions

(9o0)i (t) = IRm oo (xi, ti) I ' 90o ti +t

C IRm oo (xi, ti) I

defined as before. Each solution (g... )i exists on the dilated time interval

(-oo, (w - ti) IRmoo (xi,ti)I)

which contains the subinterval [-ai, wi] on which we have a good estimate ofthe curvature. Indeed, (8.28) implies that for all x E .M' and t E I-ai, Willone has

(t+1 t(Rmoo)i(x,t)I = Rm0

I oo (xi, ti) I IRm. (xi, ti)

(ti - Ti) I tilt l(1 - Ei) (ti + IRmoo xi,ti - TT)

aiwiti + Rm oo(xi,ti)I

(1 - Ei) (ai + t) (wi - t)Because ai --> oo, wi --* co, and Ei \ 0, we conclude that the pointed limitsolution 92co (t) , is an eternal solution that satisfies

sup IRm 20o I < 1 = IRm 200 (x2oo, 0) IA4 2x(-W,00)


The main reference for this chapter is Section 16 of [63]. Limit solutionsare obtained by the application of the Compactness Theorem we reviewedin Section 3 of Chapter 7. Limits formed from dilations about sequences ofpoints and times that are not curvature essential will be considered in thenext volume.

CHAPTER 9

Type I singularities

Suppose that (.M3, g (t)) is a solution of the Ricci flow which becomessingular at some time T < oc. In order to extract geometric and topolog-ical information about M3 from the solution g (t), one wants to performgeometric-topological surgeries on M3 as the singularity develops in sucha way that the maximum curvature of the solution is reduced sufficiently.In order to perform such surgeries, one needs a good understanding of thepossible singularities that can arise and of the possible limits of dilationsabout them. Ultimately, one then wants to argue that only geometricallyrecognizable pieces will remain after finitely many surgeries.

This is the program for proving the Geometrization Conjecture that wasdesigned by Hamilton and advanced by Perelman [105, 106, 107]. We willfurther discuss the details of this program in a successor to this volume. Inthis chapter, we will begin to explore some of the reasons why singularitiesare expected to be topologically and geometrically tractable in dimension 3,and why near a `typical' singularity, one expects to see a neck: a piece of themanifold which is geometrically close to a quotient of the shrinking roundproduct cylinder. This heuristic notion will be made precise in Section 4below. (Recall that we made a rigorous study of neckpinch singularitiesunder certain symmetry assumptions in Section 5 of Chapter 2.)

1. IntuitionSingularities of the Ricci flow (1Vl', g (t)) in high dimensions are ex-

pected to be very complex. In dimension n = 3 however, there are threeobservations that lead one to expect singularities to be relatively tractable.

The first observation is the pinching estimate of Theorem 9.4, below.This estimate says that at any point and time where a sectional curvatureis negative and large in absolute value, one finds a much larger positivesectional curvature. It implies in particular that any singularity model musthave nonnegative sectional curvature at t = 0.

The second observation which restricts the possible singularity modelsone may see in dimension n = 3 is the fact that so (2) is the only propernontrivial Lie subalgebra of so (3). This fact says that at the origin of anysingularity model, the eigenvalues of the rescaled curvature operator mustconform to either the signature (+, +, +) or else (0, 0, +). The signature(0, +, +) is ruled out by the Lie algebra structure of so (3). The other

253

254 9. TYPE I SINGULARITIES

possible pattern, (0, 0, 0, ), may be ruled out by choosing a dilation sequenceso that (8.9) holds, thereby ensuring that the model one obtains is not flat.

The third observation is the fact that a strong maximum principle holdsfor tensors. (See Chapter 4.) Because the curvature operator of any singu-larity model (M3, (t) , xc) is nonnegative at t = 0 and has either thesignature (+, +, +) or else (0, 0, +) at the single point (x,,,,, 0), the strongmaximum principle says that the curvature operator must possess either thesignature (+, +, +) or else (0, 0, +) respectively at all points x E M3 and00

times t > 0 such that the limit solution g,,,, (t) exists.

The standard example of a singularity of signature (+, +, +) in dimen-sion 3 is the self-similar solution formed by the shrinking round 3-sphere

(9.1) 4(T-t)gcan,

which we encountered back in Section 5. Singularities of signature (0, 0, +)are expected to resemble the self-similar `cylinder solution'

(9.2) ds2+2(T-t)gcaI,

on R X S2. We saw rigorous examples of such singularities and their asymp-totic behaviors in Section 5.

When viewed at an appropriate length scale, the geometry of a solutionto the Ricci flow (.M3, g (t)) that becomes singular at time T < oo closelyresembles the singularity model (M', goo (t), x.) one obtains by blowing-up the singularity, at least for points near the singularity and times justbefore T. The intuitions outlined above, therefore, support the followingheuristic picture. This picture is intended as a guide to our intuition in theremainder of this chapter.

(1) If a solution (M3, g (t)) of the unnormalized Ricci flow encountersa Type I singularity, then after performing dilations correctly andobtaining an injectivity radius estimate, one expects to see a limitwhich is a quotient of either(a) a compact shrinking round sphere (S3, g (t)), where g (t) is

given by (9.1), or(b) a noncompact shrinking cylinder (R x S2, g (t)), where g (t) is

given by (9.2).(2) If a solution (M3, g (t)) of the unnormalized Ricci flow encounters

a Type Ha singularity, then after performing dilations correctlyand obtaining an injectivity radius estimate, one expects to see aquotient of one of the following noncompact limits:(a) a steady self-similar solution (1R3, g (t)) of positive sectional

curvature,(b) a shrinking cylinder (JR x S2, g (t)) as in Case lb above, or

2. POSITIVE CURVATURE IS PRESERVED 255

(c) a cigar product (1183, g (t)), where g (t) is the self-similar solu-tion corresponding to the soliton metric introduced in Subsec-tion 2.1.

For any limit in Case 2a, one could perform a technique called dimen-sion reduction in the hope of obtaining an ancient solution. (Dimensionreduction is introduced in Section 4 below and will be discussed further in aplanned successor to this volume. It involves taking a limit around a suitablesequence of points tending to spatial infinity.) If the ancient solution oneobtains is not in fact an eternal solution which attains its maximum curva-ture, one can then take a third limit about a suitable sequence of points andtimes tending to -oo where the curvature is sufficiently near its maximum.Having done so, one expects to see either Case 2b or 2c above.

If one were in Case la, then one could conclude from compactness of thelimit that the underlying manifold of the original solution must have beenS3 or one of its quotients. In the other cases, the singularity model shouldgive local information about the original solution near the singularity justprior to its formation.

REMARK 9.1. The recent work [105] of Perelman rules out Case 2c forsingularities that form in finite time.

2. Positive curvature is preserved

In this chapter, we shall make heavy use of the maximum principles forsystems discussed in Section 3 of Chapter 4. In so doing, we will followthe methods introduced in Section 4 of Chapter 6. In order to review thesetechniques, we will first apply them to derive two results which were obtainedearlier by classical methods in Sections 1 and 3 of Chapter 6.

LEMMA 9.2 (Positive sectional curvature is preserved.). If (M3,g(t))is a solution of the Ricci flow such that the initial metric g(O) has posi-tive (nonnegative) sectional curvature, then the metrics g(t) have positive(nonnegative) sectional curvature for all t > 0 that the solution exists.

PROOF. We show that positive sectional curvature is preserved at eachx E M3; the proof that nonnegative sectional curvature is preserved isentirely analogous. Let A (IF) > i (IF) > v (P) denote the eigenvalues of any

P E (n2T*M3 ®s A2T*M3)x.

Let M (t) E (A2T*M3 ®s n2T*M3)X be the quadratic form correspondingto Rm [g]. Define the subset 1C C (A2T*.M3 ®s A2T*M3)X by

K={P:v(P)>0},and notice that M (t) E K if and only if all sectional curvatures are positiveat (x, t). It is easy to see that K is invariant under parallel translation. Kis convex in each fiber because if

p : (A2T*M3 ®S A2T*M3)X --> ](8


by

p:PH minP(V,V),VI=1

then v (IP) = p (P) and

p(SIP +(1-s)Q) > sp(IP)+(1-s)p(Q).The fact that M remains in IC under the ODE (6.31) follows from (6.32),because

dtv=v2+Aµ>0

whenever v > 0. The lemma now follows from the maximum principle forsystems, which says that if Rm (x, 0) > 0, then Rm (x, t) > 0 for all t > 0such that the solution exists.

The fact that positive sectional curvature is preserved in dimension 3 is aspecial case of the more general result that positivity of the curvature oper-ator is preserved in all dimensions. A result which is special to 3 dimensionsis the following:

LEMMA 9.3 (Positive Ricci curvature is preserved.). If (.M3, g(t)) is asolution of the Ricci flow such that the initial metric g(0) has positive (non-negative) Ricci curvature, then the metrics g (t) have positive (nonnegative)Ricci curvature for all t > 0 that the solution exists.

PROOF. The proof is analogous to the previous lemma. Here we defineIC by

IC ={IID:v(P)+It (P)>0},so that the quadratic form M corresponding to Rm [g] lies in 1C if and onlyif Rc [g] > 0. 1C is fiberwise convex, because the function

v (IF) + µ (P) =I V1 min

[P (V, V) + P (W, W) l(V,W)=0

is concave in each fiber. If v + µ > 0, we have A > (v + µ) /2 > 0 andhence

d

dt(v+µ) = v2+Aµ+µ2+Av> A(v+µ) > 0.

The lemma follows again from the maximum principle for systems.

3. Positive sectional curvature dominates

The Ricci flow prefers positive curvature, in the sense that it preservespositive curvature operator in any dimension. Moreover, we saw in Section2 that both positive sectional curvature and positive Ricci curvature arepreserved in dimension 3. Thus it is natural to investigate the extent towhich the sectional curvatures of an arbitrary initial metric tend to becomepositive under the flow. By the maximum principle, it suffices to study thecorresponding system (6.32) of ODE. We are interested in the case that thesmallest eigenvalue v is negative. To simplify and motivate the qualitative

3. POSITIVE SECTIONAL CURVATURE DOMINATES 257

study of (6.32), we first consider the subcase that = v < 0. We then havethe following system of two ODE:

dt

A2 + v2

dv 2

dt = v + Av.

For convenience, we set it = -v > 0 and consider the system

dt = A2 + 7r2

d7r

dt = A7 - 7r2

under the assumption that A > v = -7r initially. Since

dt(A + 7r) = A (A + 7r),

this condition is preserved. Similarly, it > 0 as long as the solution exists.If A < 0, then d7r/dt < -7r2 < 0. If it > A, then d7r/dt = 7r (A - 7r) _< 0

but dA/dt > 0. Hence it is non-increasing except possibly when it < A,and it follows that it < max {A, C}. To improve on this estimate, note thatdA/dt > 0, so that we may write

d7r A7r - 7r2TA T2 + 7r2

This is a homogeneous equation. The standard substitution it = A gives

d _ 2

+AdA 1+c2Using partial fractions and integrating, we obtain

which yields

log JAI ---log (1+2log 11+(1=C.

Since (= 7r/A and A + it > 0, we may write this in terms of A and it as

I + C.log 7r = A + 2 log l 7r

If it (t) is sufficiently large and positive, this leads to a contradiction unless

A (t) > 7r (t) log 7r (t) .

In particular, A is positive and much larger than it. On the other hand, if itis sufficiently small and positive, we get a contradiction unless -7r < A < 0.

This discussion motivates the following theorem, which reveals the pre-cise sense in which all sectional curvatures of a complete 3-manifold evolv-ing by the Ricci flow are dominated by the positive sectional curvatures.

Namely, large negative sectional curvatures can occur only in the presenceof much larger positive sectional curvatures:

THEOREM 9.4. Let (M3, g (t)) be any solution of the Ricci flow on aclosed 3-manifold for 0 < t < T. Let v (x, t) denote the smallest eigenvalueof the curvature operator. If infxEM3 v (x, 0) > -1, then at any point (x, t) EM3 x [0, T) where v (x, t) < 0, the scalar curvature is estimated by

R> IvI (logvj + log (1 + t) - 3).

Note that one can always achieve infXEM3 v (x, 0) > -1 simply by scalingg (0) by a sufficiently large constant. Note also that if v < -e6, we have

R > 2 Ivl log Jul.

In particular, R > I v I when v << -1. Thus whenever we encounter a largenegative sectional curvature at some point and time (x, t), we find a muchlarger positive sectional curvature at the same point and time. As we shallsee, this estimate implies that limits (if they exist) of sequences of dilationsabout a finite-time singularity have nonnegative sectional curvature.

Before proving the theorem, we establish two technical lemmas.

LEMMA 9.5. At each fixed time t, the subsets

K C (A2T*M3 ®s A2T*M3)X

defined by

trIP>-3(1+t),

and if v (IP) < -1/ (1 + t) , then

tr P > Iv (P) l (log Iv (P) l + log (1 + t) - 3)

are invariant under parallel translation and are convex in each fiber.

PROOF. Invariance under parallel translation is clear. To prove convex-ity at each fixed x E M3 and t E [0, T), first consider the region K C R2given by

v>-3(1+t),

v > -3u,u, v)

and if u > 1/ (1 + t) , then

v > u (log u + log (1 + t) - 3)

See figure (1) and notice that all three curves shown intersect when u =1/ (1 + t). The region K is bounded on the left by the line v = -3u,below by the line v = -3/ (1 + t), and on the right by the curve v =

2u (logu + log (1 + t) - 3). It is clear that K is a convex subset of 1R.

V

U

FIGURE 1. Boundary of the region K in the (u, v)-plane.

Now consider the map J : (A2T*.M3 ®S n2T*M3) --> R2 defined by

(D : IF ---> (u (IF) , v (IP)) - (Ivj , trIP)

and observe that IF E 1C if and only if 4) (IF) E K. (The condition v > -3uis satisfied automatically, because tr IP > 3v (I).) Thus the proof will becomplete if we show that (P (SIP + (1 - s) Q) E K for all IP, Q E K andS E [0, 1]. Since trace is a linear functional,

v(slln+(1-s)(Q) =sv(IF)+(1-s)v(Q).

Because v > -3u and the function -u (IF) = minjW1=1 P (W, W) is concave,

(93) v(sP+(3 - s)Q) < u(SIP +(1-s)Q) <su(P)+(1-s)u(Q).

Together, these relations show that (D (sIP + (1 - s) Q) E H, where H is therhombus with vertices (u (F) , v (IF)), (u (Q) , v (Q)), (-3v (F) , v (IF)), and(- 3v (Q) , v (Q)). Because each vertex of H lies in the convex set K, wehave HCK.

LEMMA 9.6. Given any x E M3 and quadratic form

IF E (n2T*M3 (9s )X

with eigenvalues A (P) > y (IF) > v (F), where v (IF) < 0, define

S2 (I) -v (F) - log (-v (IID))

Let M (t) correspond to Rm [g]. Then wherever S2 (M) is defined,

-11(M) > -v (IMI) .

PROOF. By (6.32), we calculate

v2 dt S2 (1M1) _ -vt

(tr M) + (tr M) dt v - v v

_ -v t (A+y+v)+(A+µ) tv=-1/3-v(A2+L2+Aµ)+(A+µ)Aµ.

Thus it suffices to prove that

7r-v(A2+112+Aµ)+(A+µ)Aµ>0whenever v < 0. There are two cases: if µ < 0, then we write

7r = (µ - v)(A2 + µ2 + aµ) - µ'3 > -µ3 > 0,

while if µ > 0, we have7F =A2(JU -v)-vµ2+iµ(µ-V)>0.

PROOF OF THEOREM 9.4. Fix any x E M3 with v (x, 0) < 0. Let M bethe quadratic form corresponding to Rm [g], and let

,C C (A2T*.M3 ®g A2T*.M3)

be the time-dependent set defined in Lemma 9.5. Then since

k (0)tr IP > -3 andtrP> v(IID)I (log{v(P)l -3) if v(P) < -1 }'

we have M (0) E IC by the hypothesis inf v 0) > -1. We claim M(t)remains in IC as long as v < 0. The theorem follows directly from the claim,which implies that

-3Rlog(-v)> lit - log 1+t) log(l+t)-3I+t

when 0 < -v < 1/ (1 + t), and

R > log (-v) + log (1 + t) - 3-vwhen -v > 1/ (1 + t).

The proof that M (t) E IC uses the maximum principle for systems. Theinequality trM > -3/ (1 + t) is preserved, because

d (tr 1071) - 3 (tr 1M1)2 >3 (A2 + µ2 + v2) > 0.

And if IvI _> 1/ (1 + t), then Lemma 9.6 implies that Q (M) > _1/ (1 + t),dtwhich is equivalent to the inequality

dt

[trMd _ log lvl-log (l + t) > 0.IvI -

COROLLARY 9.7. In dimension n = 3, every Type I limit of a Type I sin-gularity or Type II limit of a Type Ila singularity has nonnegative sectionalcurvature for as long as it exists.

PROOF. Let (,M3, g (t)) be a singular solution with blow-up time T <no. As in Section 3.1 of Chapter 8, let (xi, ti) be a sequence of points andtimes chosen such that ti / T and

Rm (xi, ti) c2 sup IRm (x, ti) I > c2c1 sup IRm (x, t)I ,

xEM3 (x,t)EJV13 X [se,ti]

where (ti - si) supXEM3 I Rm (x, ti) I --- no. Suppose that a subsequence ofthe pointed solutions (M3" q, (t) , xi) defined by

gi(t) Rm(xi,ti)I'g ti+ IRm(xi,ti)I)

converges to a complete pointed solution (M00, g00 (t) , x...) of the Ricci flow.Then by Proposition 8.20 and (8.18), g,,o (t) is an ancient solution satisfying

sup Rm (x, t) < ooxEM3 00

for all t < 0. By Corollary 9.8, the sectional curvatures of g00 are nonnega-tive.

Another consequence of the pinching estimate in Theorem 9.4 is thatancient 3-dimensional solutions of the Ricci flow have nonnegative sectionalcurvature.

COROLLARY 9.8. Let (M3, g (t)) be a complete ancient solution of theRicci flow. Assume that there exists a continuous function K (t) such thatIsect [g (t)] [ < K (t). Then g (t) has nonnegative sectional curvature for aslong as it exists.

PROOF. If (M3, g (t)) is a solution of the Ricci flow that exists at leastfor 0 < t < T with

0 > vo = inf v (x, 0) ,

xEM3

then the family of metrics g (t) defined on M3 by

Ivo[

is a solution of the Ricci flow with v > -1 at t = 0. So at any (x, Ivol t) EM3 x [0, T), Theorem 9.4 gives the estimate

R(x, IvoI t) > v(x, Ivol t)I. [log lv(x, IvoI t)I+log(1+IvoI t) -3]

wherever v (x, Ivol t) < 0. (Maximum principles work on noncompact aswell as on compact manifolds. We will provide a complete proof of this fact

in the sequel to this volume.) In terms of the solution g (t), this shows that

R(x,t) lv(x,t)l [log ((xMI) +log(1+vol t)-3]

_ v(x,t) [log lv(x,t)+log(jvo _I+t) -3]

wherever v (x, t) < 0. In particular, if v (x, t) < 0 at some point x E M3and time t > 0, then

R (x, t) > I v (x, t)l (log I v (x, t) l + log t - 3).

Now if g (t) exists for -a < t < w, we can translate time to conclude that

R (x, t) > Iv (x, t) j (log lv (x, t) I + log (t + a) - 3)

wherever v (x, t) < 0. But if g (t) is ancient, this leads to a contradiction,since lima log (t + a) = oo. Hence v (x, t) > 0 for all x and t.

4. Necklike points in finite-time singularities

Suppose (M3, g) is a closed solution of the Ricci flow which forms asingularity at T < oo, in other words, a singularity of Type I or TypeIIa. The main result of this section asserts that either M3 is a sphericalspace form or else M3 admits a `dimension reduction' somewhere near thesingularity, in the sense that at any time sufficiently close to T, we can find apoint in M3 near which the geometry is arbitrarily close to the product of asurface and a line. When the singularity is Type I, the surface arising from alimit of dilations about a subsequence of such points and times approachingT is a constant-curvature 2-sphere. In this case, the limit is a quotient ofS2 x R, and we say that M3 develops a `neck.' (See Section 5 Section ofChapter 2.)

To make the notion of dimension reduction precise, we shall say that(x, t) is a Type I c-essential point if

IRm(x,t)I>Tc t>0.

We say that (x, t) is a S-necklike point if there exists a unit 2-form 0 at(x, t) such that

lRm -R (0 ®0)1 < S IRml .

THEOREM 9.9. Let (M3, g (t)) be a closed solution of the unnormalizedRicci flow on a maximal time interval 0 < t < T < oo. If the normalizedflow does not converge to a metric of constant positive sectional curvature,then there exists a constant c > 0 such that for all T E [0, T) and S > 0,there are x E .M3 and t (-= [T, T) such that (x, t) is a Type I c-essential pointand a J-necklike point.

Observe that the theorem is equivalent to the following statement, whichwe will prove:

4. NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 263

If for every c > 0, there exist some -r < T and 6 > 0 suchthat there are no Type I c-essential 6-necklike points aftertime r, then the normalized solution g (t) converges to aspherical space form.

Our strategy will be to modify the second proof of Theorem 6.30 in orderto show that the pinching of the curvature operator improves sufficientlyunder the flow when this hypothesis holds. Recall that we considered thefunction

0

I2(9.4) f RmE

and saw that f stays bounded for E sufficiently small when (.M3, g) hasstrictly positive Ricci curvature. The first difficulty we face is that we nolonger know R > 0. But if we choose constants p > 0, c > 0 so thatR + p > c > 0 at t = 0, we have R + p > c for 0 < t < T by the maximumprinciple, because

a (R+p) = A(R+p)+2IRcl2.

Thus we may replace R by R+p in the denominator of (9.4). This introducesan additional bad term in the evolution equation for (9.4). To remedy this,we introduce the time factor (T - t)E and define

0 0

(R)2.(9.5) F _ (T - t)E

IRm P)2

2_E= [(T - t) (R + p)]-' (Rmp2

Since we must allow for arbitrary initial metrics, we shall have to workharder to show that F stays bounded. However, the multiplicative factor(T - t)E helps, because its time derivative is negative. In fact, a key step inproving the theorem will be to show that F --> 0 as t --* T if (M3, g) has no

essential necklike points. Notice that (R + p)-2 Rm12 is scale invariant ifwe ignore the constant p, and that (T - t) (R + p) is invariant under TypeI resealing.

Our first step is to derive an analog of Lemma 6.34.

LEMMA 9.10. There exists a constant C = C (g (0)) such that

IRmI <C(R+p).

PROOF. We may assume without loss of generality that v -1.

Set A = e6 and B = A/c, so that B (R + p) > A. Let N denote thenumber of negative sectional curvatures at a point (x, t). There are twocases: If -A < v, it is easy to see that

IRmI c IvI+IpI+IAI <_ v+ti +A+2NA<(1+2NB)(R+p).

If v < -A, then Jul < R/3 by Theorem 9.4; so 1 < N < 2, and we have

IRml < IvI+I.I+ Al <v+µ+A+2NIvI < (1+3N)R.

LEMMA 9.11. The function F satisfies a differential inequality of theform

tF<OF+2(1+P) (VF,VR)+H,

where

0

H+(R

(T-is-E [(Bi+B2_Ci)IRmlmI2_G2].+p)

Here, B1 _ Cip and B2 C2E IRmI are bad terms, while G1G2 = 2P are good terms. The function P is defined in (6.89).

PROOF. Recall that

IRmI2 3 [(A _A)2 + ( - v)2 + (µ - v)2]

and set

andTt

A3+µ3+v3+3Aµv- (,\2E.t+A2v+µ2,\+µ2v+v2,\+v2µ)

= 2RIRmI2(µ-v)2-µ(a-v)2-v(A-µ)2.

0

Taking cp = (T - t)E I Rml2, = R + p, a = 1, and j3 = 2 - E in Lemma6.33, we get

0£ '12 2

F=OF {- (T - t)(_2lVRmI2__IRmIat (R + p)2-E 3 T - t

0T-tE Rm2

() I3 E (,\2+µ2+v2+'\µ+\v+µv)

(R + p) -

- (2 - e) (3 - ) (T - t)E IRmI2 IVRI2

(R + p)4-E

+ 2 (2 - e)(R+ p)3e E

(VIRmI2, VR) .

Collecting terms as in Lemma 6.34, we obtain

tF = AF + 2R1+ p) (VF, OR)aa0 0 2

(R+p)(VRm)-VR®Rm

+E (1 - E) IRmI2 IVRIz

£EIRmI2IRc2-P 6IRm12 4(T-t)EpE+ 2 (T - t) (R +p)3-E - (R + p)2-E (T - t) '-c + 3 (R + p)3-E

<AF+2(1-E)

R + p(VF, VR)

+ (T-t3EE (2,IRcI2-E(R+tp))l°m2+PE_2P].RI3(R+p)-

The result follows when we estimate IRC12 < IRmI2 < C (R + p)2 and

E < 2 (ICI + ICI + vI) IRm12 < 2 IRmI IRmI2

Notice that the bad terms Bi + B2 = Cip + C2e IRmI come from the1/ (R + p)2-E factor in F, whereas the good term -Gl = -E/ (T - t) comesfrom the (T - t)E factor. The good term -G2 = -2P would be the onlyterm present if p = E = 0.

The next two lemmas constitute the second step in the proof of Theorem9.9 and show that the pinching of the curvature operator improves if thereare no essential necklike points.

LEMMA 9.12. If IRm -R (9 (9 9) I2 > S IRmI2 for some S E (0, 1), then

jIRmI2IRmj2.

PROOF. We may assume without loss of generality that IaI ? Iµi ? IvIThe hypothesis implies that

µ2+v2+pv>2

(A2+µ2+v2),

and hence that i2 + v2 > 3& 2. Since IjI ? IvI by assumption, we have

P = Q2 (µ - v)2 + µ2 (A - v)2 + v2 (A - µ)2

A2 (A - v)2 + 2(3

S)A2 (A - v)2 + v2 (A _,U)2.

Now notice thatIRmI2=A2+42+v2<3A2

and

IRml2 3 [(A - u)2 + (A - v)2 + ( - v)2J <3

(A2 + 2 + v2) < 4A2.

So if vJ < IAI /2, we have

P > A2 (.1- v)2 >5

A4 >b IRmI2 IRmI2,

2(3-b) 8(3-b) 96(3-b)while if vl > JAI /2, we get

P>A2(lG-v)2+ b A2(A-v)2+1a2(A-µ)22(3-b) 4

2(3-b)A2 [(µ-v)2+(a-v)2+(A-FL)2,

>b

Rm12IRm12,2(3-b)

because 1 > 273--6)' 0LEMMA 9.13. Assume that for every c > 0, there exist T E [0,T) and

b > 0 such that for every x E M3 and t E [r, T), either

(9.6)

or

(9.7)

cIRm(x,t)I <

IRm-R(0®B)I > bIRmI

T-t

for every unit 2-form 9 at (x, t). Then F -+ 0 uniformly as t -> T.

PROOF. By Lemma 9.11, F satisfies the differential inequality

(9.8)8t F < OF + 2 R1+ P) (VF, OR) + H,

where

H =(R+

p)sE

L(B1 + B2 - Gi) IRmI IRmI2 - G2J

The proof is in three steps.We first show that H < 0; by the maximum principle, this will prove

that F is nonincreasing. Observe that for every E > 0 and p < oo, thereexists r E [0, T) such that

1 E 1BI _Ci p T

= Git 3

3

for all t E [r,T). Now if condition (9.6) holds at (x, t) with c = 1/ (3C2),then

1I <

-E

t1=3

GIB2 = C2,- IRm(x, t)3 T

On the other hand, if condition (9.7) holds at (x, t), Lemma 9.12 impliesthere is ij = 77 (8) such that

P> 'q Rm12 IRm12,

whence it follows that choosing e < yields2c

B2 (Rml Rml2I =C2EIRm12IRm12<P=2 G2.

In either case, we have used at most 2/3 of GI and at most 1/2 of G2.Therefore

(9.9) H<- (T-Erl E jRmHHRmI2+PJ <0.(R+P)3--, 3T-t

We next improve estimate (9.8) Recall that either (9.6) or (9.7) holds at(x, t). In the first case, the inequality R < C3 IRml implies there is T < Tsuch that (R(x, t) + p) (T - t) < 2cC3 for t E T), whence we get

FH<-(T-t)(R+p) L3IRmIJ6cC3jRmjF

from (9.9). If (9.6) fails, then (9.7) holds and we have both conditions0

1Rm (x, t) I > c/ (T - t) and P > 77 1 Rm12 Rm12. Thus by our choice of T, weobtain

H < - (RTP)3-E 77

IRm121 0

Rm12I

--77 F<-77 IRrnjc F< - IRmlF.(R + p) (R + p) (T - t) - 2C3

Hence in either case we have

H < -C (8, c, C2, C3) IRmI F,

and thus

atF<AF+2(1+p) (VF, VR)- C IRml F.

Now we can show that F actually tends to zero uniformly. Since Fis bounded above, we may define Finax (t) _ SupM3 F (., t). Recall that

in dimension 3, there is a constant A < oo such that IRm12 < IRm12 <A (R + p)2. Let a > 0 be given, and consider any (x, t) with 7- < t < T.Observe that (T - t) (R + p) (x, t) < a only if

2

F (x, t) = [(T - t) (R + p)]E

(R0+

p)2 < AaE.

Thus if F (x, t) > AaE, we must have

jRm>c(R+p)>Tat

and henceOF <AF+2(1-E) (VFVR)- CcaF.at R+p T -t

So the maximum principle implies that whenever Finax (t) > Aae at somet E [r, T), we have

dt[(T - t)-Cca Finax (t)] < 0

in the sense of Jim sup of forward difference quotients. In particular, we get

d CcAal+E QdFinax_- T-t T-tif Finax (t) > Aat. Since F is nonincreasing, this implies that for all t E1,r, T), we have

Finax (t) < max AaE, Fax (T) + logT-tT - T

It follows that Finax (t) < Ac for t sufficiently close to T.

To finish the proof of Theorem 9.9, we need to verify that the solutionsatisfies an injectivity radius estimate on the scale of its maximum curvature.This follows from Perelman's No Local Collapsing result (Theorem 8.17).Armed with this fact, we now proceed to the final step of the proof.

PROOF OF THEOREM 9.9. We are finally ready to take a limit of dila-tions about the singularity at time T < oo, even though we do not knowwhether the singularity is of Type I or Type Ha. In either case, choose asequence (xi, ti) as in Section 3.1 of Chapter 8, taking care that the curva-ture at each (xi, ti) is uniformly comparable to its maximum at time ti. Forco E (0, 1), set

Mi° _ IX E M3 : IRm(x,ti)I > comaxM3IRm(.,ti)I}.

Then by Lemma 8.19, there is cl > 0 such that for all x E .Mio

I Rm (x, ti) I >cool

T - tiThat is, every point in M?° is Type I c0c1-essential. For such points, wehave (T - ti) (R (x, ti) + p) >- c > 0 by Lemma 9.10.

By the compactness theorem stated in Section 3 of Chapter 7 and Corol-lary 9.7, the pointed sequence (M3, gi (t) , xi) limits to a complete ancientsolution (M3, g"', (t) , x") of the Ricci flow having bounded nonnegativesectional curvature. Since we assumed there are eventually no Type I c-essential b-necklike points, the limit actually has positive sectional curva-ture. By construction of that limit, there is for every point y E M3 a00

subsequence of points {yi} in M3, orthonormal frames {.Fi} at (yi, ti), andan orthonormal frame Y,,, at (y, 0) such that

lim Rm (yi, ti) _ Rmoo (y, 0) ,

i oo IRm (xi, ti) I

where Rm,,, - Rm [gam] and the curvature operators are represented in therespective orthonormal frames. Because Rm., (y, 0) > 0, we have yi E Mi°for some co > 0 independent of i. Because ti --> T, Lemma 9.13 implies that

0

lim [(T - ti) (R (yi, ti) + p)]E IRm (yi, ti)122 = 0Z-'OO (R (yi, ti) + p)2

Then since (T - ti) (R (yi, ti) + p) > c > 0 and R (yi, ti) --r oo, we concludethat

0 0

IRm (yi, ti) I2 IRmoo (y, 0) I20= lim -i-'°° R (yi, ti)2 (R. (y, 0))2

Since V E M3 was arbitrary and R... > 0, this implies that Rm,,, 0) - 0.Then Schur's lemma implies that g,, (0) is a metric of constant positivesectional curvature on M. By Myers' Theorem, M3 is compact, hencediffeomorphic to M3. It follows that g (t) has uniformly positive sectionalcurvature for t close enough to T, hence that g (t) limits to a metric ofconstant positive sectional curvature. O

REMARK 9.14. Mao-Pei Tsui has suggested an alternate proof of the factthat F remains bounded if for every c > 0, there are r < T and 5 > 0 suchthat no c-essential 8-necklike points exist after time r. The argument usesthe enhanced maximum principle for systems (Theorem 4.10) and resemblesour first proof of Theorem 6.30. It is simpler than but basically equivalentto the argument given above in Lemma 9.13.

ALTERNATE PROOF THAT F IS BOUNDED. Recall that A > µ > v andthat A+µ+v+p=R+p>-c>0. Since

dt(A+µ+v+p) =A2+µ2+v2+AM +Av+µv,

we have2 2

dtlog(A+µ+v+p)=A+µA+ +v+ PAµ p

Fix a point where A > v and define

a) - (T - t)E -A-v

Then since A log (A - v) = A + v - µ, we compute_ 2 2 µv

p

We claim there is a constant r < oo depending only on g (0) such that

A <A+p+v+p-Indeed, since the left-hand side is scale invariant, we may assume withoutloss of generality that

min v (x, 0) > -1.xEM3

Let A = J. If -3A < A < 3A, thenA 3A

A+p+v+p - cIf not, then by Theorem 9.4 we have either v > -A and A > 3A or elsev < -A and 3 jvj < R < A; in either case,

A< < 3

A+p+v+pA/3+p.

Now fore > 0 to be chosen later, we may by the claim above choose TElarge enough so that when TE < t < T, we have

(1-E)pA+p+v+p

-(1-E)pr< 2(T-t)and hence

dlogy < EA -

E 1 - E p2 + v2dt 2(T-t) 2 A+p+v+p*

By hypothesis, for every c > 0 there are 6 (c) > 0 and T, < T such thatM (t) 0 X (t) for Te < t < T, where

X (t) = {P: (T - t) A (P) > c> 0 and p (P)2 + v (P)2 < j. '\ (p)2}

is the avoidance set discussed in Section 3 of Chapter 4. Choose C (e, c) < 00large enough so that M (T) EK (T), where T = max I-, T} and

(T - t)E (A (P) - v (P))K (t) -(A (P) + p (P) + v (IF) + P )

< c .

Notice that K (t) is convex. We next claim that when c = 1/2, there is e > 0small enough that at log (D < 0 whenever T < t < T and M (t) E K (t) \X (t).This claim implies that M (t) remains in K (t) \X (t), hence that -1 < C. ByTheorem 4.10, this implies that F is bounded above, because

Rm 2 3 [(A - p)2 + (A - v)2 + (l-L - v)2] < (A - V)2.

To prove the claim, notice that M (t) 0 X (t) only if A (T - t) < c orp2 + v2 > 5A2. In the first case, we have

d <EA_ E - 1-E p2 +v2 <E a- 1 <0.dt 2(T-t) 2 A+p+v+p - 2(T t)

5. NECKLIKE POINTS IN ANCIENT SOLUTIONS 271

On the other hand, if A (T - t) > c, then p2 + v2 > 5A2 and we have_ 2

dtlog <EA - 2(T-t) 1 2

EA+p+v+pC0

for E > 0 small enough.

5. Necklike points in ancient solutions

The objective of this section is to show that an ancient solution of posi-tive sectional curvature is either isometric to a spherical space form or elsecontains points at arbitrarily ancient times where its geometry is arbitrarilyclose to the product of a surface and a line. We begin with some generalobservations about ancient solutions - including the fact that every ancientsolution (of any dimension) has nonnegative scalar curvature.

LEMMA 9.15. Let (Mn, g (t)) be a complete ancient solution of the Ricciflow. Assume that Rain (t) + infxEM. R (x, t) is finite for all t < 0 and thatthere is a continuous function K (t) such that sect [g (t)]I < K (t). Theng (t) has nonnegative scalar curvature for as long as it exists.

PROOF. Recall that the parabolic maximum principle works on noncom-pact as well as on compact manifolds. Recall also that

= AR+2 JRc12 > AR+ 2R2.at

So if R ever becomes nonnegative, it remains so for as long as the solutionexists. On the other hand, if Rmin (t1) < 0, then Rmin (to) < 0 for all to < ti,and one has

R (t)> n > - nmin

n(Rmin(to))-1 -2(t-to) 2(t-to)for all t > to. Since the solution is ancient, we may let to \ -oc, obtaining

toRmin (t) > - llim2 (t

nto) = 0.

REMARK 9.16. In dimension n = 3, we can strengthen this result, as weproved in Corollary 9.8.

In Lemma 8.19, we derived a lower bound for the blow-up rate of so-lutions that develop singularities in finite time. The analogous result forancient solutions is the following:

LEMMA 9.17. Let (Mn, g (t)) be an ancient solution with nonnegativeRicci curvature. Then there exists a constant co > 0 depending only on nsuch that

ltm inf JtJ sup IRm (x, t) I > co.xEMer

PROOF. We may assume that supXEMn IRm (x, t) I is finite for t < 0.Because Rc > 0, we can estimate the growth of the scalar curvature R by

atR=OR+2Rc12 <AR+2R2.

Hence Rmax (t) _ supXEMn R (x, t) < C (n) supXEMn IRm (x, t) I satisfies thedifferential inequality

dt(Rmax)-1 > -2.

So for all t < 0, we have

Rmax (t)

Letting t -> -oc, we obtain

1

(Rmax (0))-1 - 2t

1liminf (ItI Rmax (t)) >

2

which implies the result.

REMARK 9.18. Because nonnegative Ricci curvature is preserved onlyin dimension n = 3, one usually must assume that the curvature operator isnonnegative in order to apply the lemma. (See Section 2.)

We are now ready to state and prove our main assertion. We shall saythat (x, t) is an ancient Type I c-essential point if

IRm(x,t)l -Itl >c>0.

By modifying the proof of Theorem 9.9, we obtain the following:

THEOREM 9.19. Let (M3, g (t)) be a complete ancient solution of theRicci flow with positive sectional curvature. Suppose that

sup tryR(x,t) < ooM3 X (-oo,O]

for some y > 0. Then either (.M3, g (t)) is isometric to a spherical spaceform, or else there exists a constant c > 0 such that for all T E (-00, 0] and8 > 0, there are x E M3 and t E (-00, T) such that (x, t) is an ancient TypeI c-essential point and a 8-necklike point.

PROOF. We will show that if for every c > 0 there are T E (-oo, 0] and8 > 0 such that there are no ancient Type I c-essential 8-necklike pointsbefore time T, then (M3, g(t)) is a shrinking spherical space form.

By hypothesis on the ancient solution, there is ry > 0 small enough that

K _ sup Itl" R (x, t) < oo.M3 x (-oo,o]

5. NECKLIKE POINTS IN ANCIENT SOLUTIONS 273

(When ly = 1, K corresponds to the bound in the definition of ancient TypeI singularity models.) Since the scalar curvature of (M3,g(t)) is positive,the function

0G_ ItIryE/2 IRI2

is well-defined. Since the sectional curvatures are positive, we have theestimates

G < CRE It17Ei2 < CKE ItI -ryE/2 ,

which show that G is bounded for all times -oo < t < 0 and satisfies

(9.10) lim max G (x, t) = 0.t--4-oo

Note that our choice of G is similar to taking T = 0 and p = 0 in the functionF studied in Theorem 9.9, except that we now consider the possibly smallerpower (T - t)7 = Jt[7. This modification does not significantly harm theevolution of G: we will once again be able to show that if there are noessential necklike points, then there exists E > 0 small enough that G is asubsolution of the heat equation, hence that its maximum cannot increase.The advantage of the modification is that we can then use (9.10) to concludethat G - 0, hence that (.M3, g(t)) is complete and locally isometric to around S3, hence that it is compact and globally isometric to a sphericalspace form S3/F.

If there are no ancient c-essential S-necklike points on the time interval(-oo, T], then for every x E M3 and t E (-oo, T) either

(9.11) IRm (x, t) I ItI < c

or we have

(9.12) IRm-R(9®B)I > SIRmj

for every unit 2-form 0 at (x, t). Taking cp = (-t)7E/2 RmJ2, i = R, a = 1,and ,9 = 2 - E in Lemma 6.33 and estimating as in Lemma 9.11, we get thedifferential inequality

atG < OG +2(1R E)

(VG, VR) + 2J,

where

j- IR3- [eIRm2 (ii2 - 4ItI

PJ

and P > 0 is defined in (6.39). Fix any (x, t) with t < T < 0. If (9.11) holdsthere with c < -y/8, then we have

IRm12 - 4RItI - 41t)< R CIRmI -2- < R \ItI 41tjl ` -gRItIand hence

J < -8 tIG.

If (9.12) holds, then by Lemma 9.12 there is q (5) such that

P > 77 IRm12 1Rml2,

whence it follows that taking F < rl gives

IJ < 4G.Thus in either case, G is a subsolution of the heat equation for all times-00 < t < T, because

atG < OG+ 2(1R s) (VG,VR) - -L G.

6. Type I ancient solutions on surfaces

The objective of this section is to prove that the shrinking round 2-sphere is the only nonflat Type I ancient solution of the Ricci flow on asurface. Because of dimension reduction, this fact plays an important rolein the classification of 3-dimensional singularities.

Before we classify Type I ancient solutions on surfaces, we shall establisha result which is of considerable independent interest. It follows from a gen-eralization of the LYH differential Harnack estimates introduced in Section10 of Chapter 5. We will discuss the full family of LYH differential Harnackestimates for the Ricci flow in a chapter of the successor to this volume. Fornow, we simply state the following result from [61].

PROPOSITION 9.20 (LYH estimate, trace version). If (M', g (t)) is asolution of the unnormalized Ricci flow on a compact manifold with initiallypositive curvature operator, then for any vector field X on Mn and all timest > 0 such that the solution exists, one has

0 < atR+ R+2(VR,X)+2Rc(X,X).

COROLLARY 9.21. If (Mn, g (t)) is a solution of the unnormalized Ricciflow on a compact manifold with initially positive curvature operator, thenthe function tR is pointwise nondecreasing for all t > 0 that the solutionexists. If (Mn, g (t)) is also ancient, then R itself is pointwise nondecreasing.

PROOF. Taking X = 0, we have/\

a(tR)=t1 atR+R I >0

for all t > 0 such that the solution exists. This proves the first assertion.To prove the second assertion, notice that one has

atR+tRa>0

6. TYPE I ANCIENT SOLUTIONS ON SURFACES 275

whenever the solution exists for t E [-a, w). When the solution is ancient,letting a -> oo gives the result. O

In the successor to this volume, we will also prove the following state-ment.

LEMMA 9.22. If (N2, h (t)) is a complete ancient Type I solution of theRicci flow with strictly positive scalar curvature, then N2 is compact.

Assuming this for now, we will conclude this volume by characterizingall complete ancient solutions of the Ricci flow on surfaces. We begin witha classification of all complete ancient Type I 2-dimensional solutions.

PROPOSITION 9.23. A complete ancient Type I solution (N2, h (t)) ofthe Ricci flow on a surface is a quotient of either a shrinking round S2 orelse a flat 1R2

PROOF. Since (N2, h (t)) is an ancient Type I solution, it exists on amaximal time interval (-oo, w) containing t = 0 and satisfies a curvaturebound of the form

IR(x,t)I C,, IRm(x,t)I < tfor all x E M3 and t E (-oo, 0). By Lemma 9.15, R > 0 on N2 x (-oo, w).By the strong maximum principle, (N2, h (t)) is either flat and hence aquotient of ]E82 or else has strictly positive scalar curvature. From now on,we assume the latter.

By Lemma 9.22 (below) N2 is compact, hence diffeomorphic to eitherS2 or RJP2. By passing to the twofold cover if necessary, we may assumeN2 S2. The area A of Ar2 evolves by

LA- -J z RdA= -47rx (,lam) ,

Nwhere x (N2) is the Euler characteristic. Since A (t) -> 0 as t -+ w by [60],we have

A(t) =47rx(N2) (w-t) =87r(w-t).Recall (Section 8 of Chapter 5) that the entropy

E (h (t)) I Rlog (RA) dA

= J R log [R (w - t)] dA + 47rx (N2) log [47rX (N2)]N2

is a scale-invariant and nonincreasing function of time. Because (N2, h (t))is an ancient Type I solution, we have

sup R (x, t) (w - t) < C < oo,N2 x (-oo,w)

whence it follows that the limit

(9.13) E_00 lim E (h (t))t'- 00exists and is finite.

We want to use the compactness theorem to take a limit backwardsin time and get a new compact ancient solution of constant entropy. To dothis, we need a diameter bound. Since (N2, h (t)) is an orientable positively-curved surface such that

Rmax (t) = Nax R (-, t) < C I tl -1

for t C (-oo, 0), Klingenberg's Theorem (Theorem 5.9 of [27]) implies thereexists cl > 0 independent of time such that

inj (A2, h (t)) Cl V ICIRma(t) /2

Then the area comparison theorem (§3.4 of [25]) implies there is c2 > 0independent of time such that

47rX (A2) . (w - t) = A (t) > c2 . diam (Ar2, h (t)) inj (A I'2, h (t))

> C1 C2 diam (A2, h (t)) ,

hence C3 < oo such that

(9.14) diam (A2, h (t)) < C3

for all t E (-oc, -1].Now take any sequence of points and times (xi, ti) with

R (xi, ti) = Rmax (ti)

and ti \ -oo. Consider the dilated solutions

hi (t) - R (xi, ti) - h (ti + t / .R (xi, ti)

At each fixed x E A2, Corollary 9.21 implies that the function t H R (x, t)is nondecreasing in time. Thus the scalar curvature R of h satisfies

max R (x, t) = R (xi, ti)N2 x (-oo,ti]

and the scalar curvature Ri of hi satisfies

max Ri (x, t) = 1 = Ri (xi, 0).N2 x (-00,01

Then applying Klingenberg's Theorem again, we obtain the injectivity radiusestimate

inj (A2, hi (0)) > 1/2 = -,/2-7r.

By (9.14) and the Type I hypothesis, there is C4 Goo giving the diameterbounddiam (A2, hi (0)) = VI'R (xi, ti) diam (A2, h (ti)) < C3 R (xi, ti) I ti I < C4.

Hence by the Compactness Theorem (Section 3 of Chapter 7), there is asubsequence (N2, hi (t) , xi) that limits to an ancient solution

(, h-. (0), x-00)of the Ricci flow on a compact surface of positive curvature. By (9.13), thescale-invariant entropy of the limit satisfies

E (h-,,, (t)) = Jim E (hi (t)) = lim E I h (ti + E_00,x oo i- roo \ R (xi, to /

because limi-,, (ti + t/R (xi, ti)) = -oo for all t E (-oo, 0]. In particular,the entropy of the limit is independent of time. Thus by Proposition 5.40,h_,,) (t) is a shrinking round 2-sphere. Since constant-curvature metricsminimize entropy among all metrics on S2, we have E_ , < E (h (t)). Butsince E (h (t)) is nonincreasing, E_,,,, > E (h (t)). It follows that E (h (t))E_,,,,, hence that h (t) is a shrinking 2-sphere of constant curvature.

We now consider the case that the solution is Type II. Recall that we dis-cussed backwards limits in Section 6 of Chapter 8. The self-similar solutioncorresponding to the cigar soliton was introduced in Section 2 of Chapter 2.

PROPOSITION 9.24. Let (.M2, g (t)) be a complete Type II ancient solu-tion of the Ricci flow defined on an interval (-co, w), where w > 0. Assumethere exists a function K (t) such that IRI < K (t). Then either g (t) is flator else there exists a backwards limit that is the self-similar cigar solution.

PROOF. By Lemma 9.15, we have R > 0. The strong maximum principleimplies that either R = 0 or else R > 0. From now on, we assume the latter.By Corollary 9.21, the scalar curvature is pointwise nondecreasing. Hencewe have the uniform bound R < K (0) on (-oo, 0]. Now if M2 is compact,then Klingenberg's Theorem implies that its injectivity radius is boundedfrom below by 7r/ K (0). If M2 is noncompact, we get the same boundfrom Theorem B.65. So in either case, we can take a limit backwards in timeas in Section 6 of Chapter 8. Since the solution is Type II, we obtain aneternal solution (M2, g,,, (t)) satisfying 0 < R,,,) < 1 and R (x,,,), 0) = 1. ByLemma 5.96, g (t)) is isometric to the cigar solution (R2, gE (t)). ED

COROLLARY 9.25. Let (.M2, g (t)) be a complete ancient solution definedon (-oo, w), where w > 0. Assume that its curvature is bounded by somefunction of time alone. Then either the solution is flat, or it is a roundshrinking sphere, or there exists a backwards limit that is the cigar.


The main references for this chapter are [58], [59], and [63]. The esti-mate that the curvatures tend to positive in Section 3 was proved in Theorem4.1 of [66]. Similar estimates, without the time term, were earlier provedindependently by Ivey [77] and Hamilton (Theorem 24.4 of [63]). Theorem


9.9 is essentially Theorem 24.7 of [63]; the minor extension is that one as-sumes that the point is Type I c-essential rather than that the singularityis Type I and the point is c-essential. The alternate proof we gave of theboundedness of F was suggested to us by Mao-Pei Tsui. The nonnegativityof the sectional curvatures of ancient solutions was observed by Hamilton.Theorem 9.19, which does not seem to appear explicitly in print, is analogousto Corollary B3.5 in [65]. Proposition 9.23 is Theorem 26.1 of [63].

APPENDIX A

The Ricci calculus

From the viewpoint of Riemannian geometry, the Ricci flow is a verynatural evolution equation, because it can be formulated entirely in termsof intrinsically-defined geometric quantities and thus enjoys full diffeomor-phism invariance. Nevertheless, when studying the properties of the Ricciflow as a PDE and in particular computing the evolution equations satisfiedby these intrinsic geometric quantities, it is often more convenient to use(classical) local coordinates rather than (presumably more modern) invari-ant notation. Consequently, we provide here a brief review of some aspectsof the Ricci calculus, concentrating on its application to covariant differen-tiation in local coordinates.

1. Component representations of tensor fields

If Mn is a smooth manifold and it : E --p Mn is any differentiablevector bundle, standard constructions in geometry produce its dual bundle£*, various tensor products like £ ® E* ® £*, symmetric products like £ ®S £,and exterior products like A (E). Everything we shall say below extendsreadily, mutatis mutandis, to this general situation. However, to keep theexposition concise and concrete, we will specialize to the case that £ = TM'is the tangent bundle of Mn.

Let (Mn, g) be a (smooth) Riemannian manifold. Then a (p, q)-tensorA is a smooth section of the vector bundle

TPMI = ((&pT*Mn) ® (®gTMn)

over Mn. We denote this by writing A E CO° (TTMn). For example, avector field is a smooth section of Tom' = TMn, while a covector fieldis a smooth section of Tom' = T*Mn. If U C_ Mn is an open set andcp : U --+ Rn is a chart inducing a system of local /coordinates (x1, ... , xn) on

U, we define the component representation I Ak3 - 13l,,,k9) of A with respect

to the chart cp : U ---> W' by

A=A;i...P dxi1®...®dx3P®kl ®...®ak9.

Equivalently,

A Ca

axja

dx1`1 .... dx1 ?1,....Vxjp

279

280 A. THE RICCI CALCULUS

Although there is no natural isomorphism (in the functorial sense) be-tween the bundles TM' and T*Mn, the metric

g = gij dx' 0 dxj

and its inverse1 _ ..o a

9- - 9 axi® axe

induce canonical isomorphisms TM ' -- TP,M' whenever p+q = p'+q'. Inparticular, the metric dual of a vector field X E C°° (TM's) is the covectorfield X, E C°° (T*Mn) defined by

X, = (X2917) dx2;

and the metric dual of a covector field B G C°° (T*MTZ) is the vector field6 E C°° (TM's) defined by

BV _ (Bigii) ai

In formulas involving the components of familiar tensors in local coordinates,we will often use these isomorphisms without explicit mention, for instancewriting R!` for gjkJ. j or Rke for 9ik97eRi7

2. First-order differential operators on tensors

Differentiation of functions on a smooth manifold is easy to define. IfX E C°° (TM') is a vector field and f : M' R is a differentiable function,then X acts on f to produce the function X (f ); in local coordinates

X (.f) = X i 8x1.

(Here and everywhere in this book, the Einstein summation convention isin effect.)

To differentiate tensors requires a connection. It is a standard fact ingeometry that a connection in a vector bundle is equivalent to a definitionof parallel transport, which is in turn equivalent to a definition of covariantdifferentiation. Thus in the present context, we may in particular identifythe Levi-Civita connection r of g with the covariant derivative, which isthe unique operator

V :C°°(TM')-*C°°(TM''(9 T*Mn)

with the property that

X (9 (Y, Z)) = X ((Y, Z)) = 9 (VxY, Z) +g (Y,VxZ)

for all vector fields X, Y, Z E C°° (TM'), where

VyX = (VX) (Y) E CO° (TM') .

2. FIRST-ORDER DIFFERENTIAL OPERATORS ON TENSORS 281

In terms of the Christoffel symbols (r?) induced by the Levi-Civita con-nection in the chart co : U ---> R', the covariant derivative of a vector field Xis the (1,1)-tensor with components

1Vixk=(VX)( Xk+r Xj.

In particular, it is customary in local coordinates to denote the operatorV a simply by Vi.

axz

The Levi-Civita connection also defines a covariant derivative V* forcovectors. This is the unique operator

V* : C°° (T*M') -+ C°° (T*M' ®T*Mn)

with the property that

X (B (Y)) = (VX6) (Y) + 6 (VxY)for all X, Y E C°° (T.Mn) and 0 E C°° (T*Mn), where

VX6 = (V*6) (X) E C°° (T*.M") .

More generally, the Levi-Civita connection defines a covariant deriva-tive V(n,e) on each tensor bundle TT.Mn. This is the first-order differentialoperator

V (p,q) : Co° (TT A' V) - C°° (+1M)

defined by the requirement that

X (A(Y1i...,Yp; 61,...,64)) _ (v'A) (y, Y; 61,...,84)P

-F-EA(Y1i...,VxY,...,Yp; 61,...,64)i=1

q

+> ]A(Y1i...,Yp; 61,...,VX6j,...,6q)j=1

for all (p, q)-tensors A, all vector fields Y1,. .. , Yp, and all covector fields61, ... , 64, where

(VX,q)A) (Yi, ... , Yp; 61, ... , Oq) = (V(P,q)A) (X, Y1, ... , Yp; 61, ... , 64) E R.

Note that V(°,1) = V and V(1°0) = V*. The component representation ofthe covariant derivative is defined by

V(P,qkl ...k9 . a b'.. , a . dxkl, ... , dxk9i )Ailiv

(VA)axjp

REMARK A.1. It is customary and harmless to abuse notation and de-note all V(P,q) simply by V. We adopt this convention throughout thisvolume.

The Lie derivative is defined on vector fields so that

LXY = [X, Y]

for all X, Y E C° (TM'), where [X, Y] is the unique vector field such that

[X,1'] (.f) = X (Y (f)) - Y (X (f))for all differentiable functions f : Mn --> R. If 0 is a covector field and X, Yare vector fields, the Lie derivative of 0 is given by

(Lxe) (Y) = X (e (Y)) - 0 ([X, Y])Note that

(Gx0) (Y) = dO (X, Y) + Y (0 (X)) ,

where d is the exterior derivative defined in Section 3 below. Note too thateven though the Lie derivative is independent of any Riemannian metric gon Mn, the Lie derivative of a vector field can be calculated with respect tothe covariant derivative V induced by g by the formula

(Al) .LXY = [X, Y] _ VXY - VyX.Similarly, the Lie derivative of a covector field may by calculated by

(Lx0) (Y) = (Vx0) (Y) + 0 (DYX) .More generally, the Lie derivative of a (p, q)-tensor field A satisfies

(A.2a) (LXA) (Yi,...,Yp; 01i...,9q)(A.2b) = X (A(Yi,...,Yp; Bi,...,Bq))(A.2c) ->21<i<pA(Y1i...,[X,YZ],...,Yp; 01,...,6q)(A.2d) Ei<j<gA(Y1i...,Yp; 9i,...,GXBj,...,8q)

for all vector fields X and Y1, . . . , Yp, and all covector fields 01, ... , Oq. Aspecific case of some interest for us is the observation that the Lie derivativeof the metric itself satisfies

(Gxg) (Y, Z) = g (VyX, Z) + g (Y, VzX)for all vector fields X, Y, Z. Hence in local coordinates, Gxg is just thesymmetrized first covariant derivative of X, namely

= V2x; + Vixi.(Lxg)ij = (Lxg) - ,a

A first-order differential operator of particular interest in Chapter 3 isthe divergence

5: C°° (T*Mn (9s T*Mn) -a COD (T*Mn)

defined for all symmetric (2, 0)-tensors A and all vector fields X byn

(SA) (X) (div A) (X) _ (VA) (ei, ei, X) ,

i=1

3. FIRST-ORDER DIFFERENTIAL OPERATORS ON FORMS 283

where lei}', is a (local) orthonormal frame field. If Mn is compact, theformal adjoint of S with respect to the L2 inner product

fMn ( .) da

induced by the metric g is denoted by

S* : C°° (T*M') -} C°° (T*Mn ®s T*Mn).

By integrating by parts, it is easy to verify that

(S*0) = 2Geag

for all covector fields 0, where the Lie derivative L and metric dual £$ aredefined above. Notice that this formula can be used to define S* even when.AA' is not compact.

3. First-order differential operators on forms

A p-form is a smooth section of the bundle AP (T*M? ), namely an ele-ment of

SZp (.M n) = C°° (AP (T *.M")) .

The exterior derivative is the family of operators

d=dp:Vp(M') (Ma)

defined for all p-forms 0 and vector fields Y1,

...l,

Yp

by

d0 (Yo, Y1, ... , Yp)

Eo<ip(-1)i B(yo, Y1,...,Y,...,Yp

+ E0<i<j!5P(-1)i+j

9 ([Yi,Yj] ,Yo,Y1, . - . ,Yi, ... Yj,... ,YP ,

where the symbol Y means that term is omitted. Although independent ofany Riemannian metric g on Mn, the exterior derivative may be expressedwith respect to the covariant derivative V defined by the Levi-Civita con-nection of that metric by

dB(Yo,Yi,...,Yp)=F' o(-1)z(vyio)(Yo,Yi,...,Y,...,Yp).

If Mn is compact, the formal adjoint of d with respect to the L2 innerproduct induced by the metric g is the family of operators

j = Sp+l : $Zp+1 (M') ) -' ci (Ma)

defined by the requirement that

1

(dB, 77) = (0, bra)


for all 0 E 1P (.Ma) and 77 E QP+1 By integrating by parts, it is easy toverify that

(677) (X1, ... , XP) (Vp) (ei, ei, Xl,... , Xp)i=1

for all vector fields X1,.. . , Xp, where {ei}1 is a (local) orthonormal frame

field. Note that this formula can be used to define S even when Mn is notcompact.

4. Second-order differential operators

A number of second-order differential operators are of considerable im-portance to our study of the Ricci flow.

The rough Laplacian denotes the family of operatorsA:C°°(TTM') _*Coo (TTM )

defined by

(AA) (Y1,...,Yp; e1,...,9q) (V2A) (ei,ei,Y1,...,Yp; B1, ...,9q)i=1

for all (p, q)-tensors A, all vector fields Y1,. .. , Yp, and all covector fields01 i ... , 9q, where {ei} 1 is a (local) orthonormal frame field. In local coor-dinates, we write

AAkl...kq

a a adxkl dxk9ji 7p

(AA) axi , axjl , ... ,axjp '

The Hodge-de Rham Laplacian (frequently called the Laplace-Beltrami operator) denotes the family of maps

-Ad : QP (Mn) ' Il (M')defined by

-Ad = do + bb = dp-l5p + ap+1dp,where d and b are given in Section 3 above. It differs from the rough Lapla-cian by curvature terms which depend on the index p of the space QP (.M')For instance, if 0 is a 1-form (equivalently, a covector field), then the com-ponents of d9 in local coordinates are given by the formula

Ad9 = (&2 - RzOj) dxi.

If a is a 2-form, then

(A.3) Ada = (aij + 9kP9egRijkjagp - 9W Rikati -gkERjkait) dxi A dxj.

Finally, the Lichnerowicz Laplacian, introduced in [93], is an operator

AL : COO (T*Mn (gs T*Mn) -* COO (T*.M®®s T*Mn)

5. NOTATION FOR HIGHER DERIVATIVES 285

defined so that whenever (M', g) is Ricci-parallel - that is, whenever the(3, 0)-tensor V Rc vanishes identically, as happens in particular if g is Ein-stein - the following diagram commutes:

C°O (T*Mn ®s T*Mn) AL > coo (T*M' ®s T*Ml)

C°° (T*.Mn) ti Q' (Mr) °d) Q1 (M-) c--- C- (T*M-).

If A is a symmetric (2, 0)-tensor, then the components of the LichnerowiczLaplacian ALA in local coordinates are given by the formula(A.4)

ALA = (AAij + 2gkp9tgRik1jApq - gkiRik Atj _ gktRjk Ail ) dx® ® dxi.

REMARK A.2. As we saw in Chapter 3, the Lichnerowicz Laplacian isessentially the linearization of the Ricci flow operator.

REMARK A.3. The Lichnerowicz Laplacian on symmetric 2-tensors isformally the same as the Hodge-de Rham Laplacian on 2-forms. Indeed, aneasy application of the first Bianchi identity shows that equation (A.3) isequivalent to

Ada = (Aceij + 2gkp9eq ikjapq - gkepikatj - gkeRjkCYip) dxi Adx3,

which is formally the same as equation (A.4).

5. Notation for higher derivatives

A delicate notational issue arises when considering the operatorsVk : C°° (TgMnC°° (T1Mn)ql p )

defined for k > 1. If A E C°° (TpgM'ti) and X1, ... , Xk are vector fields, weadopt the convention (which is not universally followed!) that

V 1 Vx2 ... OxkA

is the unique (p, q)-tensor field such that

(Vx,pX2 ..pxkA) (YI,...,Yp; Oi,...,6q)

_ (VkA) (XI,X2..... Xk, Y1i...,Yp; 01,...,8q)

for all vector fields Y1, ... , Y p and all covector fields 01, ... , 8q. Note that

(VkA) (XI, ... , Xk) Vx1 Vx2 ... VXkA Vx1 (Vx2 (... (VxkA)))

in general. For example, in our convention,

(A.5) Vx (VyA) _ VxVYA+VvXYA = (V2A) (X, Y) + (VA) (VxY) .


6. Commuting covariant derivatives

Recall that the (3,1)-Riemann curvature tensor is defined so that(A.6) R (X, Y) Z = (V2Z) (X, Y) - (V2Z) (1' X) E C°° (TM-)

for all vector fields X, Y, Z. Following the the notational convention of Sec-tion 5 above, we write this as(A.7) R (X, Y) Z = VXVyZ - VyVxZ.One advantage of this formula (hence of the convention adopted in Section 5)is that it makes the geometric significance of the Riemannian curvature clearby demonstrating its fundamental role in commuting covariant derivatives.By (A.1) and (A.5), one may also write equation (A.6) in the more familiarform

R (X, Y) Z = Vx (VyZ) - Vy (VxZ) - V [X y]Z.The reader is warned that this form is used without parentheses by thoseauthors who do not follow the convention we adopted in Section 5. In localcoordinates, all brackets [aa17 aa, vanish identically, so that regardless ofwhich convention one adopts, one can write unambiguously

R(axi'8xj) =Rfk(xP -(VZVj-VjV2) ().

The commutator formulas for covariant derivatives implied by equation(A.6) are collectively known as the Ricci identities. For example, if X isa vector field, then in local coordinates,

[Vi, Vj] XP _= ViVjX' - VjViXP = Rf3'kXk.

The commutator for a covector field 8 (equivalently, a 1-form) may be ob-tained from this by using the metric dual 0, yielding

[Vi, Vjl ek = ViVjOk - VjViekP P

= At [Vie Vjj()E

= 9kPRijm(9rmer) _ -Rijk9E.

More generally, if A is any (p, q)-tensor field, one has the commutatorPI...g4 PI...PQ PI...P9

[Vie V, ] Ak1...kp = ViVjAk1...kn - V?ViAkl...krq P

Pr E1 ...Pr-1 -tr+1...P9 m el... PqRijm,Akl...kp Rz3ks`4k1...ks_1 mk9+I...kp

r=1 8=1


The contents of this appendix are standard parts of classical Riemanniangeometry. For the convenience of the reader, we have reviewed them here inorder to establish an unambiguous notation.

APPENDIX B

Some results in comparison geometry

We begin this appendix by recalling basic results in comparison geometrythat are useful in the study of limits of dilations about a singularity of theRicci flow. Because these are standard results, we shall in some cases omit ormerely sketch the proofs. We assume throughout that (Mn, g) is a completeRiemannian manifold.

1. Some results in local geometry

We begin with a collection of results that hold at small length scales,where a complete Riemannian manifold (M',g) is `almost Euclidean'.

1.1. The Gauss Lemma and radial vector fields. Recall that foreach point p E Mn and tangent vector V E TPMT, there is a uniqueconstant-speed geodesic

'Yv : [0, oo) Mnemanating from p such that -yv (0) = p and -fv (0) = V. Given p E Mn, letexpP : TP.M' -* M" denote the exponential map defined by

expP (V) --ryv(1).

By the uniqueness of geodesics with given initial data, one has

7v (t)='Ytv(1),so that

'Yv (t) = expP (tV) .

Observe that (expP)* : To (TPM') -- TQM' is the natural isomorphismbetween the two vector spaces. (We may in fact regard it as the identitymap.) So by the inverse function theorem, there exists e > 0 such that

expP I a(d,e) :B(0, e) -+ expP (B(0, e)) C Mn

is a diffeomorphism, where B(V, r) denotes the open ball with center V ETPMn and radius r. In fact, even more is true; expP preserves orthogonalityto radial directions, and hence the length of radial vectors.

LEMMA B.1 (Gauss). If p E Mn and V, W E TPMn are such that(V, W) = 0, then

((expp)* (V v) , (expP)* (Wtv) ) = 0.

287

288 B. SOME RESULTS IN COMPARISON GEOMETRY

Here Vtv, Wtv E Ttv (TpMn) are the vectors identified with V, W E TpMnrespectively, and

(eXpp)* : Ttv (TpMn) -' Texpp(tv)M

Let E be as above. The radial function

r : B(0, E) -3 [0, E)

given by

defines a function

where

r(V) - IVI

f : expp (B(0, E)) -> [0, E)

f (x) - r ((exPPB(E)) (x))

The observation f (expp V) = V I shows that f is just the distance function

induced by g on expp (B(O,e)). The function f is smooth everywhere except

at p, so that V f exists on expp (B(0, e)) \ {p}.

On the other hand, there is a unit radial vector field R defined onTpMn\{0} by

RvV= VI

Define a vector field a/ar on expp (B(0, e)) \ {p} by

Since

we have

(eXpp). (Rv) .ar expp(V )

(eXpp)' (RV) = V dt ('Yv (t))

ai

Or1.

The Gauss Lemma implies the following result.

COROLLARY B.2. On expp (B(, E)) \ {p}, the vector fields V f and a/arare identical:

aVf

PROOF. For every V E B(0, e)\{d} and every Y E Texpp(v)Mn, there isa unique orthogonal decomposition

Y=a-+Z

I. SOME RESULTS IN LOCAL GEOMETRY 289

obtained by taking a - (Y, (9/ar) and Z _ Y-a (a/ar). Then (Z, a/ar) = 0and

We claim that(Y, Vf) = Y (f) = a (f) +Z(f) .

(f) and Z(f)=0.The corollary will follow from these formulas, because they imply that

a(Y,Vf)=aY'ar)

for all Y E Texpp(v),Ntn and all V E B(O,E)\{6}.To first formula of the claim is proved by the observation

ar (f) _ ((expP)* (R)) (ro (exPPB())') =R(r) = 1.

To get the second formula, we note that

Z(f)=Z (ro (exppB(e))1)

= [((exPPB())')* (Z), (r)

= (((exPPB())') (Z), R).

But

(Z, (expp)* (R) /_ \Z' a 0.

By the Gauss lemma, (expP)* maps R1 to (6/ar)1; since (expP I

is an isomorphism, it follows that ((expplB(,e))-1)* maps (,9/(9r)-L to F?-L,hence that

)=0.Z(f) = (((exPpB(e)) I.R\

1.2. Conjugate points. If cp : Nn -+ P" is a smooth map betweendifferentiable manifolds of the same dimension, we say cp. : TyNn _is singular if it is not an isomorphism. In this case we say that x E Nn isa critical point of cp and that co (x) is a critical value of V.

DEFINITION B.3. A point q E M is a conjugate point of p E M" ifq is a critical value of

expP : TP,Mn --> Mn,

namely, if q = expP (V) for some V E TpMn such that

(eXpp)* TV (TPM')_ Texpy(V)Mn


is singular. In this case, we say q is conjugate to p along the geodesic yvdefined by yv : t i > expp (tV) .

DEFINITION B.4. The conjugate radius re E (0, oo] of a point p E Mnis

r, (p) - sup {r: (expp)* is nonsingular in B(0, r)} .

REMARK B.5. exppis an immersion of the open ball B(0',re).

Intuitively, a conjugate point is one where distinct geodesics come to-gether. The following result (which uses terminology from Definition B.66)makes this notion precise.

LEMMA B.6. Let (Mn, g) be a complete Riemannian manifold, and let{I'iheN be a sequence of nondegenerate proper geodesic 2-gons,

2] -- Mn} ,ri = y : o, 1 -- Mn, y2 : [0"r

such that -y' (0) = rye (e2) = pi and rye (0) = y1 (21') qi. Suppose that thefollowing limits all exist:

P. Jim pi E Mn,

q0 = lim qi E M',i- 00

VIXI = lim ' 11 (0) lira y2 (212) E TPOQMni-.oo i-.oo

40 Jim el = lim 4.8-x00 2 00

Then 20 > 0, and q00 is a conjugate point to p00 along the limit geodesicyoo : [0, too] , Mn determined by boo (0) = V00.

PROOF. That to, > 0 follows from the lower bound for the injectivityradius of (Mn, g) in a compact set. Now suppose that q00 = expp. (&0V00)is not a conjugate point of p00 along the geodesic -yam : [0, £00] --> Mn. Then

(exnp.),, : T$.v. (Tp.Mn) - Tq.Mn

is nonsingular. Since exp : TMn --+ Mn is a smooth map, there exists aneighborhood U of p00 in Mn and a neighborhood V of £00V... in TP0Mnsuch that

(expp)* : Tyv (T PM n) -> Texpp(W)Mn

is nonsingular for all p E U and W E V C TpA4". Here we used paralleltranslation along geodesics emanating from p00 to identify Tp00Mn withTPMn for all p E U, which allows us to regard V as a subset of TpNtn. Thusit follows from the inverse function theorem that there exists a neighborhoodV C V and some e > 0 such that

( - JVinexpp s(e,o B e)

1. SOME RESULTS IN LOCAL GEOMETRY 291

is injective for all p E V. Now observe that for all sufficiently large i, wehave

piEv,t1 Yi (0) E B (f..Voo, e) C T'pi' L

Z 2 (e2) E B (QooVoo, E) C Tpi.n.

Since fi 1i (0) -f z2 (42) but exppi 4 4 (0)] = exppi [-4'Y2 (4)] = qi,this contradicts the injectivity of expel

REMARK B.7. It is important to note that this result can fail in the con-text of sequences (M', gi) of Riemannian manifolds - in particular, wheneach Fi is a nondegenerate proper geodesic 2-gon in (M', gi). The failurecan occur even if the sequence (M , g.) has uniformly bounded geometry.This fact becomes highly relevant when one wants to develop injectivityradius estimates for such sequences in order to pass to a limit (A4n ,gam).

EXAMPLE B.8. Consider a sequence {2 } - Z +

of collapsing flat tori withfundamental domains

i] x [-1/i, 1/i] c I[82

Take Oi = (0, 0), and define constant-speed geodesics

2 2ai,A: [o,/1i/iJ --->7by

ai (s) = (s, csc-1(i) s) and Qi (s) = (s, - csc-1 (i) s) .

Then length ai = length 3i = 1 for all i, but their limit in the universal coverR2 is just the segment s F--1 (s, 0) for 0 < s < 1, which has no conjugatepoints.

EXAMPLE B.9. Consider Si = { (x, y, z) : x2 + y2 + z2 = 1}. For eachi E N, let (M?, gi) denote S1 /Zi, where Zi acts on S2 by rotation by angle27r/i around a fixed axis, say the x-axis. The surface M? is an orbifold withtwo cone points of order i. Simply for the sake of visualization, isometricallyembed M? into R3 as a surface of revolution about the x-axis centered atthe origin. This embedding will be smooth except at the two cone pointswhich are at opposite ends of the surface and lie on the x-axis. Since A4?is collapsing, its length along the x-axis tends to ir. For i large enough,consider two points in M? defined as follows. Let p2 = (x1 , yZ , z,) be theunique point on Mi with xi = -1, y2 = 0, and z < 0. Let pz = (x?, yi , zZ )be the unique point on M? with x? = +1, y2 = 0, and zZ > 0. There areexactly two minimal geodesics ai and (3z joining pi and pi2 in M?. Thesegeodesics have the same length, and the angle between them at pz (and p?)tends to 0 as i -4 oc. However, the limit

-yam = lim ai = lim )3ii-.oo i-.oo


in the limit geodesic tube, which has constant curvature 1, is a geodesic oflength 2. In particular, 'y,, does not have any conjugate points along it.

1.3. Jacobi fields. We have already seen how distinct geodesics cancome together at a conjugate point. The infinitesimal version of this idea iscaptured in the concept of a Jacobi field.

DEFINITION B.10. A Jacobi field is a variation vector field of a one-parameter family of geodesics.

LEMMA B.11. expp (V) is conjugate to p if and only if there exists W ETp.Mn\ {0} such that (W, V) = 0 and (expp)* (Wv) = 0, where WV ETV (TTMn).

PROOF. Sufficiency is clear. To prove necessity, it is enough to observethat .yv (1) E Texpp(V)Mn\{0}.

PROPOSITION B.12. Let (Mn, g) be complete and connected. Then q isconjugate to p along a geodesic y if and only if there exists a nonzero Jacobifield J along y with

J(0)=J(1)=0.PROOF. By Hopf-Rinow, q = expp (V) for some V E TpMn\ {0}. Take

any W E TpM \ {0} such that (W, V) = 0. For s small, consider theone-parameter family of geodesics

yS:tHexpp[t(V+sW)].Note that ry - yo is a geodesic such that y (0) = p and y (1) = q. Thecorresponding Jacobi field is given by

Jw (t) asys (t)

S=0

(expp) * (tWty) E Ty(t)Mn_ expp (tV + stW) Lo

Since Jw (0) = 0 and

Jw (1) = (eXpp)* (Wv),the result follows from the lemma.

By using the calculus of variations, one obtains the Jacobi equation.

LEMMA B.13. J is a Jacobi field along a geodesic -y with unit tangentvector T if and only if

VTVTJ = R (T, J) T.

COROLLARY B.14. Any Jacobi field J along a geodesic y with unit tan-gent vector T admits the unique orthogonal decomposition

J = Jo + (at + b) T,

I. SOME RESULTS IN LOCAL GEOMETRY 293

where a, b E R, and J0 is a Jacobi field such that

(Jo, T) = 0.

PROOF. Since 7 is a geodesic, we have VTT = 0 and hence

d2Wt-2 (J, T (VTVTJ, T) = (R (T, J) T, T) = 0.

t=o

1.4. Focal points and the normal bundle. Focal points are a nat-ural generalization of the notion of conjugate points.

DEFINITION B.15. If t : Ek 9-4 Mn is an immersion of a smooth manifoldEk, the normal bundle of Ek is the subbundle NEk of the pullback bundlet*T.Mr whose fiber over each p E Ek is

NpEk={VETpMn:(V,W)=0forallWEt*(rl(p)Ec)}.

Consider

expp I NPEK: NpEK --3 M,.

Taking the union over all p c EK, one obtains a map of the total space

explNE : NEk Mn

defined for all pairs (p, V) with p E Ek and V E NpEk by

explNEk (p, V) - expPINPEk (V)

Notice that explNEk : NEk Mn is a smooth map between manifolds ofthe same dimension.

EXAMPLE B.16. If En = Mn, then NM' = .Mn x {0} is naturallyidentified with Mn, so that we may regard explNMn : NMn -> Mn as theidentity map.

EXAMPLE B.17. If E° = {p} is a point, then T {p} = {d}, so thatN {p} = TpMn and explN{p} = expp.

DEFINITION B.18. Let t : Ek 4+ Mn be an immersion. One says q E Mnis a focal point of t (Ek) if q is a critical value of explNEk. In particular,one says q is a focal point of Ek at p if it is the image of some critical pointin NpEk.

Notice that q E Mn is a focal point of Ek C Mn only if q V Ek.

EXAMPLE B.19. Let Mn = Rn and let En-1 = Sr -1 (p) be the (n - 1)-sphere of radius r > 0 centered at p E Mn. Then the unique focal point ofEn-1 is p.

1.5. The Rauch comparison theorem. If one has suitable boundson the sectional curvatures of a Riemannian manifold (Mn, g), one cancompare distances in M' with a those in a convenient model space.

THEOREM B.20 (Rauch). Let (.MT, g) and (.M , g) be complete Rie-mannian manifolds. Let

y: [0Q]_'Mn

7y :[0,2]--+./

be unit-speed geodesics of the same length such that

sect (II) < sect (II)

for any 2-plane H containing T 7 and 2-plane fl containing T =y. Let Jand J be Jacobi fields along y and ry respectively, such that J (0) and J (0)are tangent to ry and 7 respectively, and

IJ (0) Ig = I I (0) I9 .

Suppose that

IVTJ(0)Ig =and

(VTJ (0) , T)g = (VTJ (0) , T)9If ry (t) is not conjugate to ry (0) for any t E [0, 1], then for all t E [0, t],

IJ(t)Ig? IJ(t)I9.Note that it is often convenient to apply the theorem when one knows

that sup (sect (Mn, g)) < inf (sect (JR , g)) and takes Jacobi fields J, Jsuch that J (0) = J (0) = 0, IVTJ (O) Ig = VTJ (0) I9, and (J, T)g(J, T)g - 0. Note too that the condition (J, T)g - 0 implies in particu-lar that

0= d

dt (J,T)g = (VTJ,T)g,because y is a geodesic.

By taking (NI , g) to be the Euclidean sphere gcan) of radius

1/vlk, one obtains the following result.

COROLLARY B.21. Let K > 0, and suppose (M', g) is a complete Rie-mannian manifold with sect (g) < K. If J is a Jacobi field along a geodesicy in Mn with unit tangent T such that

J(0) = 0 and (VTJ (0), T) = 0,then

IJ (t)I >IVTr

)I sin (V'K--t) > 0

for all t E [0, it/VK-). In particular, there does not exist a conjugate pointin B (p, 7r/v) _ {q E Mn : d (p, q) < 7r/v}.

2. LOCAL VERSUS GLOBAL GEOMETRY 295

COROLLARY B.22. Let (Mn, g) be a complete Riemannian manifold withsectional curvatures bounded above by K > 0. Let ,0 be a geodesic path fromp to q in .M'l with L (/3) < ir/v'K--. Then for any points p' and q' sufficientlynear p and q respectively, there is a unique geodesic path 0' from p' to q'which is close to /0.

PROOF. By the previous corollary, the map

exppIB(p,,r/f):B(0,r-/v/K--) CTpM' M'

is an immersion. We may assume that the geodesic /3 : [0, 1] --> M" isparameterized so that 0 (s) = expp (sV) for some V E B(d, it/V) withexpp (V) = q. Since expp I

B(o,7,1,1_K)is a local diffeomorphism, there is for all

q' sufficiently close to q a unique V' E B(6, 7r/vK) such that s F-* expp (sV')is the unique geodesic path from p to q' lying close to 0. Since for allp' = expp W sufficiently close to p, the map

(eXpp)* : TpM' = TW (TpM') TpM'is an isomorphism, there exists a unique vector V" E Tp,.M" such that

expp, (V") = expp (V') = q'.

The path ,0' : [0, 1] -p Mn defined by 0'(s) expp, (sV") is the uniquegeodesic near Q joining p' to q'.

2. Distinguishing between local geometry and global geometry

In Section 1, we observed that the exponential map is always an embed-ding of a sufficiently small ball. We now study larger length scales, in orderto investigate some of the ways that the topology of a complete Riemannianmanifold (M', g) affects its geometry. In this section, interpret geodesic tomean unit-speed geodesic.

2.1. Cut points and the injectivity radius. Given p E Mn and ageodesic 7 : [0, oo) -+ Mn with y (0) = p, define

S.y = {t E [0, oo) : d (y (0) , y (t)) = t} .

A sufficiently short geodesic always minimizes distance. Its behavior atlarger length scales is described by the following elementary observation.

LEMMA B.23. Either d (y (0) , y (t)) = t for all t E [0, oo), or elsethere exists a unique t.y such that d (y (0) , -y (t)) = t for all t E [0, t y] andd (y (0) , y (t)) < t for all t > ty. In short,

either S.y = [0, oo) or else S.. = [0, ty] .

PROOF. If d (y (0) , y (t)) < t for some t > 0, then for all t >

d(y(0),y(t)) < d(-y(0),y(0)+d(y(),y(t)) <t+(t-) =t.Take t.y inf {t > 0 : d (-y (0) , y (t)) < t}.

DEFINITION B.24. If 'y : [0, oo) --p Mn is a geodesic with -y (0) = p, wecall y (ty) the cut point of p along y.

Notice that if M' is compact, there is for each p E Mn a unique cutpoint of p along every geodesic -y emanating from p.

DEFINITION B.25. The union

Cut (p) - {y (t.y) : y : [0, oo) -- .M' is a geodesic with y (0) = p}

of all cut points along all geodesics emanating from p E Mn is called thecut locus of p.

The existence of a conjugate point is sufficient (but not necessary) for ageodesic to fail to minimize distance.

LEMMA B.26. Let y : [0, 2] -+ Mn be a geodesic. If there is T E (0, 2)such that y (-r) is conjugate to y (0) along y, then there exists a proper vari-ation

{Is(t):-E<s<E, 0<t<Q}with r, (0) (0), r, (i) (Q), and I'o = y such that for some s - 0,

L (rs) < L ('y) .

Briefly: a geodesic fails to minimize distance past its first conjugate point.

COROLLARY B.27. If 7 (t) is conjugate to y (0) along y, then t > ty.

If expp does not fail to be an immersion at the cut locus, it must at leastfail to be an embedding there. This statement can be made more precise.

LEMMA B.28. -y (t) is a cut point of -y (0) along the unit-speed geodesic-y if and only if t > 0 is the smallest value such that either of the following(non-exclusive) conditions hold:

(1) The point y (t) is conjugate to y (0) along -y.(2) There exists a unit-speed geodesic,3 : [0, t] --> Mn such that 0 (0) _

-y (0), Q (t) = y (t), and L (/3) = L (-yI[o,tI), but 4 (0) 0 zy (0). Thatis, 0 and yI[o,t] are distinct geodesic paths of equal length from theircommon starting point -y (0) to their common end point y (t).

Given p E Mn, denote the unit sphere in TPMn by

Spn -1Mn={V ETPMn: IVI =1},

and define p : Sp -I Mn --> (0, oo] by

d (p, -yv (tyv )) = ty, if t.yt, existsp (V) 00 otherwise.

That is, p (V) is the distance from p to the cut point of p along -yv.

LEMMA B.29. For each p E tiln,p : Sp -1Mn _., (0, oo]

is a continuous function. That is to say:

(1) 0 - {V ESP -1M- : p (V) < oo} is an open subset of Sp-'M';(2) pro : 0 --; (0, oc) is continuous; and(3) for any V E Sp -1M'\0 and every L E (0, oo), there exists a

neighborhood U of V in Sp -1Mn such that p (U) > L for all U E U.

PROOF. The proof is in two steps.We shall first show that if a geodesic emanating from p E .Mn stops

minimizing at some point, then nearby geodesics emanating from p alsostop minimizing near the same point. If V E 0 and E > 0, the definition ofp (V) implies that

d (p, expp [(p (V) + E) V]) < p(V)+E.

Define 8 E (0, oo) by

8 p(V)+E-d(p,expp[(p(V)+E)V]).

By continuity of the exponential map, there exists a neighborhood Wv ofV in SP-'M' such that for every W E Wv, one has

d (expp [(p (V) + e) W] , expp [(p (V) + E) V]) <2

By the triangle inequality, we obtain

Id (p, expp [(p (V) + E) W]) - d (p, eXpp [(p (V) + -) V]) <2

and thus

d(p,expp[(p(V)+E)W]) <d(p,expp[(p(V)+E)V])+2

=p(V)+2<p(V)+E.We conclude that p (W) < p (V) + E for all W E Wv. This proves both thatp is upper semicontinuous at V and that Wv C 0, hence that 0 is open.

Now suppose that {Vi} is a sequence from S'-'MP such that V --+ Vbut p (Vi) -"p p (V). We shall derive a contradiction and thus complete theproof of the lemma. By passing to a subsequence, we may assume thereis p,,,) in the compact set [0, oc] such that p (Vi) --> p,, p (V). Since pis upper semicontinuous at V, we may in fact assume p. < p (V), whichimplies in particular that expp (p,,,V) is not a cut point of p along the ge-odesic expp (tV). By passing to a further subsequence, we may by LemmaB.28 assume either that each expp (p (Vi) V) is a singular point of (expp) *

or else that there are V' E Sp-'M' such that V' Vi for any i, butexpp (p (Vi) V') = expp (p (Vi) V j). The first case is impossible, since it im-plies that (expp)* is singular at p,,,V. In the second case, observe that exppis an embedding of a sufficiently small neighborhood V of V in Tp.Mn. Sofor all large enough i, each V' lies outside V fl SP-' Mn By passing to a

further subsequence, we may thus assume Vti' -f V' V. By continuity ofthe exponential map, this implies that expp (p,,,Vo) = expp (p,,V) and

d (p, expp (p,,,V,,,)) = ilim d (p, expp (p (V) V') )

= Zlim d (p, eXpp (p (V) VZ)) = d (p, expp (pc,,) V)) ,00

which contradicts the fact that expp (p,,V) is not a cut point of p along thegeodesic expp (tV).

COROLLARY B.30. Cut (p) is a closed set for each p E .M".

Given p E .M'n, define

Cp-{V ETpMn:d(p,expp(V))=IVIRecalling that yv (t) = expp (tV), we observe that

Cp={tV:V ESp-IMnand d(p,yv(t))=t}

= { tV : V E Sp-1.Mn and t < d (p, yv (try )) } ,

hence conclude that Cp is closed.

DEFINITION B.31. If p E Mn, we call aCp C Tp.Mn the cut locus inthe tangent space TpJAAn.

Note that aCp may be the empty set 0. But in any case,

Cut (p) = expp (aCp) .

Moreover, Mn\ Cut (p) is homeomorphic to Cp\aCp.

LEMMA B.32. For each p E Mn, the map

exppIintCP:Cp\aCp --> .Mn\ Cut (p)

is an embedding.

Note that if Mn is compact, then int Cp, hence Mn\ Cut (p), is homeo-morphic to an open n-ball.

DEFINITION B.33. The injectivity radius inj (p) at a point p E .Mn is

inj (p) --'.Sup {r> 0 : expp I B(o,r) :B (,r) -+.A4' is an embedding} .

The injectivity radius inj (Mn, g) isinj (Mn, g) - inf {inj (p) : p E Mn} .

When we want to take limits of a sequence of Riemannian manifolds, itis of fundamental importance to be able to estimate the injectivity radiusfrom below. A basic estimate is the following.

LEMMA B.34 (Klingenberg). If Mn is compact and sect (g) < K forsome constant K > 0, then

n I 7r 1 ( the length of the shortestin

1 l1(M, g) >- min ' 2 smooth closed geodesic in Mn J j

2.2. Lifting the metric by the exponential map. Let r, = r, (p)be the conjugate radius of p E .Mn. Then

P (, r,)n

is an immersion. Hence

9 - (exp)is a smooth metric on B(06, r,) C Tp.Mn.

9

LEMMA B.35. There are no cut points of 0 in the open Riemannianmanifold

(B(,rc),).

PROOF. Let d denote distance in B (0, rc) measured with respect to the

metric g. It will suffice to show that d (0, V) _ IV I for all V E B(0, r,). The

unit speed geodesics of (B(0, re), g) emanating from 0 are the lines defined

for V E Sp -1.Mn C TpMn and t E [0, r,) by yv : t -- tV. We have

Ly (v[ol]) _ IVI,

and hence

a (0-, V) < V1.

To see that equality holds, let w : [0, a] -+ B(0, r,) be any path with w (0)and w (a) = V. By Corollary B.2 of the Gauss lemma, we have

(ar) = R.Vgr = (eXpp'),, (V9f) = (expp'),,

Since R9

= 1, it follows that

fa

lca (t)l9 dt

>ja

Cw (t) R>9 dt =ja

(cv (t) , Vgr)g dt

=f dt[r(w(t))] dt=r(w(a))-r(w(0))=r(V)=IVI.

Hencei(m) > IVi.

Notice that the lemma implies

inj y (0) = rc.

Applying Corollary B.21, we obtain the following conclusion.

COROLLARY B.36. If (MI, g) is a complete Riemannian manifold withsectional curvatures bounded above by K > 0, then for any p E Mn,

inj y(0) > .

2.3. A Laplacian comparison theorem for distance functions.Given any smooth function f on (Ma, g), the Bochner-Weitzenbock for-mula says

20IVf12 = IVVfl2 + (Vf, V (of)) +Rc (Vf, Vf).

This formula is of fundamental importance and is used in the proof of theLi-Yau gradient estimate [91] for the first eigenfunction and the Li-Yau dif-ferential Harnack inequality [92] for positive solutions of the heat equation,among many other results. It is easily proved using Ricci calculus:

AIVf12 = 1Vivi (vjfo'f) =of (v vifvif)12 2

_ViVjV2fV3f +VzvjfViV2f

_ VjviV2fV- f +Rjkp'fVkf + VVjfV Vjf

_ (Vhf, Of) + Re (Vf, Vf) + IVVf I2 .

We say r : Mn --+ [0, oo) is a generalized distance function if

(B.1) 1Vrl2 = 1

at all points where r is smooth. Generalized distance functions have theproperty that the integral curves of Vr are geodesics.

LEMMA B.37. If r is a generalized distance function, then wherever r issmooth,

VVT(Vr)-0.

PROOF. Differentiating (B.1) shows that for all i = 1, ... , n,

0 = Vi IVrI2 = Vi (V jrVjr) = 2V3rV jVir = 2 (VvrVr)i.

COROLLARY B.38. Wherever r is smooth,

IVVrI2 >n

1 1 (Or)2.

PROOF. VVr has a zero eigenvalue, because

(VVr) (Vr, Vr) = V2rV3rViV jr = (Vr, VvrVr) = 0.

Hence the standard estimate becomes

(Ar)2 = (trgVVr)2 < (n - 1) IVVr12 .

Assume there is B E I[8 such that Rc > (n - 1) Bg. Wherever r isa smooth generalized distance function, taking f = r in the Bochner-Weitzenbock formula gives the estimate

0 = I VVrl2 + (Or, V (Ar)) + Rc (Vr, Dr)

(B.2) >n

1 1 (Ar)2 + (Vr) (Or) + (n - 1) B,

since (Vr, V (Ar)) = d (Ar) (Vr) (Or) (Ar). If we define

v+ 1 Or,n-1then (B.2) is equivalent to the inequality

(B.3) dv (r) = (Or) (v) < -v2 - B.

This suggests we compare v (r) with the solutions of the Riccati equation

dw 2

dr= -w - B, rl_i,o w = 00,

namely

(B > 0) w (r) = cot (,/Br)

(B=0) W (r) _ I

(B < 0) w (r) = v/-B coth(r).We can indeed make this heuristic rigorous in the following case.

PROPOSITION B.39. Let (M', g) be a complete Riemannian manifoldsuch that Rc > (n - 1) Bg for some B E R. If r (x) d (p, x) is the metricdistance from some fixed p E M', then on a neighborhood of p containedinJ4' \ ({p} U Cut (p)), we have the estimate

v" cot (,/Br) ifB > 0

Or< B1 i O=/r f

V-B coth (v/-Br) ifB < 0.

PROOF. Note that r is smooth on M'\ ({p} U Cut (p)), and Ar satisfies

lim rAr = n - 1.r- O+

Thus the function defined byn-1Nr ifx p

0 ifx=p

is continuous in a neighborhood U C_ M"\ Cut (p) of p and is smooth onU\ {p}. Let

y : [O, inj (p)) - Mnbe a unit-speed geodesic such that y (0) = p and 'Y = Or, and fix anyr E (0, inj (p)). By (B.3), we obtain the differential inequality

1 < du (r) (Vr) (u)1 + Bu2 1 + But

which implies

So if B > 0, we get

r < y (t) (u) dtjr

1 + B u (y (t))2

r < Jr dt ( tan-1 (u((t)))) dt =

while if B = 0, we have

tan-1 (V-B-u (-y (r)))

VP '

r dur< tdt=u(y(r)),

and finally if B < 0, we obtain

r < fr

d(_1

B tanh-1 ((y (t))) I dt = tanh-1 (v/- Bu (y (r)))

The result now follows easily from the monot/onicity of tan-1 land tanh- 1. 11

2.4. The Toponogov comparison theorem. The Rauch comparisontheorem works at infinitesimal length scales to compare the geometry of aRiemannian manifold (Ma, g) with model geometries of constant curvature.It has a powerful analog at global length scales: the Toponogov comparisontheorem.

THEOREM B.40 (Toponogov Comparison Theorem). Let (M', g) be acomplete Riemannian manifold with sectional curvatures bounded below byHER.Triangle version (SSS): Let 0 be a geodesic triangle with vertices (p, q) r),sides qr, rp, pq of lengths

a = length (qr) , b = length (rp) , c = length (pq)

satisfying a < b+c, b < a+c, c < a+b (for example, when all of the geodesicsides are minimal), and interior angles a = Lrpq, Q = Zpgr, y = Lgrp,where a,)3, y E [0, 7r]. Assume that c < 7r/v'-H- if H > 0. (No assumptionon c is needed if H < 0.) If the geodesics qr and rp are minimal, then thereexists a geodesic triangle A = (p, q, r) in the complete simply-connected space

3. BUSEMANN FUNCTIONS 303

of constant sectional curvature H with the same side lengths (a, b, c) suchthat

a>=Lrpq/3 > - zpgr.

Hinge version (SAS): Let / be a geodesic hinge with vertices (p, q, r),sides Fr and rp, and interior angle Lgrp E [0, 7r] in Mn. Suppose that Tr isminimal and that length (rp) < 7r/' if H > 0. Let L' be a geodesic hingewith vertices (p', q', r') in the complete simply-connected space of constantsectional curvature H with the same side lengths and same angle. Then onemay compare the distances between the endpoints of the hinges as follows:

dist (p, q) < dist (p', q) .

3. Busemann functions

In this section, we develop some tools to study a complete noncompactRiemannian manifold (.MI, g) at very large length scales, in order to under-stand its geometry `at infinity'.

3.1. Definition and basic properties.

DEFINITION B.41. A unit speed geodesic 'y : [0, oc) ---p Mn is a ray ifeach segment is minimal, namely, if yl[a,b] is minimal for all 0 < a < b < oo.We say y is a ray emanating from 0 E M if 7 is a ray with -y (0) = 0.

LEMMA B.42. For any point 0 E Mn, there exists a ray -y emanatingfrom 0.

PROOF. Choose any sequence of points pi E Mn such that d (0, pi)oo. For each i, choose a minimal geodesic segment yi joining 0 and pi. Let

V = dyi/dt (0) E T0Mn

Since the unit sphere in TOA4' is compact and JVil - 1, there exists asubsequence such that the limit

(B.4) V lim Vi--->00

exists. Let -y : [0, oo) -4 Mn be the unique unit-speed geodesic with -y (0) =0 and d-yl dt (0) = V. Recalling that the solution of an ODE is a continuousfunction of its initial data, we note that (B.4) implies the images -yi, --p yuniformly on compact subsets of M'. Because each segment of every yi isminimal, it follows that each segment of y is minimal.

DEFINITION B.43. If y is a ray emanating from 0 E .Mn, the pre-Busemann function b.y,8 : Mn -> R associated to the ray 'y and s E [0, oo)is defined by

b.y,s(x)=d(0,y(s))-d(y(s),x)=s-d(-y(s),x).

LEMMA B.44. The functions b.y,s are uniformly bounded: for all x E Mnands>0,

Ib7,,(x)I <d(x,O).The functions b.y,s are uniformly Lipschitz with Lipschitz constant 1: for allx,yEMn ands>0,

Ib7,s (x) - b7,s (y) I < d (x, y)

And the functions b.(,3 (x) are monotone increasing in s for each x E Mn:ifs < t, then

b..,3 (x) C b.y,t (x) .

PROOF. All three statements are consequences of the triangle inequality.The first is immediate, and the second follows from observation

Ibti,s (x) - b..,s (y) I = I d (y, 'y (s)) - d (x, 'y (s)) 1 <- d (x, y)

To prove the third, we note that d ('y (s) , y (t)) = t - s and observe that

b..,,t (x) = t - d (x, -y (t))

> s+(t-s)-d (x,'y(s))-d(y(s),y(t))s-d(x,y(s))-b.y,$(x)

The monotonicity of the pre-Busemann functions in the parameter senables us to make the following

DEFINITION B.45. The Busemann function by : Mn --> Ilk associatedto the ray y is

b.y (x) _ lim b.y,s (x) .800

Since the family {b..,,3} is uniformly Lipschitz and uniformly boundedabove, we can immediately make the following observations.

LEMMA B.46. The Busemann function by associated to a ray y emanat-ing from 0 E Mn is bounded above: for all x E Mn,

(B.5) l b-, (x) I < d (x, 0).

And b-y is uniformly Lipschitz with Lipschitz constant 1: for all x, y E Mn,

(B.6) Ib.y(x)-b.y(y)1 :5 d(x,y)

Intuitively, b.y (x) measures how far out toward infinity x is in the direc-tion of -y. One could also regard b.y as a renormalized distance function fromwhat one might think of as the `point' y (oo). For example, in Euclideanspace, the Busemann functions are the affine projections. In particular, if

y : [0, oo) - I18'l is a ray with 'y (0) = 0 and dry/dt (0) = V, then theassociated Busemann function is easily computed using the law of cosines:

by (x) = lim (d (0, 0 + sV) - d (0 + sV, x))s-+o.o

= lim I s- s2+Ix-0I2-2s(x-O,V))s--oo \

_ (x-O,V).DEFINITION B.47. The Busemann function b :.M" --> R associated

to the point 0 E M7 isb _ sup by,

7

where the supremum is taken over all rays y emanating from 0.

This definition enables us to associated a Busemann function to a point.In Euclidean space, the Busemann function associated to a point 0 E R'1 isjust the distance function, because for all x E ]RY2,

b(x)x-0,'x-01)-Ix-01=d(x,O).

LEMMA B.48. The Busemann function b associated to 0 E Mn isbounded above: for all x E M'ti,

b (x) I < d (x, O) .

And b is uniformly Lipschitz with Lipschitz constant 1: for all x, y E M',

Ib(x)-b(y)I <d(x,y)PROOF. The first statement follows from (B.5). To prove the second,

note that for any x, y E M1 and e > 0, there exists a ray y emanating from0 such that

b(x)-b(y) by(x)+e-b(y) <b.y(x)+E-by(y)Hence by (B.6),

b (x) - b (y) < d (x, y) + e.

The following result says that any sufficiently long minimal geodesicsegment emanating from a point 0 can be well approximated by a ray em-anating from 0.

LEMMA B.49. Given 0 E M', define 0 : [0, oo) --> [0, 7r] by

0 (r) = sup inf Lo (Q (0), p (0)),vES(r) PER

where S (r) is the set of all minimal geodesic segments a of length L remanating from 0, and 7Z is the set of rays emanating from 0. Then

lim 9(r)=0.r-o0

PROOF. If not, there exists e > 0, a sequence of points pi E Mn withd (pi, 0) / oc, and minimal geodesic segments Qi joining 0 and pi such that

Lo (&i (0), P (0)) >- E

for each i and all rays p emanating from 0. By compactness of the unitsphere in TO.M', there exists a subsequence such that limi__,,,,, di (0) Vexists. Let o : [0, oo) - Mn be the unique geodesic with o-,),, (0) = 0and o',, (0) = V. Arguing as in Lemma B.42, we see that a,, is a ray. Inparticular, the condition

is impossible.

Lo (Qi (0) , d (0)) > E

Recall that we already have a good upper bound for the Busemannfunction associated to a point:

b(x) <d(x,0).The previous lemma lets us construct a lower bound for b in the event thatMn has nonnegative sectional curvature.

COROLLARY B.50. If (M',g) is a complete noncompact Riemannianmanifold of nonnegative sectional curvature, then

b (x) > d (x, 0) (1 - 9 (d (x, 0))) .

PROOF. Given any point x E Mn, let a be a minimal geodesic segmentjoining 0 and x. By the lemma, there exists a ray -y emanating from 0 suchthat

L(y(0),a(0)) < 9(d(x,0)).Set y (d (x, 0)). By the Toponogov comparison theorem applied to thehinge with vertex 0 and sides a joining 0 to x and 'YI [0,d(x,o)] joining 0 toy, we have

d (x, y) < 9 (d (x, 0)) d (x, 0) ,

where the right-hand side is the length of an arc of angle 0 (d (x, 0)) in acircle in the Euclidean plane of radius d (x, 0). Thus since b.y (y) = d (x, 0),we get

6 (x)>b.y(x)>by(y)-d(x,y)>d(x,0)[1-0(d(x,0))],where we used the fact that b.y has Lipschitz constant 1 to obtain the secondinequality.

The gist of the this is that the Busemann function b associated to a point0 is similar to the distance function to 0. However, from some points ofview, b has better properties. The property we shall be most interested inis convexity.

3.2. Constructing totally convex half spaces. An important useof Busemann functions is to define arbitrarily large `totally convex subsets'in a complete noncompact manifold of nonnegative sectional curvature. Asa first step in their construction, we make the following definitions.

DEFINITION B.51. Given a ray y : [0, oc) --f Mn emanating from a point0 E Mn, the open right half space is the union

3y U B(y(s),s),sE(0,00)

where B (y (s) , s) {x E Mn : d (x, -y (s)) < s}. The closed left half spaceH. is its complement:

Ky_.Mn\3ry=Mn \ U B(y(s),s).

sE(0,00)

The motivation for the term `left half space' is the following result.

LEMMA B.52. b.y (x) < 0 if and only if x E ILL

PROOF. By Lemma B.44, b y,,5 is monotone increasing in s. So by (x) < 0if and only if b.y,,3 (x) < 0 for all s E (0, oo). But by definition

by,, (x) < 0 if and only if x V B (y (s) , s) .

Hence by (x) < 0 if and only if x 0 U8E(o,00) B (-y (s) , s). O

DEFINITION B.53. A set X C M" is totally convex if for every x, y EX and every minimizing geodesic a joining x and y, we have a C X.

Our interest in half spaces is explained by the following result.

PROPOSITION B.54. If y is any ray in a complete noncompact Riemann-ian manifold (Mn, g) of nonnegative sectional curvature, then the closed lefthalf-space IHiy is totally convex.

We shall give two proofs of the proposition, which are quite different incharacter. The first uses the second variation formula, while the second usesthe Toponogov comparison theorem.

PROOF OF PROPOSITION B.54 IN THE CASE sect (g) > 0. If My is nottotally convex, there are x, y E lfi[y and a unit-speed geodesic a : [0, f] --> Mnsuch that a (0) = x and a (.E') = y, but a Z iHly. (We abuse notation bywriting a to denote its image a ([0, .£]).) First observe that B (y (t) , t)B (-y (s) , s) for all t > s > 0, since by Lemma B.44,

pEB(y(s),s) 0<by,,(p):5 by,t(p) = pEB(y(t),t)Then note that by definition of H',, we have a fl B (y (so), so) 34 0 for someso > 0, hence for all s > so. Fix any s > max {so,1}. Since the image of ais compact, there is Ts E [0, 2] and a minimal geodesic N3 : [0, 23] --> Mn suchthat/3 (0)=a(rs) and Q3(23)=y(s),where 43_L(,C33)=d(a,y(s)) <s.

In fact, TS E (0, £), because a (0) , a (t) E H. Thus we can apply the firstvariation formula to conclude that a (r5) 1 /3s (0) at a (Ts) = /3s (0). Nowparallel translate the unit vector a (TS) to get a unit vector field U alongand define the variation vector field

V W. (a)) _ L(0s) a U (Ns (Q)) 0 < o, < L (,33)

Note that V is the Jacobi field of a family of geodesics joining a to y (s);in particular, V (/3 (0)) = a (Ts) and V (/3s (PS)) = 0. Because as is minimalamong all geodesics joining a to y (s), applying the second variation formulawith S - (/3s)* (d/da) yields

0< jL(39)(vsv2_(R(s,v)v,s)) dQ.

0

Since 17S V = - Ls

U, we have

fL(3B) 1 1

JIVsV 2 do, =

L (QS)2 d L (Qs)

0

But since sect (g) > 0, there is e > 0 depending only on a C M' such that

JL(ae)

1(R (S, V) V, S) do, > J (R (S, V) V, S) da

f1(L))2L(/3s)-1

-e L(as)Hence H. can fail to be totally convex only if

1 L(8s)-10 - L L (l3s) '

which is equivalent to the condition

e (L (/3s) - 1) < 1

holding for all s > so > 0. Since L (/3s) -+ oc as s -+ oo, this is impossible.

PROOF OF PROPOSITION B.54 IN THE CASE sect (g) > 0. If M is nottotally convex, there are x, y E IHL and a unit-speed geodesic a : [0, fl ---> ,M'such that a (0) = x and a (2) = y, but a Z IHIy. As in the proof above, thereis so > 0 such that a n B (-y (s) , s) 0 for all s > so. Since a ([0,U]) iscompact, there are for any s > so some Ts E [0, e] and a minimal geodesic/3S,T8 : [0, -* ,Mn such that l3s,re (0) = a (Ts) and /3 ,T3 (QS,T3) = -y (s),where 2S1T8 = L (,(3s Te) = d (a, y (s)) < s. Define

e = so - d (a (Tso) ,'y (so)) = so - d (a, y (so)) > 0

and notice that the triangle inequality proves for all s > so that

" 9,Ty = d (a, -y (s)) < d (a (Tso) 'Y (s))

(B.7) < d (a (Tso) 'Y (so)) + d (-y (so) , y (s)) = s - E.

Let 13s,0 : [0, £s,o] --> M' be a minimal geodesic such that J3s,o = a (0) _x and /3s,o (&s,o) = y (s), where 4,o _ L (/3s,o) = d (x, -y (s)). Becausea (0) , a (f ) E IH[.y = M UsE(0,00) B (y (s) , s), we have Ts E (0,2) and

(B.8) Cs,Te < s < 2s,o.

In particular, we can apply the first variation formula at Ts to concludethat a (Ts) 1 Qs,TS (0). Thus we have constructed a right triangle withvertices y (s) , a (0) , a (Ts) and sides 1s,o, aI[o,Tel, Ns,T5 Since sect (g) > 0, theToponogov comparison theorem implies that the hypotenuse length satisfiesthe inequality

(B.9) £s,o = L (Qs,o) < L L (NS,Tg )2 = V-7r2 + S, Ty

Recalling that Ts E [0, £] and combining inequalities (B.7), (B.8), and (B.9),we see that H. can fail to be totally convex only if

s2 <TS +Q2 <Q2+(s-E)29 e

for all s > so. Since f and E are independent of s, this is impossible.

3.3. Constructing totally convex sublevel sets. We are now readyto construct the sublevel sets of the Busemann function associated to a pointO E Mn.

DEFINITION B.55. If y : [0, oc) --4M' is a ray and s c [0, oo), theshifted ray ys : [0, oc) -> M' is defined for all s' E [0, co) by

ys(s') y(s + s)

Note that yo = y. It is worth noting some elementary properties of thehalf spaces associated to a shifted ray.

LEMMA B.56. If s < t, then Byt C ]I$.ys.

PROOF. For all r E [0, oo), we have the inclusion

B(y(t+r),r) C B(y(s+(t-s+r)),t-s+r) 9 Bys.Hence

yt _ U B(y(t+r),r) 3-y,rE(0,oo)

H

0

LEMMA B.57. Whenever 0 < s < t,

dye = {x E .Mn : d (x, By,) < t - s} .

PROOF. We prove both inclusions C and D to obtain equality.(C) If X E 3.y8, there exists r > 0 such that d (x, y (r + s)) < r. We may

assume r > t - s, because d (x, y (q + s)) < q for all q > r. Then we have

d(x,B(y(s+r),r- (t-s))) <r- (r- (t-s)) =t - s.

This proves that d (x, H < t - s, because

B(y(s+r),r- (t-s)) = B (t + (s + r - t), s + r - t) C F97t-

(D) If d (x, IE$.yt) < t - s for some x E Mh, then there exists y c 1B.yt suchthat d (x, y) < t - s. By definition of B.yt, this implies there is r E (0, oo)such that y E B (yt (r), r). Thus we have

d(x,y(s+(t-s+r))) =d(x,y(t+r))< d(x,y) +d(y,y(t+r)) < t - s+r,

and hence x E B(y(s+(t-s+r))t-s+r) C H 9

DEFINITION B.58. Given a point 0 E M' and s E [0, oc), the sublevelset of the Busemann function associated to 0 is

c,, + n 8 ,

7ER(O)

where R (0) is the set of all rays emanating from O.

Note thatC3=MTh\ U

ryEIZ(O)

is79.

The next several results explain both the name and the usefulness of C.

LEMMA B.59. For every choice of origin 0 E Mn and s E [0, oo), thesublevel set of the Busemann function associated to 0 is given by

C9={xEM?: b(x)<s},where b is the Busemann function associated to O. In particular, we have

Cs C Ct

whenever 0 < s < t, and

5C9={xEM": b(x)=s}

because b is continuous.

PROOF. Observe that bye = b.y - s, since for all x E Mn,

bye (x) = lim bryeit (x) = lim (t - d (y3 (t) , x))t-+oo t-soo

_ -s+ lim (s+t-d(y(s+t),x)) = -s+b.y(x).t 00

Hence by Lemma B.52,

C5= n ]EIL = n {xEMn:b73(x)<0}ryER(O) ryE1Z(O)

={xEMn:bry(x) <sfor all yE7Z(O)}{xEMT:b(x)<s}.

COROLLARY B.60. Ifs > 0, then CS contains the closed ball of radius sat 0.

PROOF. By Lemma B.48, we have b (x) < d (0, x), and hence

Cs=Ix EMn:b(x)<s}D Ix EMn:d(0,x)<s}=B(0,s).

COROLLARY B.61. For any choice of origin 0 E Mn, one has Us>0C5 =Mn.

PROOF. Mn = Us>0B (0, s) C Us>OCS.

PROPOSITION B.62. For every choice of origin 0 E Mn and s E [0, oo),the set C, is compact and totally convex.

PROOF. By Proposition B.54, C. is the intersection of closed and totallyconvex sets, hence is itself closed and totally convex.

Suppose Cs is not compact. Then there exists a sequence of points pi ECs with d (O, pi) --> oo as i --> oo. For each i, let Oi : [0, d (0, pi)] --> Mn bea unit-speed minimal geodesic from 0 to pi. Since C. is totally convex, eachNi C C9. After passing to a subsequence, we may assume the unit tangentvectors ,i (0) converge to a unit vector V,,,, E TOM'. Let /3,,, : (0, oo) -> Mndenote the geodesic with Qcc, (0) = V. As in Lemma B.42, we observe that/3,,) is a ray such that the images j3 -+ & uniformly on compact sets. Nowfor any t E (0, oo), consider the point (s + t). Then Pi (s + t) E C,s isdefined for all i large enough, and

Jim d(Qi (s + t)) = 0.

Since C, is closed, this implies /3,,, (s + t) E C, which contradicts the factthat / 3, , (s + t) E 13(0.)..

COROLLARY B.63. The Busemann function b associated to a point 0 EMn is bounded below.

PROOF. b (x) is continuous and CS = {x E Mn : b (x) < s} is compact.

We have seen that the totally convex sets CS exhaust Mn. The nextresult refines Lemma B.59 and gives the sense in which the level sets of bare parallel.

PROPOSITION B.64. For any choice of origin 0 E Mn and all s < t,

(B.10) C5= {xECtd(x,(9Ct)>t-s}.In particular, 0 E aC0 and

aC={xECt:d(x,aCt)=t-s}.PROOF. By Lemma B.59, we have 0 E 0C0 = {x E Mn : b (x) = 0},

because b (0) = 0. And since the distance function is continuous, the char-acterization of DCs given here follows from (B.10). So it will suffice to proveboth inclusions C and D in (B.10).

(C) Suppose x E Ct and d (x, aCt) < t - s. We want to show thatx C8. Because

d (x, U-yER(o)157t) = d (x, Mn\Ct) = d (x, 9Ct) < t - s,

there exists a ray 0 emanating from 0 such that d (x, 3,3t) < t - s. ByLemma B.57, this proves that x E Bp.,, hence that x Cs.

Suppose x E Ct\C5. We want to show that d (x, aCt) < t - s. Sincex 0 Cs = f1 E7Z(o)IH- 8 , there is a ray Q emanating from 0 such that x E I.By Lemma B.57, this proves that d (x, Il$pt) < t - s. Hence

d (x, aCt) = d (x, Mn\Ct) = d (x, U.yE7.(0)1HB.yt) < t - s.

4. Estimating injectivity radius in positive curvature

The objective of this section is to prove the following result.

THEOREM B.65. Let (M', g) be a complete noncompact Riemannianmanifold of positive sectional curvatures bounded above by some K E (0, oo).Then its injectivity radius satisfies

inl (Mn' 9) ? vf K__

Before proving the theorem, we will establish some preliminary results.Assume for now only that the sectional curvatures of (Mn, g) are boundedabove by K > 0.

DEFINITION B.66. If k E N*, a proper geodesic k-gon is a collection

r={'yi:[0,fi]--_M?:i=1,...,kJof unit-speed geodesic paths between k pairwise-distinct vertices pi E Mnsuch that pi = -yi (0) = -yi_1 (ii- 1) for each i, where all indices are interpretedmodulo k. The total length of a proper geodesic k-gon is

kL(r) L(1i)

i=1

r is a nondegenerate proper geodesic k-gon if 4PZ (- yi_1, ryi) 0 foreach i = 1, . . . , k; if k = 1, we interpret this to mean L (r) > 0.

4. ESTIMATING INJECTIVITY RADIUS IN POSITIVE CURVATURE 313

Note that a proper j-gon may be regarded as a proper k-gon for anyk > j merely by choosing extra vertices. In fact, it will be easier to dealwith limits if we work with a more general collection of objects.

DEFINITION B.67. A (nondegenerate) geodesic k-gon is a (nonde-generate) proper geodesic j-gon for some j = 1, ... , k.

If the theorem is false, there is a nondegenerate geodesic 2-gon IF in M'of total length 2A < 7r/-v/_K. Let 0 E Mh be any choice of origin, andlet Cr denote the sublevel sets of the Busemann function based at 0. ByCorollary B.61, the collection {Cr : 0 < r < oo} exhausts Mn. Thus thereis s E (0, oo) such that I' C CS. By Proposition B.62, CS is compact. Sodefine

A --' {nondegenerate geodesic 2-gons in Cs of total length < 2A}

and{geodesic 2-gons in CS of total length < 2A}.

Let S"-1CS denote the unit sphere bundle of Mn restricted to Cs. Observethat

ACE C (Sn-1CS x [0, 2A]) X (Sn-1C8s x [0, 2A]),

because any r E ' is described uniquely by data (p, V, p', V', fl, whereyi is the geodesic path determined by y1 (0) = p, j (0) = V E Tp.Mn, andL (y1) = 2 E [0, 2a], while y2 is the geodesic path determined by rye (0) = p',Y2 (0) = V' E Tp,MT, and L (y2) = .E' E [0, 2A]. (V may be chosen arbitrarilyif f = 0, likewise for V' and 2'.) It is easy to see that 8 is closed in the inducedtopology, hence is compact.

LEMMA B.68. Let (Mn, g) be a complete noncompact manifold with sec-tional curvatures bounded above by K. Let A C 8 be defined as above. ThenA is closed, hence is compact.

PROOF. Suppose {ai} is a sequence from A such that ai - a,, E E.Equivalently,

(pi, Vi, 4, pi, V', Qs) - (P., V., too, Po ' Voo, Qoo)

We first claim a.. ) is not a degenerate proper 1-gon. This can happenonly if £ = I?' = 0, hence only if p,, = p' . But since C, is compact, we00see that this is impossible, because

0 < inj (Cs) < inj (pi) = d (pi, cut (pi)) < max {fi, Q'i} \ 0

as i -->oo.We next claim that ac.. is not a degenerate proper 2-gon. This can

happen only if the path /3 from p,,, to p' # p0c, is the path p' from p'to per, traced in the opposite direction. Then for i sufficiently large, thereare distinct but nearby geodesics /i from pi to pi and ,62 from pZ to pi,such that max {L (/3i) , L (/3i') } <

3But this is impossible, because

the sectional curvature hypothesis implies expp, : B(0, 7r/v,-K-) - Mn is animmersion.

Now by the claims above, we know a. is a nondegenerate proper 1-gonor 2-gon, hence belongs to A. It follows that A is a closed subset of thecompact set HE.

This compactness property implies there is ,0 E A whose total lengthrealizes L (0) = infrEA L (F). The extra hypothesis of positive sectionalcurvature guarantees that Q is smooth:

LEMMA B.69. Let (M', g) be a complete noncompact Riemannian man-ifold of positive sectional curvatures bounded above by K > 0. Then any,3 E A such that L (/3) = infrEA L (F) is a smooth geodesic loop.

PROOF. By choosing a vertex at L (/3) /2 if necessary, we may supposewithout loss of generality that /3 is a nondegenerate proper 2-gon corre-sponding to the data (p, V, £, p', V', Q'). Let y denote the path from p to p',and let y' denote the path from p' to p. We will show that ,8 is smooth atp', hence at p by relabeling.

Define a 1-parameter family

{/3t:0<t<4 =min{t,P}}of nondegenerate proper 2-gons /3t by taking pt = -y (2 - t) and letting ytdenote the truncated path 'I [o,t_t]. Then by Corollary B.22, there is aunique geodesic yt near -y' joining p't to p. By Proposition B.62, CS istotally convex; since p, p't E C3, it follows that yt lies in Cs, hence thatQt E A for all t E [0, 2o). Notice that ,0 can fail to be smooth at p' only ify (2) y' (Q'). But the variation vector field of Qt is a Jacobi field V alongy' with V (0) = -'y (Q) and V (41') = 0. Thus by the first variation formula,we take a one-sided derivative to obtain

d L (at) =(V (0) (0) + (L)) _ - (y (i) , y' (o)) - (y (L) , (L))

t=o

= - (- (4) ,'Y (0)) - I < 0,

because 1yJ = ry'J = 1. This contradicts the minimality of L (,8) in A unless/3 is smooth at p'.

PROOF OF THEOREM B.65. We have already shown that the result canfail only if there is a smooth geodesic loop )3 of length less than 7r//contained in a compact totally geodesic set Cs based at 0 E Mh. Let abe any ray emanating from 0. For t E (1, oo) to be chosen later, join theloop 0 to the point a (t) by a minimal geodesic yt : [0, 2t] -3 M', whereyt (0) E /3, yt ($t) = a (t), and it _ d (a (t), /3). Since /0 is smooth, we canapply the first variation formula to conclude that yt 1 /3 at yt (0) E ,3. Asin the first proof of Proposition B.54, we shall obtain a contradiction byapplying a second variation argument to yt among curves joining /3 to a (t).

Parallel translate Q (yt (0)) to get a unit vector field U along yt, anddefine

0<T<2t.V ('Yt (T))it

T U ht (T))itThen V is a Jacobi field with V (yt (0)) (yt (0)) and V (yt (it)) = 0.Because yt is minimal among geodesics from 3 to a (t), the second variationformula applied with T - yt yields

0< o (vTv2 - (R(T,V)V,T)) dr.J0

Noting that VTV = - 1 U, we calculate that the first term isit

1et

IVTV12dT=2t f dT

Since sect (g) > 0, there is e > 0 depending only on, 3 c C3 c M, such that

JQt (R (T, V) V, T) d-r > J 1 (R(T,V)V,T) d-r

1(ft-T 2dT Eit-1

>Jo \ £t )

it

Hence if there is a smooth geodesic loop R of length less than 7r/vrK- con-tained in C, we must have

0 <1+E-EPt

itholding for all choices of t E (1, oo). Since it -+ oo as t -* oo, this isimpossible.


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Index

almost flat manifold, 17ancient solution, 28

ancient, 235ancient Type I essential point, 272asymptotic analysis of singularities, 40avoidance set, 102, 270

BBS derivative estimates, 116, 201, 223Berger collapsed sphere, 11Bianchi identities, 72Bochner-Weitzenbock formula, 144, 195,

300Busemann function

associated to a point, 305associated to a ray, 304

cigar soliton, 24, 168collapse, 11Compactness Theorem, 168, 231, 240component representation of tensors, 279conformal change of metric, 106conjugate point, 289conjugate radius, 290connection, 280covariant derivative, 280critical point, 289critical value, 289cross curvature tensor, 87cross-curvature flow, 88curvature blow-up rate

blow-up rate, 234lower bound for, 240

curvature controlled sequencelocally, 238

curvature essential sequence

locally, 238curve shortening flow, 160cut locus, 296

in the tangent space, 298cut point along a geodesic, 296

degnerate neckpinch, 63, 237, 244

DeTurck trick, 79differential Harnack estimates, 132, 143,

148, 274differential Harnack quantity

for curvature of variable sign, 146, 155,169

for positive curvature, 144dimension reduction, 237, 250, 255, 262divergence, 69, 75, 282

formal adjoint of, 75, 283doubling-time estimate, 138, 225

Einstein operator, 83elliptic differential operator, 72elliptic PDE

local solvability of, 91Elliptization Conjecture, 3entropy, 277

for a surface with mixed curvature, 135for a surface with positive curvature,

133for positive curvature, 141

eternal solution, 24eternal, 235

evolution of length, 70exterior derivative, 282, 283

formal adjoint of, 283

fixed point of Ricci flow, 21focal point, 293

Gauss Lemma, 287Gauss-Bonnet Theorem, 162, 166generalized distance function, 300geodesic k-gon, 150, 290

proper, 312geometric structure, 4geometrically atoroidal manifold, 2Geometrization Conjecture, 3global injectivity radius estimate

323

324

on the scale of its maximum curvature,239

gradient Ricci soliton, 22on a surface, 112

Gromov-Hausdorff limit, 12, 15, 232

Haken manifold, 3half spaces, 307harmonic coordinates, 90harmonic map flow, 85harmonic map Laplacian, 85Heisenberg group, 15Hodge star operator, 188homogeneous metric, 4homogeneous model, 4Hopf fibration, 11Hyperbolization Conjecture, 3

immortal solution, 34immortal, 235

incompressible surface, 2injectivity radius

at a point, 298of a Riemannian manifold, 298

injectivity radius estimate, 150on the scale of its maximum curvature,

268irreducible manifold, 2isometry group of a Riemannian mani-

fold, 5isoperimetric constant, 157isoperimetric ratio, 157isotropy group, 4, 5, 7isotropy representation, 5

Jacobi equation, 292Jacobi field, 292

Kahler-Ricci soliton, 25, 34, 38Kazdan-Warner identity, 106, 118Klingenberg injectivity radius estimates,

142, 143, 153, 166, 276, 277, 298Kulkarni-Nomizu product, 176

Lambert-W function, 37Laplacian

conformal change of, 109Hodge-de Rham, 92, 284Lichnerowicz, 69, 92, 174, 284rough, 284

Laplacian comparison, 301Lie algebra square, 184, 187Lie derivative, 75, 282local injectivity radius estimate

INDEX

scale of noncollapse, 240local singularity, 38locally homogeneous metric, 4

maximal model geometry, 4maximum principle

enhanced tensor versions, 101scalar versions, 93strong, 103tensor versions, 97

Milnor frame, 9minimal existence time, 226model geometry, 4moving frames

calculations in, 26, 106Myers' Theorem, 173, 192

Nash-Moser implicit function theorem,79

necklike point, 262neckpinch singularity, 39, 237normal bundle, 293normalized Ricci flow, 212

normalized, 1on a surface, 105

null eigenvector assumption, 97, 100

parabolic dilation, 49parabolic dilations, 168, 239parabolic PDE

subsolution of, 95supersolution of, 93, 95

pinching behavior, 38Poincare Conjecture, 3, 173point picking arguments, 236, 243, 245,

247

porous media flow, 31potential of the curvature, 112

trace-free Hessian of, 112, 128pre-Busemann function, 303Prime Decomposition Theorem, 2prime manifold, 2principal symbol

of a linear differential operator, 71of Ricci linearization, 72

rapidly forming singularity, 63, 65Rauch comparison theorem, 294ray, 303reaction-diffusion equation, 24, 39, 110,

175relative curvature scale

relative, 242Riccati equation, 301

Ricci flow, IRicci identities (commutator formulas),

286Ricci soliton, 22, 128

on a surface, 112Ricci-De Turck flow, 80Rosenau solution, 32

Schoenflies theorem, 158Schur's Lemma, 194Seifert fiber space, 2self-similar solution, 22, 112separating hypersurface, 157shifted ray, 309singular map, 289singularity models, 235

backward limit, 250Type I-like, 242Type II, 246, 247Type II-like, 245Type III, 250Type X-like, 235

singularity time, 233singularity types

classification of, 234slowly forming singularity, 64, 65Spherical Space Form Conjecture, 173stability of geodesic loops, 151sublevel set of a Busemann function, 310

Toponogov comparison theorem, 302Torus Decomposition Theorem, 2total symbol

of a linear differential operator, 71totally convex set, 307Type I essential point, 262

Uhlenbeck trick, 180Uniformization Theorem, 105unimodular Lie group, 7

variation formulas, 67

INDEX 325

Witten's black hole, 24

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