link.springer.com978-3-0348-8613-0/1.pdf · a proof of lichnerowicz' formula for completeness,...

A Proof of Lichnerowicz' Formula

For completeness, we include here Lichnerowicz' proof for the formula of DR(g). It is taken from [69].

In order to get the correspondence between our notations and definitions and his, note

that 8i ylg = v'9. r{j (here and in the following 8i = 8~').

PROOF:

The covariant derivative V' j is defined as follows:

for 1-1 tensors

and analogously for tensors of other type with a similar + rT term for every upper index and a - rT term for every lower index. Covariant derivatives of the metric are o.

156 A. Proof of Lichnerowicz' Formula

As for all other indices, we can raise and lower the index of the covariant derivative: vj := gj/cV/c.

Our definition of the divergence agrees with the usual definition

The Riemann curvature tensor is defined by

The Ricci curvature tensor is defined by

Rf36 = R'" f3",6

It is symmetric.

The scalar curvature is defined by

There are different conventions used in the literature. Ours is the same as in Lichnerowicz

and in [77].

The Ricci tensor and consequently the scalar curvature differ in sign from Spivak's con

vention [103]. He uses Rf3'Y = R"'f3'Y",.

We have already seen that if 9 is changed in direction h, then

(For any function of the metric tensor, let D f := D f(g)h.)

Now

Dr ",f3,p = H 8",hf3p + 8f3hap - 8phaf3 )

HVahf3p + r~f3hC7P + r~ph/3C7 + V f3hap + r~ah"p + r~pha,,- V phaf3 - r~hC7f3 - r:f3haC7)

HVahf3p + V f3hap - V phaf3 ) + r~f3hC7P .

-h'Ypraf3,p + g'YP [HVahf3p + Vf3hap - V phaf3 ) + r~f3hC7P] -h;r~f3 + hJr~f3 + HVa h; + Vf3h~ - V'Yhaf3) HVah; + V f3h~ - V'Yhaf3 ) .

157

Therefore,

Note that X is a tensor although r is not. The difference of the Christoffel symbols with respect to two metrics is a tensor, and therefore so is the derivative of a Christoffel symbol with respect to the metric. This is obvious from the explicit formula, too. Therefore, the covariant derivative of X is a reasonable concept. Use

to get

DR"'fh6 = V.,X~ - V6X;" .

Contracting a with 'Y and inserting the formula for X gives

DR{36 ~V",(V{3h6" + V6h~ - V"'h{J6) - ~V6(V",h~ + V{3h: - V"'h",(3) = HV", V{3h6" + V",V6h~ - V",V"'h{36 - V6V{3(tr h)) .

Therefore

DR = D(gf36 R{J6) = -h{J6 R{36 + 9{36 DR{36

= _h{36 R{J6 + ~ (V", V6h: + V'" V{3h~ - V'" V"'(tr h) - V{3V (3(tr h)) -h{36R{J6 + SgSgh - .6.g(tr h) .

This formula holds quite generally, but in 2 dimensions, the first term can be considerably simplified.

We claim that in 2 dimensions, h",{3 R",{3 = tR. trgh.

The conclusion is done in conformal coordinates. Let

911 = 922 = P, 912 = 0

Then, direct calculations show:

R1- R2 __ 4(Jnp) 1- 2- 2p

(A.l)

B On Harmonic Maps

Introduction

There are now various criteria guaranteeing the existence of harmonic maps between

surfaces, and, in general, between Riemannian manifolds. The first major breakthrough

was the Eells-Sampson paper [28], mentioned in theorem 3.1.1. As a special case of

this result one obtains the existence of a smooth harmonic map from (M,g) to (M,go), go EM_I, and gEM an arbitrary metric. The techniques of Eells-Sampson employ

the heat flow and therefore involve the study of non-linear parabolic partial differential

equations. For a proof of the Eells-Sampson theorem using the Palais-Smale condition via a perturbation argument see Uhlenbeck [119],[120]. For two dimensional surfaces M, N the existence of a smooth harmonic map between M and N in every homotopy class for the

case 7r2(N) = 0 was proved by several authors, namely Lemaire [66], Sacks-Uhlenbeck [95] and Brian White [121], and these proofs involve technical analytical results along with the replacement technique of Morrey [78].

Jiirgen Jost [57], on the other hand, has given a direct proof of the existence of a har

monic diffeomorphism. We have already encountered the basic compactness idea behind

this proof in lemma 3.2.3, namely that diffeomorphisms with bounded Dirichlet energy

are equicontinuous (however, the fact that the limit of a minimizing sequence of diffeo

morphisms is also a diffeomorphism is non-trivial).

The advantage of the heat flow technique is that it works in arbitrary dimensions and in

the case 8M #- 0 where one assumes Dirichlet boundary data.

We present here our own proofs of existence, uniqueness and smooth dependence. The

existence prooffollows the spirit ofthe general approach to Morse theory developed in [111]

and we shall say more about this shortly. Our proof of existence can be modified to work,

like that of Eells-Sampson, in arbitrary dimensions and in the situation where the domain

159

has non-empty boundary and Dirichlet boundary data are used, when the range is compact and has negative sectional curvature. The case of harmonic maps satisfying Dirichlet boundary conditions was first treated by Hamilton [49] by heat equation methods and for harmonic maps of surfaces by Lemaire [67]. However the most elegant way to treat the general problem of harmonic maps with Dirichlet boundary data is due to Hildebrandt, Kaul, Widman [54],[55],[56] and Giaquinta and Hildebrandt [41]. For an overview of these and other results see the survey paper of Hildebrandt [53].

A harmonic map S : (M,g) -t (M,go) is a critical point of Dirichlet's energy for 9 and go fixed. The Euler-Lagrange equation for harmonic maps is described in section 3.1. How can we find a critical point for Eg? The classical way is to produce an absolute minimum through the direct method of the calculus of variations, the method employed by Jost, Lemaire, Sacks-Uhlenbeck and White.

The idea of Morse theory, as generalized to Hilbert manifolds by Palais and Smale [89], [100], is the method of gradient descentj i.e. follow the flow of a gradient vector field until it leads you to a critical point. This method has, until now, never been made to work in providing a proof of the existence of harmonic mappings. The approach of Eells-Sampson is to follow the trajectories of the heat equation to obtain existence. This method, as beautiful as it is, and as influential as it has been for geometrical problems, requires somewhat more sophisticated methods in non-linear partial differential equations. Our goal is to show that the gradient method (which in reality is the method of ODEs to solve a PDE) works when viewed from the correct prospective. In doing so, we rely only on the linear theory of elliptic equations, the fundamental ezistence theorem of local solutions to ordinary differential equations and the Sobolev embedding theorems.

In the presentation that follows we fix two metrics 9 and go on M with the scalar curvature of go negative (for Teichmiiller theory we need only that go EM_I, but the proof for general go of negative scalar curvature is the same). Our goal is to first show that a harmonic map S(g) : (M,g) -t (M,go) exists, then to show that is is unique and smoothly dependent on g, and finally that S(g) is a diffeomorphism. We begin with:

Existence

For the convenience of the ,reader we review the Sobolev theorems. Let L:(M, m.d ) denote the space of maps from M into Euclidean space of dimension d which have all partial derivatives (in the sense of distributions) up to kth order in Y. (Other notations used for the same spaces are W""p or H""P j in our notation note that H"'(M) = L~(M,m.).) In the case dim M = 2, the Sobolev theorems take the following form:

160 B. On Harmonic Maps

1. If k - ~ ~ l- ~, and p' < 00 there is a continuous inclusion of Lt(M,rn.d ) into

L~' (M, rn.d ). If k - ! > l- !, and k > l this inclusion is completely continuous, or p p

compact.

2. If k - ~ > l then there is a completely continuous (= compact) embedding of

Lt(M, rn.d ) into Cl(M, rn.d ), the space of l times continuously differentiable functions.

For more details the reader should consult [3], [62], [75].

To begin our proof we must introduce the relevant infinite dimensional manifolds of maps.

Let k = 2, or 3 and let p > 2. Assume that M is embedded in rn.d • Let Lt(M, M) consist

of those maps in Lt(M, rn.d ) which map M to M. (This is well-defined since by the Sobolev theorems these maps will be continuous.) Then standard techniques [90] show

that Lt(M, M) is a smooth COO manifold and in fact a COO submanifold of Lt(M, rn.d ).

Again by the Sobolev embedding theorem there is a continuous inclusion of L~(M, M) into C1(M, M), the C1 self maps on M, and for k = 3 a continuous inclusion of L~(M, M) into C2(M, M), the C2 self maps of M. In both cases Lt(M, M), k = 2,3 are smooth manifolds. It is important to understand their tangent spaces.

The tangent space to Lt(M, M) at y consists of the Lt vector fields (3 over u, i.e. those

(3 E Lt(M,rn.d ) such that (3(z) E Tu(z)M for all z E M.

We shall now assume that (M,go) is isometrically embedded in rn.d . For p EM let

be the orthogonal projection. Recall from section 3.1 that S (M, g) -+ (M, go) is

harmonic iff

(AS)(p) = II (S(p)) (AgS)(p) = 0 , (B.1)

where Ag is the Laplace-Beltrami operator. Define the non-linear Laplacian A by equa

tion (B.1); S is therefore harmonic iff AS = o. In local conformal coordinates (z1,z2) on

(M,g),

AS - ~!!-.- 8So. d - Vii 8zl 8zl ' a = 1, ... , . (B.2)

For vector fields (3 over u we can define the linear Laplacian A (u is assumed fixed) by

(B.3)

161

where the kth component of the covariant derivative f& is given by

(BA)

where t~j are the Christoffel symbols of go.

The spirit of our approach to Morse theory for the calculus of variations taken in [111] was not to stress the gradient nature of the vector field whose trajectories are to lead you to a critical point, but to find the "right" vector field by solving an appropriate linear PDE on the given manifold of maps, in this case Lt(M, M).

We are now ready to define our vector field (3 on Lt(M, M) whose trajectories will lead us to a harmonic map. Again fix u E Lt(M, M).

Consider the linear partial differential operator

(B.5)

Here Zl are conformal coordinates such that gij = >.8ij • One checks easily that (3 ~ Eu({3) maps Lt vector field over u E Lt(M, M) to LL2 vector fields over u. Moreover (3 ~ Eu({3) is a linear self-adjoint second order operator, and therefore by standard elliptic theory the Fredholm alternative holds; i.e. (3 ~ Eu({3) is surjective iff it is injective. Another way of saying this is that the operator (3 ~ Eu({3) is a linear Fredholm operator of index zero.

Theorem B.1 For u homotopic to the identity, the map (3 ~ Eu({3) is an isomorphism of the Lt vector fields over u to the LL2 vector fields over u.

PROOF: By the previous remarks we need only show that Eu({3) = 0 implies (3 = o. Suppose Eu({3) = o. Then denoting the m.d inner product simply by (.,.) we have

f (Eu ({3), (3) dlLs = 0 . M

Integrating by parts and using the fact that the curvature is negative we obtain the two equations

(B.6)

and

(B.7)


Since u is C1 and homotopic to the identity, its degree (which is a homotopy invariant) is 1. By Sard's theorem, u has regular values (i.e. points y such that for every z E u-1(y), if any, Du(z) is an isomorphism). The degree being 1, these points y do have pre-images. Let us consider some point Po for which Du(Po) is an isomorphism.

Since (M, 90) has negative curvature equation (B.6) immediately implies that I3(Po) = o. Equation (B.7) on the other hand implies that the function p 1-+ Ilf3(p)II:a4 is constant. Thus 13 == 0, proving theorem B.1. •

We now define our vector field 13 on Lt( M, M) by

(B.B)

Since u 1-+ C,. and u 1-+ ~u are smooth, u 1-+ f3(u) is a smooth (COO) vector field on Lt(M, M). Moreover it is easy to see that f3(u) = 0 iff u is harmonic. Thus the zeros of 13 are precisely the harmonic maps.

Theorem B.2 If S E Lt(M, M) is harmonic, the Frechet derivative Df3(S) of 13 at S, is an isomorphism (in fact the identity isomorphism) ofTsLt(M, M), the space of Lt vector fields over S.

PROOF: f3(S) = 0 implies that for h E TsLt(M, M) the Frechet derivative Df3(S) : TsLt(M, M) --+ TsLt(M, M) satisfies the equation

1 ( 8S) 8S 1 ( 8S) 8S ~Df3(S)h + >.,'R Df3(S)h, 8zL 8zL = ~h + >.,'R h, 8zL 8zL

This last calculation is standard and we leave its verification to the reader. Let 'P = Df3(s)h - h. Then we have

which as in theorem B.l implies that 'P = o. • This last theorem says that each harmonic map is a non-degenerate critical point of Dirichlet's energy Eg in the sense introduced in [111].

Theorem B.3 The derivative of Dirichlet's energy Eg in the direction 13 is positive ezcept at a critical point, i.e. DEg(u)f3(u) ~ 0 and equals zero iJfu is harmonic.

PROOF: - J {~U, (3)m.4 dp,g

M

- J (~f3+ i'R(f3,~)~,f3)dp,g ~ 0 M

163

since go EM_I. The same argument as was used in theorem B.2 now shows that if D Eg( 1.£ )f3( 1.£) = 0 then f3( 1.£) = 0 and hence 1.£ is harmonic. •

Since f3 is a smooth vector field, given any initial point 1.£0 E Lt(M, M) we know by the fundamental existence theorem of ODE's that f3 has a flow Uti t-(uo) < t < t+(1.£o) with

dUt I Tt=f3(Ut) , Utt=o=1.£o.

Our goal is to show that t-(uo) = -00 for all initial conditions 1.£0 E L~(M,M), and Ut converges L~(M, M) to a harmonic map as t --+ -00. That this actually happens is suggested by the following two theorems:

Theorem B.4 Dirichlet's energy Eg(ut) strictly decreases as t decreases, unless 1.£0 is harmonic.

PROOF: ftEg(Ut) = DEg(Ut)~ = DEg(ut)f3(Ut), and by theorem B.3 DEg(ut)f3(Ut) > 0 unless 1.£0 is harmonic. •

Theorem B.5 It := ~Ut satisfies the pointwise exponential equation

Dlt - f, at - t

PROOF:

~~Ut ~~+i'R(~,~)~ ~f3 + in. (f3, ~) ~

= aUt·

Thus we have

(B.g)

•


PROOF: Dropping the subscript mel from the norm we have

d 2 d 2 (D It) II I 2 dt 118Utll = dt IIltll = 2 at' It = 2 It I

•

Corollary B.7 As t -+ t-(Uo), 118utll2 remains pointwise bounded by 118uoW. II t -+ -00, then 118utW -+ 0 pointwise. •

We now continue our proof of the existence of harmonic mappings. Since the theorems we have so far give only pointwise estimates on the non-linear Laplacian But and not on any norm involving the second derivatives of Ut, we cannot yet conclude that t-(uo) = -00.

The next step is therefore to work towards such a norm estimate. This will be theorem B.13. Let e(u) denote the energy density of a map u E Lt(M, M). Recall that in conformal coordinates e(u) = t(I:z,I:z)lR •. Then we have the following inequality which will be fundamental to our existence proof. For harmonic maps it is known as the Bochner identity [28) and was used for the existence of harmonic maps by Eells and Sampson.

Theorem B.8 1 l(D 8U) '2 8ge(u) ~ K(g)e(u) + ~2 8zl8U, 8zl

where K(g) is the Gauss curvature and gii = ~5ii.

PROOF: For convenience let us use complex notation

8 8 .8 2-=--.- , 8z 8z 8y

8 8 .8 2-=-+.-8z 8z 8y

(z, y) = (Zl, Z2). Again, using the Einstein summation convention, and again dropping the subscript ]Rei from the inner product we see that

4 8 8 = ~ 8z 8z e(u)

= 4 8 { ~.1 (8U 8u) 2 (D 8u 8U)} ~ 8z - ~2 8zl ' 8zl + ~ 8z8zl ' 8zl

165

h 2 D - D . D 2 D D . D d \ 0 \ C " . h" I were oz - 0'" + z oy' oz = 0'" - z oy' an I\'i;:::: ozl\. ontmumg we get t IS equa to

4 { (2AzAZ Azz) / 8u 8U) 2Az / D 8u 8U) 2Az / D 8u 8U) :x ~ - ~ \8xl ' 8xl - V \8z8xl ' 8xl - V \8z8xl ' 8xl

2/D8u D8U) 2/DD8u 8U)} +:x \8z8xL ' 8z8xl +:x \8z8z8xl ' 8xl

But t (AzAz/ A3 - Azz/ A2) = ~K(g), K(g) the Gauss curvature of the metric g. Using the fact that for a vector field v over U

( D D D D) (8U 8u) 8x 8y - 8y 8y v = R 8x' 8y v,

we get

(B.10)

Now if u : M _ M, we can take the covariant derivative V Du of Du : T",M - Tu(",)M. So Du E Hom(TM,u*TM) and VDU E Hom(TM,Hom(TM,u*TM)) ~ Hom(TM @

T M, u*T M). V Du is defined by

VDu = A:~dx'" @ dx~ @ 8~'YL ' where

( 82U'Y 8u'Y 8u' 8ui 0 )

A:~ = 8x"'8x~ - 8xp r~~ + 8x~ 8x'" rJ. and r~~ and tJ. are the Christoffel symbols of 9 and go respectively. Thus equations (B.IO) can be rewritten as

l~ge(u) = ~K(g)e(u) + 4~2 (R (:;., I;z) ;;., ;;,) + ~IV Dul2 + 4~2 (rE" o~r ;;r' ;;)

where IV Dul2 = (go)'i9"'P9P.~ A~~A~p."

Using the fact that go has negative curvature we see that

1 1/DD8u8U) 2~ge(u) ~ K(g)e(u) + A2 \ 8xl 8xT 8xr' 8xl (B.ll)


This ends the proof. • The concept of a covariant derivative employed here is a bit more subtle than the usual textbook definition, which is good enough everywhere else in this book. It is, however, not really necessary to understand the geometrical meaning of V Du here. The only thing we use is that the term called IV Dul 2 in the above formula is non-negative.

Lemma B.9 Let tp : M -+ IR be a Coo test function. Let Uo be an initial value for the gradient flow ~ = {3(Ut), t E]t-(uo),O]. Let eo = Eg(Uo) and Cl = sup lI~ut(p)1I

(t,p)

sup lI~uo(p)lI. We then have the following inequality: p

-~ 1 (Vge(Ut), Vgtp) dp,g ~ -C2 1 e(Ut) Itpldp,g - 3~1 1 Itpldp,g - Cl(2cO)1/2I1VgtpIlL' M M M

where IIVgtpllL2 is the L2-norm of the gradient of tp.

PROOF: Multiplying equation (B.ll) through by tp and integrating by parts we get

f K(g)e(ut)tpdp,g - f lI~utWtpdp,g M

+ f (~Ut, l7) (~) tpdp,g M

- f (~Uh l7) t (3;z) dp,g M

> - f IK(g)1 e(ut) Itpldp,g - Cl f Itpldp,g M

- f l (lI~utW ·I;:t 1+ IIl7W I~I) ·1~lltpldp,g - cl(2eo)1/2I1 VgtpIlL'

> - f IK(g)1 e( Ut) Itpldp,g - ~Cl f Itpldp,g M M

-l f e(ut} {>'~:'>'~} Itpldp,g - cl(2eo)1/2I1Vgtpllp

Covering (M, g) with a finite number of coordinate charts Ui , Wi, Ui :J Wi such that U Wi = M we see that we can bound (.~! + ,\~)/,\3 by some positive constant on each Wi. Therefore we can bound 11 ,\2+,\2

"2 e( ut) "',\3 1/ Itpl dp,g M

by C f e(ut)ltpl dp,g where c is some positive constant depending on (M,g). Combining M

this with the expression f IK(g)1 e(Ut) Itpl dp,g we can bound the sum of the first and the M

167

third terms immediately above by -C2 J e( ut) 1<p1 dJ.£g. Therefore the above inequality can M

be written as

-~ J (Vge(ut), Vg<p) dJ.£g ~ -C2 J e(ut) 1<p1 dJ.£g - ~Cl J 1<pldJ.£g - cl(2eo)1/21IVg<pIIL2 M M M

• This permits us to prove:

Theorem B.IO Let Uo, C2, Cl and eo be as in lemma B.9. Then for t Ejt-(uo),Oj and for any ii, 1 < ii < 2 we have

(B.12)

where the constant C3 depends only on Uo, the manifold (M,g), the constants eo, Ct, C2 and the Sobolev constant c~ coming from the embedding of Lr into Lao, j + ~ = 1; i.e.

1I<pllao ::; cp 11<pIILf '

11<pllao the Lao norm of<p.

Corollary B.ll For any ii, 1 ::; ii < 2 the L~ norm of e(ut) is bounded.

PROOF OF COROLLARY: An easy exercise in the use ofthe Sobolev embedding theorems shows that J e(ut)dJ.£g < 00 and (B.12) imply that e(ut) is bounded in Lf.

M

PROOF OF B.10: By lemma B.9 we have

J (Vge(ut), Vg<p) dJ.£g < 2C2eo 11<pllao + 3Cl 11<pllao area(M) M

+ 2Cl(2eo)1/21IVg<pIILl>' area(M)2P/(iI-2)

< (2C2Cj1eo + 3clcparea(M) + 2Cl(2co)1/2area(M)2P/(P-2») 11<pIILf

< C311<pIILf. 1

If <P is chosen so that 11<pIIL!' ::; 1 we see that 1

J (Vge( Ut), Vg<p) dJ.£g ::; C3 M

for all such <po This clearly implies that

•


Corollary B.12 For any P, 1 :S p < 00 the first derivatives ~ are bounded uniformly in LP for t Elt- (uo), Ol in any fixed conformal coordinate chart.

PROOF: By the Sobolev embedding theorems, given any p there is a q, 1 < q < 2 such that there is a continuous inclusion of LUM,IR) into LP(M,IR). Thus e(ut) are bounded in LP for any p ~ 1 and hence ~ are bounded in LP. •

We have now found the analytical key for producing a harmonic map, namely

Theorem B.13 For any P, 1 :S p < 00 and t Elt-( uo), Ol the flow Ut remains bounded in the L~ norm.

PROOF: Recall that u E L~(M, M) is harmonic if in a conformal coordinate system

Ie Ie Ie 8u; 8ui ( ~u) = ~gU + r .. (u)-- = 0 l, 8x'" 8x'"

(B.13)

where ~gUIc is the Laplace-Beltrami operator of the kth component of u : M -t IRd and (~u)1e is the kth conmponent ofthe non-linear Laplacian. Along the flow Ut we have

(B.14)

Now -~g+I, I the identity, is a strictly positive operator. But by corollaries B. 7 and B.ll the right hand side of (B.14) is bounded in LP for any p. Therefore by standard elliptic estimates Ut is bounded in L~ for any p. •

Our next goal is to show that sup {I 1,B(ut}(p) I I I p EM, t Elt-(uo),Ol} < 00; I.e. the vector field ,B is bounded along the trajectories. We begin with

Lemma B.14 Let u E L~(M, M). Then for all tangent vectors X E TuL~(M, M), there exists a constant c > 0 such that

where X (a priori a vector field over u) is considered as a map from M into the ambient space IRd.

169

PROOF: We argue by contradiction. If not, there is a sequence Xm, f IIXml12 dp,g = 1 M

with

l~m2tf {II~~;W -(R(Xm'%:L) :;,Xm)}dZdY. M

Thus ~ £ 11~:i" W dz dy~O, ~ (R (Xm'~) ~,Xm) ~ 0 and £ IIXnl12 dp,g = 1.

Using the fact that

DX 8X (au) 8zL = 8zL - DII(u) 8zL X (B.15)

where II(p) : JRd ~ TpM is the orthogonal projection, we see that Xm is bounded in the L~ norm as a map from Minto JRd. By standard functional analysis Xm has a subsequence, say Xm again, which converges weakly to some X in L~(M, JRd). Since by the Sobolev embedding theorem, the inclusion of LHM, JRd) into L~(M, JRd) := L2(M, JRd) is compact, we may assume that Xm converges in the L2-norm to X.

However, this implies that

at almost all points p E M. This implies that X = 0 contradicting the fact that

IIXII£> = 1. •

This lemma can be strengthened so that the constant C can be selected to be fixed along a trajectory Ut, t E]t-(uo), 0]. Thus we have:

Lemma B.15 Let Ut be a trajectory of f3, t E]t-(uo), 0]. Then there is a constant C > 0 such that for all tangent vectors X t E Tu.L~(M, M)

PROOF: We again argue by contradiction picking a subsequence Xtm such that IIXtm 11~2 = 1 and


By theorem B.13 Utm is bounded in the L~ norm for any p. By the Sobolev embedding

theorem we can extract a subsequence, again called Utm which converges a1 to U E L~,

in particular Utm are bounded a1 . This allows us to conclude, as in B.15 that Xtm is

bounded in LHM, rn.d). Thus we may choose a subsequence Xtm which converges weakly

to X in L~ and strongly in L2. As before we conclude that X = 0 and Ilxlli. = 1, a contradiction. • As a direct consequence of B.15 we have

Lemma B.16 The vector field f3 is bounded in L2 along any trajectory Uti fort E]t-(uo), 0]' with the bound independent of t.

PROOF: Recall that the derivative DEg(ut) of Dirichlet's energy is given by

2( £ ( !f7 /$ ) dx dy

- f (t::..Ut, f3) dILg M

-£ (t::..f3,f3) dILg - 2( £ (n(f3, ~)~,f3) dx dy

E {f (f!z, f!z) dx dy - f (n (f3, ~) ~, f3) dx dy } l M M

where we write f3 := f3( Ut).

But

Thus

it £ (~:~, ~~) dxdy < it £ ~ (II~:~ 112 + 11~~ln dx dy

Eg(Ut)+~~JII~~lr dxdy . M

for all t E]t-(uo),O]. Applying the inequality of lemma B.15 completes the proof. •

Theorem B.17 Along a trajectory Uti t E]t-(uo),O] the vector field f3 is bounded in

supremum norm by <711f311£2 I where <7 is a positive constant depending only on Uo and (M,g).

171

PROOF: f3 is a vector field in L~(M, M), therefore it is 0 2 • Fix t and consider IIf3W :=

11f3( Ut)(p) 112 as a real valued 0 2 function of p EM. Taking the Laplace-Beltrami operator we obtain

where again 9ij = ).bij.

Since 90 has negative curvature we get that

But sup II~ut(p)11 ~ Cl := sup II~uo(p)11 by corollary B.6. Applying the Schwarz inequal-(p,t) p

ity we see that

A standard fact from linear PDE states that if a 0 2 function cp : M ~ ill. satisfies ~gCp ~ -Ocp then the sup norm of cp is bounded by a constant C times the L2-norm where C depends only on M (see for example the paper by F. Tomi [109]). Putting cp = 11f3112 + Cl we conclude that

sup 1 1f3(ut)(p) 1 1 ~ C 11f3IIL 2 , P

where C depends only on M and Cl and is therefore independent of t.

Now using lemma B.16 we can immediately conclude

Lemma B.IS There is a constant C4 independent oft such that

sup 1 1f3(ut)(p) 1 1 ~ C4 , t E]r(uo),O] . p

•

Our next to final step is the following

Lemma B.19 Let p = 2m, m ~ 1. Then ~Ut is bounded in the Li norm with the bound

independent of t E]t-( uo), 0].


PROOF: Since

and both ~ and ~Ut are bounded in supremum norm it suffices to prove that the intrinsic quantity >.-1/2~~Ut is bounded in V. Let It = ~Ut and

Therefore differentiating with respect to t we obtain

~ = m~(t) + m f >.~ (~;:, ~;:) m-I (R (p, :;) It. ~Llt) dJLg M

Since both It and p are bounded in sup norm by some universal constant, as is the intrinsic quantity >.-1/28u/8;cL, we may conclude that

Thus we see that

d~ 1 IIDlt112m-1 dt ~ m~(t) - Cs f >.m-1/2 8;cL dJLg M

By the Schwarz inequality

1 liD! 11 2m_1 f J 1 liD/, 11 2m }(2m-I)/2m 1 >.m-1/2 8;c: dJLg ~ C 11 >.m 8;c: = c~(tt ,

Consequently we arrive at the inequality

d~ dt ~ m~(t) - ~~(tt

Let 1/J'"I := ~ where 'Y = I~'" Then

Thus

'Y • ~~ ~ m1/J - ~1/J"'"I-'"I+1 .

But a'Y - 'Y + 1 = 0 and so we arrive at the differential inequality

d1/J (m) dt~ -:; 1/J-~ , 1/J ~ 0 .

2m-1 a=---<l

2m

(B.16)

173

We only have to show that 'I/J is bounded for all negative time; then CfJ will also be bounded, finishing the proof. We claim that the bound for 'I/J is max( Ce'Y / m, 'I/J( 0). This bound is independent of t E)t-(Uo), 0). If 'I/J(O) > Ce'Y/m, then according to (B.16) 'I/J will decrease as t decreases, either forever (which finishes the proof) or until 'I/J(to) ~ Ce'Y/m for some to ~ o. This is already the situation in the other case 'I/J(O) ~ Ce'Y/m. But then, no t < to can satisfy 'I/J(t) > Ce'Y/m, because in this case ~.interval [Ll,t~) would exist on which 'I/J ~ Ce'Y/m but 'I/J(t~) = Ce'Y/m, 'I/J(Ll) > Ce'Y/m. Integrating (B.16) over this interval leads to a contradiction. •

'We already know (theorem B.13) that Ut, t E)t-(Uo), 0) is bounded in L; for any p with the bound independent of t. Combining formula (B.13) and lemma B.19 we can immediately conclude that Ut is bounded in L~ for any p = 2m again with the bound independent of negative time. We state this formally as

Theorem B.20 For any p = 2m, the flow Ut of {3 is bounded in L~, with the bound independent of negative time.

Now let us consider the L~ norm of the vector field {3. We have

Theorem B.21 The vector field (3 is bounded in the L~ norm along a trajectory Ut, t E)t-(Uo),O) for any p = 2m, with the bound independent oft.

PROOF: The elliptic equation for {3 is in conformal coordinates gi; = ).Oi;

where again (3 = (3(Ut). If ll(p) : JR." --+ TpM is the go-orthogonal projection then

1 8 8 t1{3 = ~ll 8zlll 8zl {3 .

Using the fact that ll(u(p»{3(u)(p) = (3(u)(p), (B.1S) can be written as

)'t1{3 = g. - n 2ll(ut) (~) (ll~) (3

- nll(ut) [nll(ut) (~)] (~) (3

- nll(ut) [ll~] (3 - nll(ut) (~) ~

- nll(ut) (~) [~- nll(Ut) (~) {3]

(B.17)

(B.1S)


We know that II,8W is bounded in sup norm and from the proof of lemma B.16 we see

that the intrinsic quantity '* II~II is bounded in L2. From equation (B.15) and the fact

that * II~II is bounded in sup norm we can conclude that * II~II is bounded in L2. Therefore we may rewrite equation (B.18) as

(B.19)

where 6.g ,8 is the Laplace-Beltrami operator on the vector ,8 (,81, ... ,f3") and n is bounded in L2. Equation (B.19) is essentially an elliptic equation in each component,8i with the right side ni bounded in L2. Therefore, since -6.g + I is a strictly positive invertible operator, we may conclude from standard elliptic estimates that ,8 is bounded in the L~ norm. From the Sobolev embedding theorems it follows that *~ is bounded in L~. Since 6.Ut is in L~ it follows that the n in (B.19) is in L~. Thus,8 E L~. Again

using the Sobolev inequalities we conclude that *~ E Vi for any p. Since 6.Ut E Lr for any p = 2m, looking once again at (B.19) we see that ,8 E L~, p = 2m. •

Theorem B.22 The flow Ut E L~, p = 2m > 2 is defined for all time.

PROOF: From theorem B.21 ,8 is bounded in the L~ norm and a standard result [64J from ODE is that if a vector field on a complete manifold (L~(M, M) is complete, since M is compact) is bounded along a trajectory Ut, t EJt-(Uo), OJ, then necessarily t-(uo) = -00.

We have now come to the result we have worked towards:

Theorem B.23 For any metric g on M and any metric go on M with negative scalar curvature there exists a Coo smooth harmonic map S : (M,g)-+(M,go) homotopic to the identity.

PROOF: By theorem B.22 the flow Ut goes for all negative time. From the equation

we see that II6.ut112 -+ ° as t -+ -00. We also know that Ut is bounded in L~ for any p = 2m. If p > 2 then the inclusion of L~ into C2 is compact. If tm -+ -00 we can therefore extract a subsequence say tm such that Utm converges C2 to some C 2 map S. Clearly S must be harmonic. Write the equation for harmonic maps in conformal coordinates as

175

or -ll.g 5 + 5 = 0'

where 0' is C1. Again, repeated application of elliptic regularity for -Il.g + I yields that 5 E Coo. •

Uniqueness of Harmonic Maps

Consider Dirichlet's energy Eg as a real valued smooth map on L~(M, M). Then the fact that go E M_1 (or simply that the scalar curvature of go is negative) implies that every critical point of Eg is a (non-degenerate) minimum. To show uniqueness one could proceed as follows:

1. Show that Eg : L~(M, M) --+ IR satisfies the axioms for a Morse theory as described in [111]

2. Since L~(M, M) is connected one can then use a mountain pass argument to show that the existence of two non-degenerate minima implies the existence of at least one other critical point which is not a minimum, establishing a contradiction.

This procedure would involve checking that the Morse theory of [111] holds in the case at hand. Although this is true, there is, fortunately, a much shorter and simpler way of establishing uniqueness.

Suppose we have two harmonic maps 50 and 51 mapping (M, g) to (M,go). Since 50 and 51 are homotopic there is a smooth homotopy F: M x I --+ M with F(:z:,O) = 50(:z:) and F(:z:,I) = 51 (:z:). The negative curvature of (M,go) implies that given any:z: E M with 50(:z:) f. 51(:z:) there is a unique geodesic t ~ 5(:Z:j t), 5(:z:,0) = 50(:z:) and 5(:z:, 1) = 51(:z:) homotopic to t ~ F(:z:,t) joining 50(:z:) and 51(:z:). For details see [76]. Interestingly, the proof of uniqueness involves a Morse theoretic argument on the manifold of paths joining 50(:z:) and 51 (:z:).

Now let us suppose that our geodesics 5(:z:,t) have unit velocity with 5(:z:,0) = 50(:z:). Then there is a unique non-negative time T(:Z:) such that 5 (:z:, T(:Z:» = 5 1 (:z:). If 50(:z:) = 51 (:z:) define T(:Z:) = o. From the fact that for each :z: E M the exponential map exp", : T",M --+ M is a local diffeomorphism it follows that T is continuous and smooth wherever it is positive.

Let us suppose for a moment that a miracle occurs and T : M --+ IR+ is constant! Without loss of generality assume that T == 1 and that, as before (M, go) C IRd, isometrically. Since


t >-+ S(z, t) has unit length for all z

(B.20)

and (B.20) holds for all :z: E M, where 11·11 denotes the Euclidean IRd norm.

Thus the Laplace-Beltrami operator can be applied to (B.20) to yield

II {)SI1 2 ({)S {)S) t:.g at = t:.g at' at = 0 . (B.21)

Writing (B.21) in conformal coordinates 9ij = ).bij we have

o 2"IID 8s112 2 (Dt:.S 8S) 2(n(8S 8S) 8S 8S) X'L a;zat + at 'at + X a;z, at a;z, 8t

1 " II D 8S 112 d (t:.S 8S) 1 (n (8S as) 8S as) X LJ a;zat + ;it 'at + X a;z, at a;z, at . l

(B.22)

Hold :z: fixed, take the integral of (B.22) with respect to t over the interval [0,1]. Since

j 1£ (t:.S, ~n dt o

we see that for each z E M

(t:.S)(z, 1), ~;(z,1)) - (t:.S)(z,O), ~;(z,O))

= (t:.SI)(z), ~;(z, 1)) - (t:.So)(z), ~;(z,O)) o .

1 JIll D {)S11 2 J1 1 ( ({)S {)S) {)S {)S) o = ~ ~ 0 {):z:l at dt + 0 ~ n {)zl' at {):z:l' at dt

But both integrals are non-negative, which implies that (n (l!z, ~n l!z, ~;) == O.

Let us now use the fact that S is a diffeomorphism (to be established shortly and independently). Then this last equality implies that ~; = O. This clearly contradicts the fact

that II~;W = 1. Thus So(z) = Sl(Z).

If we could show now that T is constant we would be done. This follows by applying the maximum principle and the next

Lemma B.24 (t:.gT)(Z) ;:::: 0 whenever T > O.

Before proving this lemma let us see how it implies that T is constant. IT T(Z) > 0 for all z, then T is a globally defined subharmonic function on M. The maximum principle

177

states [91],[39] that T cannot have an interior positive maximum (i.e. there cannot be a point Zo such that T(ZO) ~ T(Z) for all Z with strid inequality holding somewhere). Since M is compact and all points are interior points, T must be constant and by the preceding argument So(z) = S1(Z),

Suppose now that the set r = {z I T(Z) = O} = {z I So(z) = S1(Z)} =F 0. Let U be a component of M \ r. Then T > 0 on U and ll.,T ~ 0 on U. Again by the maximum principle for the Laplacian the maximum of T must occur on au c r. Thus T == 0 on U and hence T == 0 on M implying once again that So( z) = S1 (z).

We now give the proof of lemma B.24.

PROOF OF B.24: S(Z,T(Z» = Sl(Z) and S(z,O) = So(z) . From these two equations it follows that

(ll.S)(Z, O) = (ll.So)(z) = 0

and (in conformal coordinates)

OlD aS1 (ll.S1)(Z) = 'X azl azl =

1 D as 2 D as aT { 1 D as ( aT ) 2 1 as ( a2T ) } 'X azl azl + 'X at azl . azl + ~ 'X at at azl + 'X at azl2

(B.23)

(B.24)

with the right hand side of (B.24) being evaluated at t = T(Z). However since t 1-+ S(z, t) is a geodesic ft~~ = 0 and we may rewrite (B.24) as

(B.25)

Now equation (B.22) holds whether or not T is constant. Let us integrate (B.22) over the interval [O,T(Z)] to obtain:

.,.("') 1 II D as 112 ( as) 1"'("') .,.("') 1 ( (as as) as as) ~ [ 'X azlat at + ll.S, at 0 + [ 'X 'R azl' at azl' at dt == 0 .

(B.26) Consider the term

( as) 1"'("') ( as) ( as) ll.S'Bt 0 = ll.S(Z,T(Z» , at (Z,T(Z» - ll.S(z,O)'Bt(Z,O) (B.27)

But by (B.23) the second term on the right of (B.27) vanishes, and by (B.25) the first term on the right equals


But

(D 8S 8S) (D 8S 8S) 1 8 118S112

at 8zl ' at = 8zl at ' at = '2 8zl at = 0

and

_ ~~ 1I~~1I2. ~;2 = - ~~:~;2 . Consequently equations (B.26) and (B.27) yield the beautifully simple equation

= E 1 lPT l X87f

= ~ l

> 0

T(Z) 1 II D 85112 f X a;z8t dt+ o

and lemma B.24 is established.

fT(Z) 1 ('R. (85 85) 85 85) dt X a;z'8t a;z'8t o

We have therefore proved (using the existence result theorem B.23):

•

Theorem B.25 For any metric 9 on M and any metric go on M with negative scalar curvature there emts a unique GOO smooth harmonic map

S: (M,g)-~{M,go)

homotopic to the identity. •

Smooth Dependence

The vector field /3 on the manifold L~(M, M) defined by equation (B.3) actually depends smoothly on two parameters, namely the metrics 9 and go. Let us, as we did with Dirichlet's energy, consider the metric go as fixed and 9 as a variable parameter. Then /3 is, in reality, a smooth function of two variables 9 and u E L~(M, M)j and we take this into account in our notation in writing /3(g,u) in place of /3(u).

If u = S is a harmonic map, then the derivative of /3 with respect to u at S, written now as Du/3(S) is by theorem B.2 the identity map on the tangent space TsL~(M, M).

In some local coordinate system for the tangent bundle T L~( M, M) about the point S, /3(g,u) = (u,Y(g,u)) where Y is the "principal part" of /3. Then if E = T5L~(M,M) we may view Y as a Goo map on M X W into E, where M are all metrics 9 on M, W

179

a neighbourhood of ° in E, with ° corresponding to S. If S is harmonic from (M,g') to (M,go) then

Y(g', 0) = ° with the derivative of Y with respect to the "second variable" u at (g',O), DuY(g',O) : E -+ E an isomorphism. In this case the implicit function theorem on Banach spaces [2] says that for some neighbourhood W' C Wand for all 9 sufficiently close to g', there is for each such 9 a unique zero S(g) to Y (g, S(g)) = ° and S depends smoothly on g.

Theorem B.25, already gave us global uniqueness, which is more than the local uniqueness we obtain from the implicit function theorem. However we do obtain the smooth dependence of the harmonic map Son g. Consequently we may now strengthen B.25 to

Theorem B.26 For any metric 9 on M and any metric go on M with negative scalar curvature there exists a unique Coo smooth harmonic map S(g) : (M, g)-+(M, go) homotopic to the identity and with 9 f-4 S(g) being Coo smooth.

The Map S(g) is a Diffeomorphism

The proof that S(g) is a diffeomorphism, originally due to Schoen-Yau and Sampson, is easily available in greater generality in their papers [97], [96] and in the lovely book by Jiirgen Jost [57] whose presentation we follow.

We shall therefore be a bit more sketchy than in the preceding parts of this appendix, preferring to outline the main points of the proof.

Dropping the 9 from the notation for S we want to show that any harmonic map S : (M,g) -+ (M,go) homotopic to the identity is a diffeomorphism. In this we need only that the scalar curvature of go is negative. Write the metrics g and go in local conformal coordinates as

Adzdz

and

pdwdw.

Recall that the equation for a harmonic map S (cf. equation (3.1)) can be written as

(B.28)

Define the functions


and

L :== 18S12 == ~IS!12 .

Let K1 and K2 be the Gauss curvatures of (M,g) and (M,go) respectively in conformal coordinates

and

2 0 0 K1 == ----logA

AOZOZ

2 0 0 K2 == ----logp < 0

AOZOZ -The following lemma follows from a straightforward calculation.

Lemma B.27 At points where H or L, respectively are non-zero we have the identities

Th~refore

ilg log H = 2K1 - 2K2(H - L)

ilg log L == 2K1 - 2K,(H - L).

ilglog(H/L) == -4K2(H - L) .

PROOF: a straightforward calculation.

(B.29)

(B.30)

(B.31)

• The quantity H - L appearing in B.29 - B.31 is geometrically significant; it is the Jacobian

determinant of the map S, which we denote by J( S).

We also observe that the product 1

HL == A,CPq;

where cpdz' == pS.Szdz' is a holomorphic quadratic differential on (M,g). From this it follows that if either H or L vanish on an open set they must vanish identically. We also note that in our situation H cannot vanish identically since this would imply that J(S) $ 0 and therefore that deg S $ 0, contradicting the fact that deg S = 1. Thus the zeros of H must be isolated.

Lemma B.28 Suppose f is a C 1 function on (M,g) with isolated zeros such that

where w is C 1 • Then locally f = egh where 9 is C 1 and h is holomorphic.

181

PROOF: Find a C1 function g with g. = w (locally) and set h = e-g f. Then h. = 0, i.e. h is holomorphic. •

As a direct consequence of this and the preceding remarks we have

Theorem B.29 Near each isolated zero Zj of H we have the ezpansion

for some aj > 0 and some non-negative integer ~.

PROOF: P 2 P -

H = ~IS%I = ~S%SZ .

Let f = S%. Then from (B.28) it follows

f.= fw

where w = -7S~. By lemma B.28

(B.32)

(B.33)

(B.34)

where h is holomorphic. Each zero of H is a zero of S% and also of h. Since h is holomorphic it has a power series expansion about Zj. The theorem then follows from the explicit expressions (B.33) and (B.34). As a direct consequence we obtain

Theorem B.30

- ~ ni = 2 J K1dp,g - 2 J K 2(H - L)dp,g , M M

(B.35)

PROOF: Integrate expression (B.29) over M with respect to the volume measure p,g, by first deleting small discs about the zeros Zi, and integrating over M minus the union of these small discs. The right hand side of (B.29) is continuous on M and therefore the limit of the integral of the right hand side of (B.29) over M \ (U discs) as the discs shrink to 0 is clearly the right hand side of (B.35). log H, however has a singularity at each Zi.

Integrating ( f 8 g log H) by parts yields a sum of integrals about the boundaries M\(Udiscs)

of these small discs. Using expansion (B.32) and going to the limit we obtain the left hand side of (B.35). •

As a consequence of theorem B.30 we obtain


Corollary B.31 For a harmonic map S homotopic to the identity, H > o.

PROOF: Consider expression (B.35). By the Gauss-Bonnet theorem

2 J K1dl'g = 411"X(M), M

X(M) the Euler-characteristic of M.

(B.36)

In the second expression we may make a change of variables y = S( z). Of course, we must not and do not use here that S is a diffeomorphism. The change of variables argument needs only deg S = 1. Thus

J K2(H - L)dl'g = J K2J(S)dl'g = J K2dl'90 = 211"X(M) , M M M

the last equation following again from Gauss-Bonnet.

Inserting (B.36) and (B.37) into (B.35) we have

-Eni=O. i

Thus all ni = 0 and H > o.

(B.37)

•

Lemma B.32 Let S : (M,g) ~ (M,go) be a harmonic map homotopic to the identity with K2 ::; o. Then the functional determinant

J(S) = H - L ~ o.

PROOF: In a region where J(S) = H - L < 0, one would have L > H> 0 and therefore log( H / L) < o. In addition,

{);.glog(H/L) = -4K2(H - L) ::; 0

wherever J( S) ::; o. Thus log( H / L) is superharmonic where J( S) ::; o. Therefore by the maximum principle [91) for the Laplacian log( H / L) cannot have a non-positive interior minimum where J(S) ::; o.

Since log(H/L) = 0 on the boundary of {z E M I J(S)(z) ::; O} and log(H/L) < 0 in the interior, the set {z E M I J(S)(z) < O} must be empty.

This allows us to prove the main result of this section.

183

Theorem B.33 Let 5: (M,g) ~ (M,go) be a hannonic map homotopic to the identity

with K2 ::; o. Then 5 is a diffeomorphism.

PROOF: We know from Lemma B.32 that J(5) 2:: 0 on M and by corollary B.31 that H > 0 on M. Suppose J(5)(Zo) = o. Then H(zo) = L(zo) > o. Thus again by (B.31)

Ilg log(H/L) = -4K2J(5) (B.38)

in some open neighbourhood about zoo

Since J(5) 2 0, K2 ::; 0 and L(zo) > 0 there are positive constants Cl and C2 and a neighbourhood U of Zo with

holding in U.

Therefore applying these facts to (B.38) we have

(B.39)

in U.

Again applying the strong maximum principle, for elliptic second order equations [91],[39], this time to (B.39) we see that log(H/ L) cannot assume a non-positive minimum in the interior of U, unless log(H/L) is constant on U. But since H(z) 2:: L(z) on U,

log( H / L)( z) 2:: 0 on U and equals zero when z = zoo Thus log( H / L) == 0 on U. This says that the set of z where J(5)(z) = 0 is open as well as closed. Thus ifthis set is non-empty it must be all of M implying that 5 is constant and deg 5 = 0, which contradicts the fact that 5 is homotopic to the identity. •

Putting all the results of appendix B together we arrive at the following conclusion, which we have used extensively in our development of Teichmiiller theory, namely

Theorem B.34 For any metric 9 on M and any metric go on M with negative scalar

curvature there exists a unique COO smooth harmonic map 5(g): (M,g) ~ (M,go) homo

topic to the identity. Moreover 5(g) is a diffeomorphism and 9 1-+ 5(g) is Coo smooth.

This concludes appendix B.

C The Mumford Compactness Theorem

We prove here the compactness theorem for the moduli space (lemma 3.2.2). In its original form the theorem is due to Mumford [79). We present another proof given by Tomi and the author [110) using only basic geometric notions instead ofthe uniformization theorem.

Theorem C.l Let M be a closed connected smooth surface, and {gn} be a sequence of smooth metrics of curvature -Ion M such that all their closed geodesics are bounded below in length by a /ized positive bound. Then there ezist smooth diffeomorphisms r of M which are orientation preserving if M is oriented, such that a subsequence of {r· gn} converges in Coo towards a smooth metric.

PROOF OF THE MUMFORD THEOREM: Since on a negatively curved surface there a.re no conjugate points along any geodesic, it follows that every geodesic arc is globally minimizing (with fixed end points). Therefore, any two geodesic arcs with common endpoints cannot be homotopic with fixed endpoints; otherwise, by a common Morse-theoretic argument (see Milnor [76]), there would exist a non-minimizing geodesic arc joining these endpoints.

Another way to see this is as follows: Two homotopic geodesic arcs on M with common endpoints would give rise to two geodesic arcs with common endpoints in the universal cover M of M, which is a negatively curved plane. If these two arcs have no interior points in common they bound a region of the type of the disc to which Gauss-Bonnet can be applied yielding a contradiction: the total curvature is negative, whereas the integral along the boundary of the region is positive. If the two arcs do have interior points in common, the same argument applies to shorter segments of these arcs.

185

Hence we may conclude that a lower bound i on the length i" of the closed geodesics of g" implies a bound on the injectivity radii p" of M" = (M,g"), p" ?: p?: i/2.

It follows on each open disc BR(p), P E M" and R :5 p, one can introduce a geodesic polar coordinate system. By a classical result in differential geometry the metric tensor associated with gn in these coordinates assumes the form

(C.l)

where r denotes the polar distance.

For the area of BR(p) we obtain from (C.l) the simple estimate

The genus of the manifolds Mn being fixed, the total area of M" is determined by the Gauss-Bonnet formula if R(gn) = -1. It follows that there is an upper bound, only depending on R, for the number of disjoint open discs BR(p) in M". Let us now take R = ~p, and let N(n) be the maximal number of open disjoint discs of radius ~R in Mn. By passing to a subsequence we can assume that N(n) = N holds independently of n. It follows that, for each n E 1N, we can find points pj E Mn, j = 1, ... , N, with the property that the discs B1/ 4R(Pj) are disjoint while the discs B1/ 2R(Pj) cover Mn. Let us now denote by IH the Poincare upper half plane. We pick an arbitrary point (0 E ill, e.g., (0 = i, the imaginary unit, and introduce geodesic polar coordinates on B4R(Pj) C Mn and on B4R«(0) C ill, respectively. The corresponding metric tensors assume the same form (C.l) in each of both cases, and we may therefore conclude that there exist orientation preserving isometries

<pi: B4R(pi)--7B4R«(0), <pi(pi) = (0 .

Let then In denote the set of all pairs (j, k), 1 :5 j, k :5 N, such that

By passing to a subsequence we can assume that In = I is independent of n. For (j, k) E I, the transition mappings

are well-defined local isometries of IH. Before proceeding further with the proof we first want to show that any such local isometry in fact extends to a global one.

186 C. The Mumford Compa.ctness Theorem

Lemma C.2 Let I : U --+ H be a C 1 orientation preserving isometry on an open con

nected subset U 01 the hyperbolic plane. Then

I(w) = -~-:-:-~=, A,B,C,D E IR ,

and AD - BC = 1.

PROOF: The class of maps w f-+ ~:t~, AD - BC = 1 with real coefficients are the group of isometries of the Poincare metric. Thus we must show that a local isometry is also a global isometry.

I is orientation preserving. Now an easy calculation shows that I must be holomorphic and has to satisfy the non-linear condition

1!,(w)1 = 1m f(w) Imw

(C.2)

One can check that every map of the form w f-+ ~:t~ as above satisfies the condition (C.2), and the set of maps from a fixed domain to itself and satisfying (C.2) form a group. Therefore, by composition with an appropriate element of the three dimensional conformal group of ill we may assume that I satisfies the following additional conditions: I is defined in a neighbourhood of i E H, and

!,(i) = 1, Im/(i) = 1 (C.3)

Now, writing w = u + iv, 8w = H8u - i8,,) and using (C.2), we have

(log f')' 2 {Re (log I')} w = 2 {log Ifill w

(1m I)w Vw -if' i = --=-+-.

Iml v Iml v

By (C.3), this implies (logf'),(i) = 0 and hence f"(i) = o. Similarly

[ "f' ] . }" "f' "f' ·2

(log /')" = - I~ I w + [; L = ;~ I - (I~ 1)2 ( T ) + 2~2 (C.4)

Again we see that (log f')" = 0 and hence f"'( i) = o. Proceeding inductively we obtain

(log/')(nJ(i) = 0

for all n E IN. In the induction process, only the derivatives of 1m I in the second term of the right hand side of (C.4) and those of v in the third term contribute, but these contributions cancel. All other contributions vanish by the inductive hypothesis

187

rei) = ... = f<n}(i) = o. Thus, since f is holomorphic in a neighbourhood of i, it follows that f' is constant; therefore few) = w. Since we normalized f by the isometry group of IH, this proves that our initial map f must be in this isometry group, and the proof of the lemma is complete. •

For (j, k) E I we have Pi: E B4R(P'j), and hence qjk := tp'j(Pi:) E B4R(eO), since tp'j is an isometry. It is obvious from the definition that

(C.S)

We are now going to construct a limit manifold of the sequence Mn = (M, gn). For this purpose we prove

Lemma C.3 The family of transition mappings (Tj,.)nEIN is compact for each

(j, k) E I.

PROOF: By Lemma C.2, each Tj,. is a global isometry of IH and there is a fixed compact subset K ofIH and points qjk E K such that (C.S) holds. By composition with a conformal map of IH onto the unit disc B C IR2 we may assume that each Tik is a conformal map of B onto itself and (suppressing the indices j, k) that there are points pn strictly staying away from 8B such that also Tn(pn) stays uniformly away from 8B. Each Tn is of the form

n( ) _ -1 W - an TW-"'n , 1- a.,w

where lanl < 1, Idnl = 1. It suffices to show that lanl stays strictly below 1. If not, we can assume an -+ a, lal = 1, and dn -+ d, Idl = 1. The limit map then

T(W) = d W -_a = ad{aw -= I} = -ad 1- aw 1- aw

collapses the disc onto a point on 8B which is a contradiction. • We can now continue with the proof of Mumford's theorem. Passing to a subsequence we can by Lemma C.3 assume that

(C.6)

We now define a limiting manifold if as the disjoint union of N discs BR(eO) c H, labelled as B l , .. . , BN with the identifications

P E B j equals q E Bk (j,k)EI and P=Tjk(q).

188 C. The Mumford Compactness Theorem

It is clear that M is a differentiable manifold carrying a natural Riemannian metric which

on each Bi coincides with the Poincare metric. We claim that M is compact. Assume to

the contrary that there were a point q E 8BR((o) such that q f/. Tilo (BR((O» for some j and all k with (j, k) E I. Then it would follow that, for sufficiently large n, we have

q rt. Tji. [B(3/4)R((O)] which means that (cpi)-l(q) rt. B(3/4)R(p~). This, however, would imply that

B(1/4)R [(cpitl(q)] n B(1/4)R(Pk) = 0 for k = 1, ... , N ,

contradicting the choice of N as the maximal number of disjoint discs in Mn of radius (1/4)R. The remainder of the proof rests upon the following

Lemma C.4 There are ditfeomorphisms r : M-+Mn, r(Bi ) C B2R(pi) such that

cpi 0 r-+id in Coo on each Bi , as n -+ 00 . (C.7)

The proof of this lemma is somewhat technical and presented below.

Let us first quickly finish the proof of Mumford's theorem assuming the lemma. Denoting by 9 the Poincare metric, we have from (C.7) that

r"CPj*g-+g as n -+ 00

on each Bi . Since, however, cpi was an isometry between 9 and gn on Mn, this means that

r*gn-+g as n -+ 00

on M. Choosing now any (symmetric) diffeomorphism I: M-+M, we obtain

which proves Mumford's theorem. • We now come to the proof of Lemma CA.

For the prooflet us consider the manifold Mn as the disjoint union of N-balls B 2R((O) c H labelled as Bf, ... , BN with the identifications

:z: E B~ equals y E Bj iff (i,j) E I and :z: = T.j(y) .

We denote this model of Mn by Mn. It then suffices to show that there are diffeomor

phisms r : M-+Mn, r(B.) c B:, such that In -+ id (as n -+ 00) on each B •. We shall do this by a Morse theoretic argument.

189

Since M is an oriented closed surface there is a Coo Morse function .,p : M -IR with distinct critical values Co > Cl > ... > Cm and such that the level sets .,p-l(Ci) contain only one non-degenerate critical point Wi'

We may use a partition of unity to construct a sequence of functions .,pn : Mn - IR such that on each Bi, .,pn _ .,p in Coo. To see this let rpi be the natural coordinate charts on M induced by the inclusion of Bi into JR, so that rpi 0 rpi1 = Tii' Furthermore, let hi} be a partition of unity on M with respect to the coordinate cover {Bi }. Define.,pn : Mn - IR by

.,pn(p) = E7Ii (rpilrpj(p)).,p (rpilrpj(p)) i .

.,p"(p) = E7Ii (rpi1rZj(u)).,p (rpi1rji.(u)) j

As n _ (X) this converges Coo to

E7Ii (rpi1rilc(U)).,p (rpi1rjlc(u)) = E7Ij (rpi;I(U)).,p (rpi;I(U)) =.,p (rpi;I(U)) , j j

which (after viewing .,p" as defined on Mn) proves the result.

Consequently, for large n, .,p" has non-degenerate critical points {wj} "near" the {Wi} N

on U Bl . By trivial modifications of .,pn we may further assume that .,p" has the same l=I

critical values Co > ... > Cm as does .,p and that wi = Wi for all j.

Furthermore we can assume that in a small disc about each Wi (in the B/s) .,pn and .,p actually agree.

Let (M")" = {zl.,p"(z)::; a} and (M")" = {zl.,p"(z) ~ an with (M)", (M)" defined similarly in terms of.,p. Let e > 0 be small enough so that (Mn)co_2e and (M)CO-2e contain only Wo as its only critical point and (M")cl-2e and (M)cl-2e contain only Wo

and WI.

Let G be a fixed metric on M which agrees with the Euclidean metric on a neighbourhood of the {Wj}' As in constructing the .,p" we can easily find a sequence of metrics G" on M" so that Gn agrees with G on a neighbourhood of the (Wj) (in UBi) and G" - G as n _ 00. Let V.,p" and V.,p denote the gradients of .,p" and .,p with respect to these

metrics, and X" and X the normalized fields lI:j:1I and lI:jll' 11·11 denoting the norms with

respect to G" and G. Of course X" and X are defined only on Mn - U Wi and M - U Wj

190 C. The Mumford Compactness Theorem

respectively. We shall define a mapping In : (Mn)c,-2e-+M which is a diffeomorphism of a neighbourhood of (Mn)c,_< to a neighbourhood of (M)c,-e'

Let Do be a "small disc" about Wo for which the Morse lemma holds for tPn and tP about woo Thus there exists a map Q from a neighbourhood of 0 in m.2 to a neighborhood of Wo so

that tPn 0 Q(z) = Co - z: - z~ = tP 0 Q(z). Thus we may take Do = {zltPn(z) :::: Co - e}. By the Morse lemma it follows that both (M)co-e and (Mn)co_e are diffeomorphic to a ball and hence to each other. The idea of our proof is to now proceed down the critical points to show that (Mn)c;_< is diffeomorphic to (M)c;-e for j = 1"", m. It is enough to indicate how this is done for j = 1. Let p E aDo, and let O';(t) and O'p(t) be the flows of the vector fields xn and X respectively with 0';(0) = p = O'p(O). It follows immediately

that tPn (O';(t)) = Co - e + t = tP (O'p(t)) and that O';(t) (resp. O'p(t)) as t decreases either

converges to WI or drops into (Mn)c,-2< (resp. (My,-2<). Let U be the unstable manifold

of WI for the flow of xn. Then it follows that every q E (Mn)c,-2< \(UUwo) can be written as O'p(t) for some t E m. and p E aDo. Define the map

by !n (O';(t)) = O'p(t), P E aDo and !n(wo) = woo Since xn and X agree on Do (in some coordinate system) it follows that In is the identity in a neighbourhood of wo, with respect to the above coordinate system, and is thus smooth everywhere it is defined. It also follows from our construction that

tPn(w) = tP (In(w))

and so !n takes level sets to level sets and also that !n -+ id as n -+ 00 (on U Bj).

Now let us assume that we are in a coordinate neighbourhood WI of WI where Morse's lemma holds for tPn and tP and where tPn == tP. The situation is as depicted in the figure below.

Let D~ and DI, D~ C DI be two strips as in figure C.l. Let TJ be a COO function which is 1 on WI \ DI and 0 on D~. Define a new map r : (Mn)c,_2< -+ M by

It is clear that for sufficiently large n, In is a diffeomorphism. Taking now the initial values of our trajectories to lie on (tPn)-I(CI - 2e) and tP-I(CI - 2e), we can proceed inductively to extend In to a diffeomorphism of Mn onto M. This completes the proof of Theorem C.1.

191

Figure C.l:

D Proof of the Collar Lemma

A stronger version of the collar lemma can be found in [92]. The original references are

[48] and [59]. The weaker result given here does not give a lower bound for the area of the collar as l -+ 0 in contrast to [92]; its advantage is that it avoids some topology.

Let us repeat here the lemma 3.2.1 to be proved, except for a slight change of notation and

a rescaling which changes R == -1 (K == -~) to K == -1. This corresponds to changing

all lengths l into l . v'2:

Lemma D.I Let 0: be a (non-trivial) closed geodesic of length l on a surface (M,g) where gEM_I. Then there exists a neighbourhood U of 0: in M which is isometric to

the following set Tj '" in the hyperbolic plane: T = {(r,9) 11:::; r:::; el , 90 :::; 9:::; 7r - 90 }

and", identifying (1,9) with (e l , 9). Here, land 90 satisfy the estimate

490 1 + coshl cot - > --:-:--:--

2 - sinhl

PROOF: It is obvious that some collar can be put around the closed geodesic. What is not trivial is the bound for 90 . Therefore let 80 be the infimum of all 90 such that the collar given in the hyperbolic plane projects injectively onto the manifold. We are going to work with this 80 exclusively and drop the tilde again. This gives us the following picture: The segment from i to iel in III has length l and projects to the closed geodesic 0:

on M. The geodesics in III orthogonal to the imaginary axis do not intersect at all in III,

and the projections on M of their parts inside T do not intersect either due to the choice of 90. But there are two points Ql, Q2 on the boundary of the collar (i.e. 9 = 90 or 9 = 7r - 90 , and 1 :::; r :::; el ) which project to the same point Q on M. Let A,?2 be their foot points of their respective (hyperbolic) perpendiculars onto the imaginary axis. The two perpendiculars together project to a geodesic segment [PIQP2] on M. There is no "angle" at Q in this segment, for else a shorter segment [P1,P2,] from 0: to 0: in the

193

same homotopy class on M could be found (some segment by cutting short at Q, and an even shorter geodesic segment by minimizing the length in this homotopy class). Such a shortest geodesic segment would also have to be perpendicular on a by the same short-cut argument and would hence arise from the same construction as did [PI ,P2,j. This would contradict the choice of 80 • We are looking for a lower bound for the length of the segment [Pl'P2,j.

We do not know whether QI and Q2 are on the same component of the boundary of the collar, 8(QI) = 8(Q2), or on different components, 8(QI) = 7r - 8(Q2). Moreover we cannot guarantee that PI = P2 • This latter property can however be achieved by doubling the surface M: cut M along a and glue two copies together thus obtaining a new manifold, 2M.

We have now two closed geodesics a and {3 = PIQP2Q'PI intersecting orthogonally at PI on 2M. The length of a is t. By construction, for any point on {3, the shortest (on M)

perpendicular to a is a segment of {3. Let us consider this situation in the universal cover: a can be lifted to a geodesic segment a from A, i.e. ie- i to B = i. Then {3 can be lifted to a geodesic segment from B to c. Continuing this way, the path a{3a-I{3-1 on 2M can be lifted to a path of segments with corners at A,B,C,D,E all of which project to Pl. We claim that the hyperbolic lines DE and AB do not intersect. As we shall prove quantitatively later, this is the property which bounds the length of {3. But the reader can immediately glance at the figure to see that for short {3 the lines would intersect.

194 D. Proof of the Collar Lemma

'.

Figure D.l:

So suppose now that DE and AB intersect in F (between A and B or not, and between D and E or not, as the case may be). The divergence of geodesics due to negative curvature actually rules out the possibility that F is between A and B. For constant curvature this calculation needs only elementary hyperbolic geometry and will be given below. We intend to show under this assumption that the projection of the quadrangle F BC D covers all of 2M and derive a contradiction from this. To this end let P E 2M be any point not on /3. Drop a shortest perpendicular 'Y from P to a (its footpoint being G) and construct a lift l' of 'Y as follows: Choose a over G on the segment [AB] if this makes L.( aB, aF) a positively oriented angle. Else choose a over G on the segment [CD].

We have one of the two situations sketched in figure D.l. 'Y being a shortest perpendicular

to a, it cannot intersect /3 (unless it is contained in /3, which we excluded). Therefore the line aF cannot intersect either of the lines BC or FD. Nor can 'Y intersect a except at its endpoint G. Therefore the segment [aF] cannot intersect either of the lines CD or AB. Tlpts F must be inside the quadrangle F BC D whose projection therefore covers all of 2M. Its area is 211' - 3· i - (angle at F) < i. But the area of 2M is 2· 211'(2g - 2) ~ 811', so we get a contradiction that leaves us with the conclusion that DE and AB do not intersect.

Now look at formula (D.1) and figure D.2 below. It not only proves f' > i, hence F is not

between A and B, but it also gives the condition for F not to exist which is our actual

195

situation. It is sinh i sinh d :::: 1. Now d is twice the length of {3, i.e. 4ln cot,.. This immediately gives the bound and leaves us only with the task of providing the details of hyperbolic trigonometry used for deriving formula (D.1). One makes use of the law of sines and the two laws of cosines in hyperbolic trigonometry.

Figure D.2:

law of cosines, t::..iJ6 D coshs coshdcoshi sin cp sin .,p 1

=--=--sinh i sinh d sinh s cos X - sin cp sin .,p + cos cp cos .,p cosh s

law of sines, t::..iJ6 D law of cosines, t::..DP iJ

sinh i' sinh s =

cos.,p sin X law of sines, t::..DPiJ

Calculate sinh2i' from the last of these equations in terms of d and i using the first three equations to eliminate the other variables: the horrible formula simplifies considerably by noting that cosh2dcosh2i- 1 - sinh2d = cosh2dsinh2i.

The result is sinh i' cosh d -- - > coshd> 1 sinhl - .1 "2 2 V 1 - sinh i sinh d

(D.1)

This ends the proof of our version of the lemma.

E The Levi-Form of Dirichlet's Energy

We present here a direct computation of the Levi-form of Dirichlet's energy E on T(M) using the Abresch-Fischer coordinates introduced in section 4.3. The result is:

Theorem E.1 The Levi-form of E,

2 -" 8 E a73 ~ 8za 8zf3 [gle e

where ea = "'fa + ipa is given in the Abresch-Fischer coordinates by

! L / {h . h }g( Z )(VgSl, VgSl)dJLg 2 l M

-E/ {IIV~w"W + IIV~Wi"W}dzdy r M

+;;£ go(z) (~(:;,w") w", :;) dzdy

+ ;; £ go( z) ( ~ ( :; , wi") wi", :; ) dz dy

(E.1)

(E.2)

where (ZI, Z2) = (z, y) and h is the horizontal lift of h* = E eah;, h; a basis for T(g]T(M) a

over C defining the local coordinates za, zf3. Moreover w" = DS(g)h, DS(g) the derivative of S with respect to 9 in the direction h, ~ the curvature tensor of (M,go) and where i denotes the multiplication by i for the complex structure on T(M) induced by the complex structure on A. Finally V tb denotes covariant differentiation with respect to go "along S".

PROOF: We must compute

D2i;[g](h*,h*) + D2i;[g](ih*,ih*)

for any h* E TIBj(M_1 /Do), h* = Eeah:.

197

(E.3)

For 9 a Riemannian metric on M, let E(g) = Eg (S(g)). Let ff : A-+M_1 be the map which assigns the Poincare metric 9 to e E A and .,p a complex coordinate system for A about Jo = ff-1(g). Therefore (E.3) is equivalent to computing

(EA)

for all H E 'J-lTT(JO) where cp = ff o.,p and the derivatives are computed at 0 E TJoA.

(E.5)

We would like to compute Dcp(O)H and D2cp(0)(H, H), the first and second derivatives of cp at the origin and evaluated at Hand (H, H) respectively. Let S2 be the space of symmetric Coo 0-2 tensors and SiT(g) denote the trace for divergence free "symmetric two tensors with respect to g. Then from (2.6) we know that Dff(J) : TJA-+TgM_1 C S2 is given by

Dff(J)i = pg + h

where 9 = ff( J), h = -( J i)~ and

/lp - p = 6g6g h ,

/l the Laplace-Beltrami operator on functions.

Let Lg = /l - I, I the identity. Then

p = L;1(6g 6g h) .

If h is divergence free then p = o. From equation (4.9) it follows that

(i) D.,p(H)(i1) = i 1JO(I + H)-l - (I + H)Jo(I + H)-li1(I + H)-l

(ii) D.,p(O)(it} = -2Joi1

(iii) D2.,p(0)(ib i 2) = 2Jo(i1i2 + i 2i 1).

Therefore

Dcp(H)i1 = Dff(J) 0 D.,p(H)i1

(-J D.,p(H)i1)~ + pg

(-J D.,p(H)i1)~ + p(J). ff(J) ,

(E.6)

(E.7)

198 E. The Levi-Form of Dirichlet's Energy

9 = ~(J), J = (I +H)Jo(I +R).-1 and p(J) = L;1 (5,5, (-J D,p(H)jl)) where, as usual ~ denotes lowering an index via the metric g.

Now ,p(0) = Jo and D,p001) is a trace free divergence free tensor, whence it follows that p(Jo) = o.

Let us first consider the term

in expression (E.7) for which we would like to compute the derivative in the direction J2 •

But

(-J D,p(H)jl)~ = - ((1 + H)Jo(1 + H)-1 D,p(H)jl)~ = - ((1 + H)Jo(1 + H)-ljlJO(I + H)-1 + jl(I + H)-I)~ .

For H = 0 we obtain -2jl = D,p(O)k The derivative of

at 0 in the direction of j2 is easily computed to be

(E.8)

Consider now the map - l l H f-+ W(J)ilAj = (Aj)~ (E.9)

where A is a fixed 1-1 tensor. The derivative of this at 0 in the direction j2 is

(E.I0)

In the case A~ = -2jl we see that this is equal to

Adding this and (E.8) together we find that the derivative of

at 0 is the bilinear map

(jl,j2) f-+ 201j2 + j2jl)~ Thus in order to complete our computation of the derivative of

199

we must consider the second term in the final expression (E.7) on the derivative of the map

h·~ p(J)fj,(J)

at the point Jo. Since p(Jo) = 0 we need only calculate Dp(JO)j2. Let X = J D1/J(O)j1, Y = D1/J(O)j2 and 9 = fj,(Jo). Then since X and Yare trace free divergence free it follows that

(E.1l)

where Dgbg(Y) is the derivative of the divergence operator bg with respect to 9 in the direction Y. Thus we have our formula for D2rp(O), namely

D2rp(O)(i1, j2) = 2(i1 j 2 + j2j1)~ + L;1 (bg(Dgbg)(Y)X) ,

where X = J D1/J(O)j1, Y = J D1/J(O)k

Lemma E.2

PROOF: Since (Dgbg)(Y)X = -bgDxY, DxY the derivative of Y with respect to X, then by (5.15) it follows that

1 (Dgbg)(X)X = -"2 *dl' ,

I' a real valued function on M, and *dl' the Hodge dual. Thus

1 bg(Dgbg(X)X) = -"2bg(*dl') = 0 .

This gives us

Theorem E.3

We are now ready to complete the proof of Theorem E.1. By formula (E.5) we must compute the sum of D2 E(Drp(O)H, DrpoH), DEoD2rp(O)(H, H), D2 E(Drp(O)JoH, Drp(O)JoH), and DE. D2rp(O)(JoH, JoH).

Now for h E SiT(g), h* E T[g]T(M), DE(g)(h) = DE[g]h, by lemma 3.1.4 we see that for k arbitrary

200 E. The Levi-Form of Dirichlet's Energy

where K = (k)1 and KT is the trace free part of K. Therefore

DE(g)D2rp(0)(H,H) = -2E j g(z) (H2)TVst, VSl) dp.tJ . lM

Lemma E.4 If H E TJA is divergence free then H2 = p.l where p. is a non-negative function which vanishes at most at finitely many points of M.

PROOF: Write H in conformal coordinates gi; = >.8i ; as H = (: ~,.). Then >.a - i>.h is a holomorphic quadratic differential on M and thus as 4(genus M)-4 zeros (genus M > 1). H2 = (a2 + b2)I = p.l, p. = t trace H2, which concludes the proof of the lemma. Consequently we see that

DE(g)(Drp(O)(H),Drp(O)(H» = D2(E 0 rp)(O)(H,H) .

If h = Drp(O)(H) = (-2H)~ then by formula 3.5

D2 E(g)(h, h) = ~ E j(h. h)g(z)(VtJSl, VtJSl)dp.tJ - D2 EtJ(S)(w", w") . lM

Therefore

D2(E 0 rp)(H, H) + D2(E 0 rp)(JoH, JoH) =

(E.12)

t E J {h . h }g( Z )(VtJSl, VtJSl)dp.tJ - D2 EtJ( S)( w", w") - D2 EtJ( S)( wi", wi h) lM

since if ih = (-2JoHh, then {ih. ih} = {h. h}.

As we know the second variation of EtJ, namely

we arrive at the conclusion of Theorem E.1.

Remark E.1 The similarity between the formula (E.1) for the Levi-form in AbreschFischer coordinates and formula (6.8) is no accident. It is a consequence of the fact that Abresch-Fischer coordinates_ almost satisfy the condition of Deligne, Griffiths, Morgan and Sullivan (el. theorem 6.2.1). See remark 6.2.2.

F Riemann-Roch and the Dimension of Teichmiiller Space

In this chapter we briefly give the background material from Riemann surface theory that

enables one to state the famous theorem of Riemann-Roch and to compute the dimension

of the space of holomorphic quadratic differentials on a Riemann surface.

Let (M, c) be a surface of genus greater than one with an associated complex structure c.

For ease of exposition we shall suppress the c from the notation (M,c) for the rest of

the chapter. Given such a structure c we clearly have the notion of a meromorphic

function on M. From the maximum modulus principle it follows that the only holomorphic functions on M are constants. Nevertheless, we can have holomorphic differentials.

Let w be a complex valued one form on Mj i.e. for x E M, w(x) : T",M -+ C is linear over

the real. In a local coordinate chart with local variables designated by x and y we can represent w by

w = P dx + Q dy + i( P dx + Q dy)

We say that w is holomorphic differential if "locally" w can be written as

w(z) = J'(z)dz

dz = dx + i dy, and where f is a holomorphic function. It is a well known fact (thm 10.3

of [104]) that the complex dimension of the vector space of holomorphic differentials is equal to the genus of M. One also has the obvious notion of a meromorphic differential

by requiring f to be meromorphic. A complex I-form on M which is either holomorphic or meromorphic is called an abelian differential.

It is easy to check that the order of a zero or a pole of either a meromorphic function

or an abelian differential is well-defined. We are interested in specifying to some extent

202 F. Riemann-Roch and the Dimension of Teichmiiller Space

the location and orders of poles of both meromorphic functions and abelian differentials

on M.

Let P1 , P2 , •• • , Pn be points on M and all a2," ., an be integers. The symbol

is called a divisor. The integer a" is called the order at Pie' By the degree d[a] of a divisor

a we mean the sum

If 1 is a meromorphic function not identically zero on M and w ¢: 0 is an abelian differ

ential we define the divisors (J) and (w) of 1 and w by

(I) - pa, pa'Q-f3, Q-f3l -1"'" 1 00 '1

where the zeros of 1 are P1, .. . , Pie with orders a1, .. . , a", all ai ~ 0, and the poles of 1 are Q1," . ,Q1 with orders /31>" . ,/31, all /3i ~ 0 and

where 1'1, ... ,Pr are the zeros of w of orders (}1, ... ,~ and Q1, ... , Q. are the poles of w

of orders /31, ... , /3 •.

Since for a meromorphic function, the sum of the orders of its zeros is equal to the sum

of the orders of its poles it follows that d(J) = 0 for any I.

The following is a basic result in Riemann surface theory.

Theorem F.l Ilw is an abelian differential then d(w) = 2 genus(M)-2.

Note that d(w) > 0 in our case where genus(M) > 1. Consequently every abelian differential must have a zero.

A divisor a = Pt' ... p;. is called integral if aj ~ 0 for all j. If b = Qf' ... Qf' then the

quotient divisor alb is defined by

a lb - pa, pa'Q-f3, Q-f3, -1 00

'" 1 00 '1'

By ~ we mean the divisor p1- a , ••• Pha ,. If alb is integral we say that b divides a or

that a is a multiple of b.

203

Define by L(a) the vector space of merom orphic functions on M whose divisors are integral multiples of a and by O(a) the vector space of abelian differentials whose divisors are integral multiples of a.

A beautiful relationship between the dimensions of these vector spaces over the complexes is given by

Theorem F.2 (Riemann-Roch)

dimL (~) = dimO(a) + d[a]- genus(M) + 1

For a proof the reader may consult any number of elementary texts on Riemann surfaces, see for example [104], [37].

In addition to permitting us to speak about meromorphic functions and abelian differentials a complex structure on M allows us to speak about holomorphic or meromorphic quadratic differentials.

A complex valued quadratic differential on M is a complex valued symmetric 0-2 tensor Q. Thus for each q E M,

is bilinear and symmetric. Locally Q can be expressed as

Q is said to be holomorphic if it can be expressed in a local complex coordinate system as

Q(z) = rp(z)dz 2 •

with rp holomorphic. Let Q( M) denote the complex linear space of holomorphic quadratic

differentials on M. The following theorem on the dimension of Q( M) over the complexness is the principal result we will need from elementary Riemann surface theory.

Theorem F.3 dime Q(M) = 3genus(M) - 3. Therefore dimlR Q(M) = 6genus(M) - 6.

PROOF: Let wo( z )dz2 be a holomorphic quadratic differential, say for example the product of two holomorphic abelian differentials. Then it follows from F.1 that if ao is the divisor of

Wo then d[ao] = 4genus( M) - 4. If w( Z )dz2 is any other holomorphic quadratic differential

204 F. lUemann-Rocb and tbe Dimension of Teicbmiiller Space

then w( z )dz2 / wo( Z )dz2 is a meromorphic function. H a denotes the divisor of wand since aaol is the divisor of a meromorphic function it follows that d[a] = d[ao] = 4 genus(M)-4.

For arbitrary w let ,.,(z) = w(z)dz2/wo(z)dz2 • Then w = ,.,. Wo where U.,) is an integral multiple of aol . It therefore follows that the elements of Q(M) are in one to one correspondence with L(aol ). By the Riemann-Roch theorem

dimL(aj)l) = dimO(ao) + d[ao]- genus(M) + 1

H T = cp(z)dz E O(ao) is non-zero, d(T) ~ d[ao] = 4 genus(M)-4. But by F.l, d(T) =

2genus(M) - 2 which is impossible. Thus dimO(ao) = o. Hence

dime Q(M) = dim L(aol ) = 4 genus(M) - 4 - genus(M) + 1 = 3 genus(M) - 3 I.

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214 Index of Notation

We include a list of some notations here. It does not contain every single item, but should include the basic symbols and where they occur. An asterisk in the column "where?" means that the notation is used throughout the book and no reference for a definition is needed. Otherwise, main occurrences and places of definitions are indicated.

symbol

a,b A,A'

ClI ClI •••

C

C CI, Ch,Coo

C

d[a] 'D 'Do

Df(g)h DX(Y)

DyX D at e

E,Eg

t,E E,EG

£. Cu

f :F

what?

divisors manifold of almost complex structures constants a complex structure on M the set of complex structures on M 1, k, 00 times continuously differentiable the set of complex numbers degree of a divisor the set of Coo diffeomorphisms of M - " - homotopic to the identity derivative of the function f at 9 in direction h for vector fields X, Y the function p f-+ DX(p)Y(p), extrinsic

" covariant derivative, extrinsic energy density Energy functional, Dirichlet's energy Dirichlet's energy on M, T(M) Wolf's form of Dirichlet's energy functional Wolf's form of Dirichlet's energy on T(M) the differential operator defined in usually an element of'D a real valued function on a complex manifold a metric on M a metric / induced metric on E a (Hilbert) Lie group a 0-2 tensor field, tangent vector to M,M_I' T(M)

where?

app. F def. 1.1.1

* def. 0.2, sec. 1.1 p.10

* * app. F p. 10 p. 10

* ch.5

sec. 5.3 sec. 2.1,3.1, app. B sec. 3.1, app. B ch.3 sec. 3.1 sec. 6.3 sec. 6.3 (B.5), app. B

* ch. 6

* * /thm. 5.1.1 chs.4,5

*

Index of Notation 215

symbol what? where?

H horizontal component p. 99 H a 1-1 tensor field on M * 1lI the upper half plane (hyperbolic plane) *

H'(M) functions on M of Sobolev class H' ch.O

1{.·(T:M) tensor fields on M of class H', p times contravari- ch.O ant and q times covariant

I unit matrix or identity map on some vector space * id identity map * J an almost complex structure on M, i.e. an element def. 1.1.1

of A' J J aco bian determinant p.180 K a 1-1 tensor field on M * K- sectional curvature sec. 5,4

Lx Lie derivative in direction X (1.3) Lt(M,IRd ) Sobolev space p.159

£, ll. - id p.115 M Riemann surface of genus ;::: 2 *

M,M' manifold of COO or H' metrics on M sec. 1.2

M_t.M~l manifold of COO or H' metrics on M with scalar secs. 1.2, 1.6 curvature -1

N some manifold * N Nijenhuis tensor def.4.1.3

Of, a! the action by f on some space * p a point on a manifold; or a positive function in the *

context of conformal coordinates p total space of a principal bundle sec. 4.1

P,P' positive functions on M sec. 1.3

Q(M) space of holomorphic quadratic differentials on M app. F pr a projection map *

216

symbol

R R,ft

R(M) ill. S

S2 SiT

T

TT

T",M T(M)

tr trg u

u

v

X(M) X,Y,Z

z

Index of Notation

what?

scalar curvature

Riemann curvature tensor

Riemann moduli space

the set of real numbers

a harmonic map

symmetric 0-2 tensors

space of transverse (=divergence free) traceless

0-2 tensors, i.e. the tangent space to T(M) ~ M-dVo "slice" the group of 2x2 matrices of determinant 1, two

sheeted cover of automorphisms of upper half

plane

trace free part

transverse (=di vergence free) traceless

tangent space at x to M Teichmiiller space

trace of a 1-1 tensor or linear map

trace with respect to the metric g a function from M to M an open set; domain of coordinate function or of its inverse

vertical component

space of Coo vector fields on M vector fields complex coordinate on M

where?

p. 24

sec. 5.4

def.0.5

* p. 64

ch.O

p.45

pp.47-57

*

p. 19, sec. 1.3

p. 45

* def.0.6

* * app. B

*

p. 99

* * *

symbol

OJ

(3(u) rt Oij

Og AU)

A

II

O',O'EE

0'( t) E

<J>,~ cp,'if;

X(M) n n

nI;,n (., .)

(., ')wp ((., .)) (((., . )))

'\7g

'\7, V flg fl

.-, -.

---+ :

( . )

Index of Notation

what?

the map X f-+ Lx] the vector field over u defined in

Christoffel symbol

Kronecker delta

divergence

Lefshetz number of f conformal factor III the context of conformal

coordinates

given] E A, certain functions on M projection map in a bundle

projection map in a linear space

a section in a hundle, Earle-Eells section

a geodesic

base space of a principal bundle

almost complex structure on A, T(M) coordinate map or its inverse

Euler characteristics of M lattice in C Kahler form on A Kahler form on E resp. T(M)

inner product (specified in the context)

Weil-Petersson metric L2-inner product for 1-1 tensors and on M alternative L2-inner product on M-l gradient with respect to 9

connection, Levi-Civita connection

Laplace-Beltrami operator

non-linear Laplacian of a map M ---+ M or linear

Laplacian of a vector field over u

equal by definition; the colon is on the side being

defined converges to something, the name of the limit is

defined

the Hodge dual of a differential form w

L2-inner product density of 0-2 tensors

(in SiT ~ T M-d

where?

sec. 1.4

(B.8)

* * sec. 1.4

p.39

*

sec. 5.4

* * sec. 3.4

sec. 2.1

sec. 4.1

ch.5

* * p. 8 sec. 5.1 sec. 5.1

* (2.7) pp. 19,56 sec. 2.5,2.6

* sec. 5.4

sec. 1.4

p. 160

*

*

ch. 5,6, app. E p. 72, ch. 6, app. E

217

218

The Maps Used in the Construction of Teichmiiller space

e / /

/ /

/

A/Vo ~ T(M) ~

/

s x Vo /

/

----.J The fat arrows denote V-equivariant diffeomorphisms or V-equivariant

bijective maps

- - - ~ The dashed arrows denote maps defined only on a neighbourhood of some point(s) which are diffeomorphisms from such a neighbourhood

to their image.

------;l::» surjective maps

c ) natural inclusions

The diagram is commutative.

Index

Abresch-Fischer coordinates on A 91, 93 Abresch-Fischer coordinates on T(M) 95 Almost complex principal bundle 86

Almost complex structures 14, 93

Almost complex structure on A 83, 102

Baerll Banach Lie group 53

Collar Lemma 74, 192 Complex structure 6

Complex structure on T(M) 88f Conformal coordinates 19 157 Courant Lebesgue lemma 76 Curvature tensor: see Riemann curvature

tensor

Diffeomorphism 7

Dirichlet's energy 30, 63, 73, 76, 137

Divergence 26 Divisor 202 Domain of holomorphy 124 Ebin-Palais theorem 41

Earle-Eells section 81

Eells-Sampson theorem 65 Gauss-Bonnet theorem 25 Geodesics 36

Harmonic map 64,174,179,183

Hodge dual 106, 143, 199 Holomorphic equivalence 7 Holomorphic mapping 7, 89

Holomorphic quadratic differentials 46, 203 Horizontal distribution 108

Horizontal vectors 55

Hurwitz' formula 70

Hyperbolic plane 9, 74 Kahler 98, 102, 105 L2-metric on A 56 L2-splitting of metrics 19

L2-splitting of vector fields 27

L2-splitting of 0-2 tensors 28

Laplace-Beltrami operator 27

Lefshetz Fixed Point theorem 39 Levi-Civita. connection 103

LevI-form 123, 196 Lie derivative 27 Lie group 53 Manifold 6

Mumford compactness theorem 75, 184 Natural connection on A 102

Newlander-Nirenberg theorem 86

Nielsen 11 Nielsen realization problem 152

Nijenhuis tensor 84 Orient able 6

Orientation 6 Pluri-wbharmonic function 123, 137

Poincare's theorem 25, sec. 1.5

Principal bundle 54, 86 Pseudo convex 124

Quasi-conformal mapping 46 Ricci curvature of Teichmiiller space 121

Riemann curvature tensor 105, 111 Riemann moduli space 10

Riemann-Roch theorem 46,203

Riemann surface 6 Riemannian metric on M 18

220

Riemannian metrics 18

Royden 90

Scalar curvature 24 Schoen-Yau theorem 66

Second variation of Dirichlet's energy 73,

129 Section 81 Sectional curvature 111

Sectional curvature of Teichmiiller space

120, 121 Sobolev embedding theorem 13, 159

Sobolev spaces 12, 159

Stein manifold 125

Teichmiiller metric 60

Teichmiiller moduli space 10

Vertical vectors 55 Volume preserving diffeomorphisms 109,

111

Weil-Petersson metric 60, 61, 102, 105

INDEX

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