conference board of the mathematical sciences · kahler manifolds 26 (a) kahler geometry 26 (b)...

Conference Board of the Mathematical Sciences REGIONAL CONFERENCE SERIES IN MA THEM A TICS

supported by the National Science Foundation

Number 57

PRESCRIBING THE CURVATURE OF A RIEMANNIAN MANIFOLD

by Jerry L. Kazdan

Published for the Conference Board of the Mathematical Sciences

by the American Mathematical Society

Providence, Rhode Island

http://dx.doi.org/10.1090/cbms/057

Expository Lecture s

from th e CBMS Regional Conferenc e

held a t th e Polytechni c Institut e o f Ne w York

January 6-10 , 198 4

Research supporte d i n par t b y Nationa l Scienc e Foundatio n Gran t MC S 82-01140.

1980 Mathematics Subject Classifications. Primar y 35Jxx , 53Cxx .

Library o f Congres s Cataloging in Publication Dat a

Kazdan, Jerry L. , 1937 -

Prescribing th e curvatur e o f a Riemannian manifold .

(Regional conferenc e serie s in mathematics; no. 57 )

"Expository lecture s fro m th e CBM S regional conferenc e hel d a t th e Polytechni c

Institute o f Ne w York, January 6-10 , 1984" —

Bibliography: p .

1. Riemannia n manifolds . 2 . Curvature. I . Conference Boar d of th e Mathematica l

Sciences. II . Title. III . Series.

QA1.R33 no . 57 [QA649 ] 510 s [516.3'62 ] 84-2827 4

ISBN 0-8218-0707-2 (alk . paper )

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Contents

Preface v

Notation vi i

I. Gaussia n Curvature 1

1. Surfaces i n R3 1 2. Prescribing the curvature form o n a surface 3 3. Prescribin g the Gaussian curvature on a surface 3

(a) Compact surface s 4 (b) Noncompact surface s 6

II. Scala r Curvature 9 1. Topological obstructions . . 9 2. Pointwise conformal deformation s an d th e Yamabe problem 1 1

(a) Mn compac t 1 2 (b) M" noncompac t 1 5

3. Prescribing scalar curvature 1 6 4. Cauchy-Riemann manifold s 1 7

III.Ricci Curvature . 1 9 1. Loca l solvability of Ric(g ) = R ij 2 0 2. Loca l smoothnes s of metrics 2 1 3. Global topologica l obstructions 2 1 4. Uniqueness , nonexistence 2 3 5. Einstein metrics on 3-manifolds 2 3 6. Kahler manifolds 2 6

(a) Kahler geometry 2 6 (b) Calabi's problem and Kahler-Einstei n metric s 2 7 (c) Another variationa l problem 3 1

IV. Boundary Value Problems 3 3 1. Surfaces with constant mean curvature and Rellich' s problem 3 3 2. Some other boundary value problems 3 6

(a) Graphs with prescribed mean curvature 3 6 (b) Graphs with prescribed Gauss curvature 3 8

3. The C 2 + a estimat e at the boundary 3 9

Some Open Problems 4 7

References 5 1

iii

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Preface

These notes were the basis fo r a series of te n lectures given from Januar y 6-10 , 1984 a t Polytechni c Institut e o f Ne w Yor k unde r th e sponsorshi p o f th e Con -ference Boar d o f th e Mathematica l Science s an d th e Nationa l Scienc e Founda -tion. Th e lecture s wer e aime d a t mathematician s wh o knew eithe r som e differen -tial geometr y o r partia l differentia l equations , althoug h other s coul d hopefull y understand th e lectures.

Ostensibly, th e primar y proble m addresse d her e i s t o understan d th e variou s curvatures o n a Riemannia n manifold . Often thi s questio n ca n b e reduce d t o solving som e nonlinear partia l differentia l equations . But fro m th e viewpoin t o f a geometer, thes e question s ar e onl y portal s t o see k a deepe r understandin g o f Riemannian manifolds . On th e other hand , a n analys t may find th e geometry dul l and stil l be delighted with the nonlinear partia l differentia l equation s one is led t o understand. Th e questions are rich enough to serve as motivation fo r man y varie d tastes.

One goa l o f thes e lectures was to state what i s currently known an d no t know n about a variety of problems tha t involv e the curvature of a Riemannian manifold . My ow n inclinatio n wa s especially t o emphasize areas of curren t ignorance . Wit h this i n mind , a t th e end o f thes e notes I have collected a partial lis t o f som e ope n questions tha t ar e drawn fro m th e main body of the lectures.

Generally, I trie d t o giv e th e essentia l idea s behin d variou s proofs , an d hav e given references bu t certainly not detailed arguments . The only places where ther e are detaile d discussion s ar e wher e th e materia l i s eithe r difficul t t o extrac t fro m the literature , or where it has not bee n written down . This is especially true of th e last sectio n o f thes e notes, where I give Caffarelli's simplificatio n o f a n importan t recent estimat e o f Krylo v concernin g th e boundary regularit y fo r som e nonlinea r elliptic equations . While thi s material i s a bi t technical , i t appear s nowher e in th e literature an d ma y hel p spee d progres s i n resolvin g variou s boundar y valu e problems tha t aris e in geometry .

There i s a considerabl e overla p betwee n thes e note s an d a serie s o f lecture s I gave i n Japa n i n Jul y 1983 . Bu t thes e note s hav e a muc h greate r emphasi s o n geometric issues , whil e th e note s [K4 ] fro m th e lecture s i n Japa n gav e a mor e detailed an d elementar y expositio n o f th e idea s fro m partia l differentia l equa -tions. Becaus e of thos e earlier notes—which wer e distributed a t th e NSF/CBM S

v

VI PREFACE

lectures—I wa s less inhibited abou t using ideas from differentia l equations . The books [Au4, GT], and the appendix to [Be2] may help fill this gap.

I wil l be pleased if these lectures make the problems and ideas discussed more accessible to nonexperts.

It is a pleasure to thank Craig Evans and Neil Trudinger for generously sharing their idea s on th e boundary regularit y in th e last section , and t o Lui s Caffarell i for the use of his contributions in that same section. I also benefited fro m Denni s DeTurck's many suggestions for improvements.

Many people helped make this conference enjoyable and rewarding to me. The main organizationa l detail s wer e superbl y execute d b y Edwar d Miller , Rober t Sibner, and especially Lesley Sibner. I am grateful to them.

Finally, I would like to thank Frank Warner for getting me interested in these problems fifteen year s ago.

Notation

All manifolds ar e assumed to be smooth and connected. The notation we use is generally standard . C k+a, 0 < a < 1, is the space of function whos e /cth deriva-tives satisfy a Holder condition with exponent a, C" is the space of rea l analytic functions, an d th e Sobole v spac e HQ(SI) i s the completio n o f smoot h function s with compac t suppor t i n fl i n th e nor m (/a|Vw| 2)1/2. Ou r Laplacian , A , o n real-valued function s ha s the sign so that Aw = +u" o n R 1. The Ricci curvature of th e standard uni t spher e S" with metric g in R n+1 i s (n - l)g , an d it s scalar curvature i s n(n — 1) . If a is a one-form o n a Riemannian manifold , the n a* i s the dua l vecto r field . I n classica l tenso r notatio n w e use a semicolon t o denot e covariant differentiation. u„ is the volume of the unit sphere S" in R" +1.

vu

48 SOMH OI'l. N PROBLEM S

curvature - i . Whic h function s K(x) ar e Gaussian (o r scalar ) curvature function s of complet e pointvvis e conformal metrics ? This involves solving

A// = - 1 - A'(.v)* 2" o n H\

zAL'L_ZilA„ + S o u - S ( x ) u ( " ' 2)/{" 2) o n H\ n : > 3,

where S {) = ~n(n - 1 ) i s th e scala r curvatur e o f //" . Ne w phenomen a ar e expected sinc e o n H" on e ca n uniquel y solv e th e Dirichle t proble m fo r A w = 0 with continuou s boundar y values , wherea s o n R " th e onl y bounde d harmoni c functions ar e constants. Blan d an d Kalk a [BK] have some results . See also [AvM].

One shoul d als o conside r related boundar y valu e problem s o n compac t mani -folds wit h boundary . Cherrie r [Ch] has some results.

5. Recentl y ther e have been some result s (by Oliker , and Treiberg s and Wei ) on graphs ove r S 2 wit h prescribe d Gaussia n o r mea n curvature . Th e nex t simples t case ma y b e graph s ove r th e torus . T 1 *- > R\ bu t ther e ar e man y ope n question s here.

6. Harmoni c spinor s an d th e Dira c operato r hav e bee n importan t i n findin g topological obstruction s t o compac t (o r complete ) manifold s havin g metric s wit h positive scala r curvatur e (se e [GL1-GL3]) . I s ther e som e wa y t o us e harmoni c spinors t o construct metric s wit h positiv e scala r curvature ? Ther e i s a simila r question concernin g harmonic 1-form s and positiv e Ricc i curvature .

7. Doe s ever y noncompac t manifol d hav e a complet e metri c wit h constan t negative scala r curvature? This is known t o be true fo r compact manifolds .

8. Solv e th e analogu e o f th e Yamab e proble m fo r complet e noncompac t manifolds, a s well as some version on compact manifold s wit h boundary .

9. Doe s ever y compac t M'\ n ^ 3 , admi t a metri c wit h negativ e Ricc i curva -ture? Ga o an d Ya u hav e announce d tha t ther e i s alway s a negativ e Ricc i curvature metri c o n an y compac t M 3. Wha t abou t th e analogou s questio n fo r n > 4 and fo r complet e metrics on noncompac t A/" ?

10. The only know n obstruction s t o Einstei n metric s on compac t manifold s M n

are whe n n < 4 . Ar e ther e an y i n highe r dimensions ? On e expect s th e answe r i s "yes".

11. Whic h symmetri c tensor s R l} ar e locall y th e Ricc i tenso r o f a metric ? DeTurck [D2 ] prove d tha t R lf i s locall y a Ricc i tenso r i f R lf i s invertible , bu t found som e obstruction s i f R {/ i s no t invertible . Th e natura l gues s i s tha t R t/ i s locally a Ricc i tenso r i f ther e i s some metri c g so tha t th e second Bianch i identit y is satisfied locally . This is unresolved eve n i n the real analytic case.

12. I f a compac t M" ha s a metri c wit h positiv e Ricc i curvature , ca n i t b e deformed t o an Einstei n metri c with positiv e Ricci curvature? Hamilton' s theore m [Hal] shows thi s i s true if n = 3 .

http://dx.doi.org/10.1090/cbms/057/05

SOME OPE N PROBLEM S 49

13. I f a compac t M 3 satisfie s th e assumption s o f th e Poincar e conjecture , doe s it admi t a metric with positive Ricci curvature? I f so , then b y Hamilton' s theore m [Hal] the Poincare conjecture i s resolved .

14. Whic h compac t Kahle r manifold s wit h th e Cher n clas s c x > 0 hav e Ein -stein-Kahler metrics ? I s Futaki' s obstructio n [Fl ] th e onl y one ? Thi s i s closel y related t o problem 3 above.

15. Calab i ha s aske d when , o n a compac t Kahle r manifol d m wit h metri c g 0, there i s always a cohomologous metri c g = g () 4 - </>_ minimizing th e squar e o f th e scalar curvatur e J(g) = / S 2(g) dV. Fo r example , i f c\(M) > 0 an d doe s not admit a Kahler-Einstei n metric , the n suc h a metri c minimizin g J i s a candidat e for a "nice " metri c o n M. Levin e [Le ] ha s example s wher e n o minimu m exists . When does the minimum exist ?

16. Minima l surface s hav e recentl y show n thei r valu e a s a n analogu e o f geodesies i n provin g result s i n geometry . I s ther e a fruitfu l generalizatio n o f suc h ideas a s "geodesi c flow" , "compariso n theorems" , o r th e "cu t locus " t o minima l surfaces?

17. B y considering th e Dirac operator o n auxiliar y vecto r bundle s over compac t manifolds, Gromo v an d Lawso n [GL1 ] extende d Lichnerowicz' s positiv e scala r curvature obstruction s t o a muc h large r clas s o f manifolds . Ca n on e als o us e harmonic 1-form s o n auxiliar y vecto r bundle s t o exten d Bochner' s classica l obstructions t o positive Ricc i curvature ?

18. (a ) I n th e recen t wor k o f Brezi s and Coro n [BC ] on th e Rellic h proble m o n surfaces wit h constan t mea n curvature , they onl y conside r map s fro m th e uni t disc. Wha t ca n on e sa y i f on e seek s surface s o f othe r topologica l typ e an d removes th e assumption tha t th e mean curvature i s a constant?

(b) Ther e ar e tw o spherica l cap s i n R 3 tha t spa n a circl e an d hav e constan t mean curvatur e H < 1 . Thes e ar e th e tw o obviou s solution s o f th e Rellic h problem. Ar e these the only solutions ?

19. Minima l graph s oi codimensio n on e are fairl y wel l understood . Lawso n an d Osserman [LO ] hav e foun d tha t fo r highe r codimensio n ne w phenomen a aris e concerning th e homotop y clas s o f th e boundar y values . On e shoul d understan d this muc h better . Surel y thes e same issues arise fo r th e Dirichle t proble m fo r mor e general quasilinea r ellipti c systems, but nothin g i s known.


References

[Ab] U . Abresch , Constant mean curvature tori in terms of elliptic functions, J . fu r di e rein e und angewandt e Math . 37 4 (1987) , 169-192 .

[Aul] T. Aubin, Metriques riemanniennes et courbure, J. Differentia l Geom . 4 (1970), 383-424. [Au2] , Meilleures constantes dans le theoreme d'inclusion de Sobolev et un theoreme de

Fredholm non lineaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal. 32 (1979), 148-174.

[Au3] , Equations du type Monge-Ampere sur les varietes kahleriennes compactes, C. R . Acad . Sci. Pari s Ser . A 283 (1976), 119.

[Au4] Nonlinear analysis on manifolds, Monge-Ampere equations, Di e Grundlehre n de r Math. Wissenschaften , vol . 252, Springer-Verlag, New York , 1982.

(Au5) , Reduction du cas posit if de / ' equation de Monge-Ampere sur les varietes kahleriennes compactes a la demonstration d* une inegalite, J. Funct. Anal . 57 (1984), 143-153.

[Au6] , Equations du type Monge-Ampere sur les varietes kahleriennes compactes, Bull . Soc . Math. Franc e 10 2 (1978), 63-95.

[Ave] A . Avez , Valeur moyenne du scalaire de courbure sur une variete compacte. Applications relativistes, C R . Acad . Sci . Paris Ser. A 256(1963), 5271-5273.

[Av] P . Aviles , Conformal complete metrics with prescribed nonnegative Gaussian curvature in R 2 (t o appear).

[AvM] P . Avile s an d Rober t McOvven , Conformal deformations of complete manifolds with negative curvature (to appear) .

[Az] D . G . Azov , On a class of hyperbolic Monge-Ampere equations, Uspckh i Mat . Nauk . 3 8 (1983), 153-154; Englis h transl . in Russian Math . Survey s 38 (1983), 190-191 .

[Ba] Alfred Baldes , personal communication . [BaC] A. Bar i e t J . M. Coron, Une theorie des points critiques a Vinfini pour Vequation de Yamabe et

le problem de Kazdan- Warner, C. R . Acad. Sci . Paris (to appear). [BB] L . Berard-Bergery , La courbure scalaire des varietes riemanniennes, Seminair e Bourbalci ,

Lecture Note s i n Math. , vol . 842, Springcr-Verlag, Berlin an d Ne w York, 1980 , pp. 225-245. [B] Melvy n Berger , On Riemannian structures of prescribed Gaussian curvature for compact 2-mani-

folds, J . Differentia l Geom . 5 (1971), 325-332. [Bel] Seminair e Arthu r Besse , Geometrie riemannienne en dimension 4 , CEDIC/Fernan d Nathan ,

Paris, 1981. [Be2] Arthu r Besse , Einstein Manifolds, Ergebniss e de r Math. , 3 Folge , Ban d 10 , Springer -

Verlag, 1987 . [BK] J. Bland an d M . Kalka , Complete metrics conformal to the hyperbolic disc (io appear). [BI] D . Bleecker , The Gauss-Bonnet inequality and almost geodesic loops, Adv. i n Math. , 1 4 (1974),

183-193. [Bo] J. P . Bourguignon. A review of Einstein metrics. Lectures in China (to appear) . [BE] J . P . Bourguigno n an d J . P . Ezin , Scalar curvature functions in a conformal class of

metrics and conformal transformations, Trans . Airier . Math . Soc . 30 1 (1987) , 72 3 736 . [BC] H . Brezi s an d J-M . Coron , Multiple solutions of H-systems and Rellich's conjecture, Comm .

Pure Appl . Math . 37 (1984), 149-187 . [BL] H. Brezi s an d E . Lieb , A relation between pointwise convergence of functions and convergence of

functionals, Proc . Amer. Math . Soc . 88 (1983), 486-490.

51


52 REFERENCES

[BN] H . Brezi s an d L . Nirenbcrg , Positive solutions of nonlinear elliptic equations involving critical Soholev exponents, Comm . Pur e Appl. Math . 36 (1983), 437-477.

[CNS] L . Caffarclli , L . Nirenber g an d J . Spruck , The Dirichlet problem for nonlinear second order elliptic equations I . Monge-Ampere equation, Comm. Pur e Appl. Math . 37 (1984), 369-402.

[CI] E . Calabi , On Kahler manifolds with vanishing canonical class. Algebrai c Geometr y an d Topology, Princeto n Univ . Press , Princeton, N. J., 1955 , pp. 78-79.

[C2] , Extremal Kahler metrics. Semina r o n Differentia l Geometr y (S . T. Yau , ed.) , Ann. o f Math. Studies , vol. 102 , Princeton Univ . Press , Princeton, N. J., 1982 , pp. 259-290.

[C3] , Extremal Kahler metrics. II , Differential Geometr y an d Comple x Analysi s (Chavcl an d Farkas, eds.) . Springer-Verlag, Berlin , 1985 , pp. 95-114.

(CY] S . Y . Chen g an d S . T . Yau , The real Monge-Ampere equation and a/fine flat structures, Proc. 1980 Beijin g Sympos . o n Differentia l Geometr y an d Differentia l Equation s (S . S . Cher n an d W u Wen-Tsun, eds.) . Vol. 1 , Science Press , Beijing, and Gordo n & Breach, New York . 1982 , pp. 339-370 .

[CYlJ Sun-Yun g A . Chan g an d Pau l C . Yang , Prescribing Gaussian curvature on S 2 (t o appear).

[CY2] , Conformal deformation of metrics on S2 (t o appeal*) .

[Ch] P. Cherrier, Problemes de Neumann non lineaires sur les varieties Riemanniennes, J . Funct . Anal . 57(1984), 154-206 .

[Chi ] Wenxion g Chen , Scalar curvatures on S 2, Trans . Amer . Math . Soc . (t o appear) .

[Ch2] , Scalar curvatures on Sn (t o appear) .

[Del] P . Delanoe , Plongements Radiaux S n «— • Rn+l a courbure de Gauss positive prescrite, Ann. Scient . Ec , Norm . Sup. , 4 eser, 1 8 (1985) , 633-649 .

[Dl] D . DeTurck , Metrics with prescribed Ricci curvature. Semina r o n Differentia l Geometr y (S. T . Yau , ed.) , Ann . o f Math . Studies , vol . 102 , Princeto n Univ . Press , Princeton , N . J. , 1982 , pp . 525-537.

[D2] , Existence of metrics with prescribed Ricci curvature: local theory. Invent . Math . 6 5 (1981), 179-208 .

[1)3] , Nonlinear Douglis-Nirenberg systems in geometry. Lectur e note s Universit a d i Roma , 1982.

[DK] D . DeTurc k an d J . L . Kazdan , Some regularity theorems in Riemannian geometry, Ann . Sci . Ecole Norm . Sup . (4) 1 4 (1981), 249-260.

[DKo] D . DeTurc k an d N . Koiso , Uniqueness and non-existence of metrics with prescribed Ricci curvature, Ann . Inst . H . Poincare Sect . A (N.S.) 1 (1984), 351-359.

[ELI] J . Eell s and L . Lemaire, A report on harmonic maps, Bull . London Math . Soc . 10 (1978), 1-68 . [EL2] , Selected topics in harmonic maps, CBMS-NS F Regiona l Conf . Ser . i n Math. , No . 5 0

Amer. Math . Soc , Providence , R. I. , 1983. [ES] J . F . Escoba r an d R . M . Schoen , Conformal metrics with prescribed scalar curvature,

Inventiones 8 6 (1986) , 243-254 . [E] L . C . Evans , Classical solutions of fully nonlinear convex second-order elliptic equations, Comm.

Pure Appl . Math . 35 (1982), 333-363. [Fl] A . Futaki , An obstruction to the existence of Einstein Kahler metrics, Invent . Math . 7 3 (1983) ,

437-443. [FM] A. Futak i an d S . Morita , Invariant polynomials characteristic to compact convex manifolds and

compact group actions (to appear). [G] L . Z . Gao , The construction of negatively Ricci curved manifolds, Math . Ann . 27 1 (1985) ,

185-208. [GY] L . Z . Ga o an d S . T . Yau , The existence of negatively Ricci curved metrics on three-

manifolds, Inventione s 8 5 (1986) , 637-652 . [GT] D . Gilbar g an d N . S . Trudinger, Elliptic partial differential equations of second order, 2nd ed. ,

Die Grundlehren de r Math . Wissenschaften, Vol . 224, Springer-Verlag, Berlin , 1983. [Gi] E. Giusti , On the equation of prescribed mean curvature, Invent . Math . 46 (1978), 111-137 . [GW] R . Green e an d H . Wu , Whitney's imbedding theorem by solutions of elliptic equations and

geometric consequences, Proc . Sympos . Pur e Math. , Vol . 27 , Par t 2 , Amer. Math . Soc , Providence , R . I., 1975 , pp. 287-296 .


REFERENCES 53

[GLl ] M . Cjromo v an d H . B . Lawson , Spin and scalar curvature in the presence of a fundamental

group. I , Ann . o f Math . (2 ) 11 1 (1980) , 209-230 .

[GL2] , The classification of simply connected manifolds of positive scalar curvature, Ann . o f

Math . (2 ) 11 1 (1980), 423-434 .

[GL3] , Positive scalar curvature and the Dirac operator on complete Riemannian manifolds,

Inst. Haute s Etude s Sci . Publ . Math. , 5 8 (1983), 83-196 .

[Gu] H . Guggenehimer , Uber vierdimensionale Einsteinra'ume, Experiment a 8 (T>52) , 420-421 .

[Ha l ] Richar d S . Hamilton , Three-manifolds with positive Ricci curvature, J . Differentia l Geom . 1 7

(1982), 255-306 .

[Ha2] The Ricci curvature equation, MSR I Report , Berkeley , Ca. , 1983.

[ H a 3 ] R icha r d S . H a m i l t o n , Four-manifolds with positive curvature tensor, J . Diff . G eom . 2 4

(1986), 153-179 .

[He] E . Heinz , On the non-existence of a surface of constant rectifiable boundary, Arch . Rationa l

Mech. Anal . 3 5 (1969), 244-252 .

[Hi] S . Hi ldebrandt , On the Plateau problem for surfaces of constant mean curvature, Comm . Pur e

Appl. Math . 2 3 (1970), 97-114 .

[HI ] N . Hitchin , Harmonic spinors. Adv . i n Math . 1 4 (1974) , 1-55 .

[H2] , Compact four-dimensional Einstein manifolds, J . Differentia l Geom . 9 (1974) , 435-441 .

[HTY] W.-Y . Hsiang , Zhcn-huan g Ten g an d Wen- u Yu , New examples of constant mean curvature

immersions of (2k ~ 1 ) spheres into Euclidean Ik-space, Ann . of Math . (2 ) 11 7 (1983), 609-625 .

[JL] D . Jeriso n an d Joh n M . Lee , A subelliptic, nonlinear eigenvalue problem and scalar curvature on

CR manifolds (t o appear) .

[Kl] J . L . Kazdan , Another proof of Bianchi's identity in Riemannian geometry, Proc . Amcr . Math .

Soc. 8 1 (1981) , 341-342 .

[K2] A remark on the preceding paper of Yau, Comm . Pur e Appl . Math . 3 1 (1978) , 413-414 .

[K3] , Deformation to positive scalar curvature on complete manifolds. Math . Ann . 26 1 (1982) ,

227-234.

[K4] , Some applications of partial differential equations to problems in geometry. Survey s i n

Geometry Series . Tokyo Univ. , 1983.

[K5] , Positive energy in general relativity, Seminair e Bourbaki , vol . 1981-1982 , exposc e 593 ,

Asterisque, Soc . Math. France , Paris , 1982 , pp. 92-93 .

[ K p l ] N . K a p o u l e a s , Constant mean curvature surfaces in Euclidean three-space ( t o a p p e a r ) .

[ K p 2 ] , Closed constant mean curvature surfaces in Euclidean three-space ( t o appeal") .

[KVV1] J . L . Kazda n an d F . W . Warner , Curvature functions for compact 2-manifolds, Ann . o f Math .

(2 )99(1974) , 14-47 .

[KW2] Curvature functions for open 2-manifolds, Ann . of Math . (2 ) 99 (1974), 203-219 .

[KW3] , Scalar curvature and conformal deformation of Riemannian structure, J . Differentia l

Geom. 10(1975) . 113-134 .

[KW4] , Existence and conformal deformation of metrics with prescribed Gaussian and scalar

curvatures, Ann . of Math . (2 ) 10 1 (1975), 317-331 .

[KVV5] , A direct approach to the determination of Gaussian and scalar curvature functions.

Invent. Math . 2 8 (1975), 227-230 .

[KW6] , Prescribing curvatures, Proc . Sympos . Pur e Math. , Vol . 27 , Amer . Math . Soc ,

Providence, R . I. , 1975 , pp. 219-226 .

[KVV7] , Remarks on some quasilinear elliptic equations, Comm . Pur e Appl . Math . 2 8 (1975) ,

567-597.

[KN] C . Keni g an d W.-M . Ni, On the elliptic equation Eu - k : + Ke 2u = 0 (to appear) .

[Krl] N . V . Krylov , Boundedly nonhomogeneous elliptic and parabolic equations, Izv . Akad . Nauk .

SSSR Ser . Mat . 46 (1982) , 487-523 ; Englis h transl . i n Math . USSR-Izv . 2 0 (1983), 459-492 .

[Kr2] , Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izv . Akad .

Nak. SSS R Ser . Mat . 47 (1983), 75-108 ; Englis h transl . i n Math . USS R Izv . 22 (1984), 67-97 .

[LO] H . B . Lawso n an d R . Osserman , Non-existence, non-uniqueness, and irregularity of solutions to

the minimal surface system. Act a Math . 13 9 (1977), 1-17 .

[LP] J o h n M . Le e an d T h o m a s H . Pa rke r , The Yamaha Problem, Bull . Amer . M a t h . Soc .

(NS) 1 7 (1987) , 3 7 - 9 1 .

[Le] Mar c Levine , A remark on extremal Kahler metrics , J . Diff . G e o m . 2 1 (1985) , 73 - 77 .


54 REFERENCES

[LI] A . Lichnerowicz , Geometric des groupes de transformations, Dunod , Paris , 1958.

[L2] , Spineurs harmoniques, C . R . Acad . Sci . Paris Scr . I Math . 25 7 (1963), 7 -9 .

[L i l ] C h a n g - S h o u Lin , The local isometric embedding in 72 3 of two-dimensional Riemannian

manifolds with Gaussian curvature changing sign cleanly, C o m m . P u r e Appl . M a t h . 3 9 (198G) ,

867 -887 .

[Li2] , The local isometric embedding in R 3 of 2-dimensional Riemannian manifolds with

nonnegative curvature, J . Diff . G e o m . 2 1 (1985) , 213 -230 .

[Liol] P-L . Lions , Sur les equations de Monge-Ampere. I , Manuscript a Math . (1983) , 1-43 .

[Lio2] , Sur les equations de Monge-Ampere. II , Arch. Rationa l Mech . Anal , (t o appear) .

[ M M ] M a r i o J . Micalle f a n d J o h n Doug la s M o o r e , Minimal two-spheres and the topology of

manifolds urith positive curvature on totally isotropic two-planes ( t o a p p e a r ) .

[ M e l ] R . McOwcn . On the equation A w + Kc lli = f and prescribed negative curvature in R 2, J . Math .

Anal . Appl . 10 3 (1984). 365-370 .

[Mc2] Conformal metrics in R- with prescribed Gaussian curvature and positive total curvature

(to appear) .

[MY] N . Mo k an d S . T . Yau , Completeness of the Kahler-Einstein metric on bounded Riemann

domains and the characterization of domains of holomorphy by curvature conditions, Proc . Sympos . Pur e

Math. , Vol . 39. Par t 1 , Amer. Math . S o c , Providence , R.I. , 1983, pp. 41 -59 .

[M] J . Moser , On a nonlinear problem in differential geometry. Dynamica l System s M . Peixoto , ed.) .

Academic Press , Ne w York , 1973.

[Ni l ] Wei-Min g Ni , On the elliptic equation A w + K(x)e 2u = 0 and conformal metrics with prescribed

Gaussian curvatures. Invent . Math . 6 6 (1982), 343-352 .

[Ni2] On the elliptic equation A w + K(x)u {"*2)A"~2) = 0 , its generalizations and applica-

tions in geometry, Indian a Univ . Math . J . 31 (1982), 493-529 .

[N] L . Nirenberg , The Weyl and Minkowski problems in differential geometry in the large, Comm .

Pure Appl . Math . 6 (1953) , 337-394 .

[O] V . I . Oliker , Hypersurfaces in R" 4 l with prescribed Gaussian curvature and related equations of

Monge-Ampere type, Comm . Partia l Differentia l Equation s 9 (1984) , 807-838 .

[P I ] A . V . Pogorelov , Extrinsic geometry of convex surfaces, " N a u k a " , Moscow , 1969 ; Englis h

transl.. Math . Monographs , vol . 35, Amer. Math . Soc. , Providence, R . I. , 1973.

[P2] , The multidimensional Minkowski problem (Englis h translation ) Winston , 1978 .

[Pok] S . Pokhozaev , Eigenfunctions of the equation A w + A/(w ) = 0 , Dokl . Akad . Nauk . SSS R 16 5

(1965), 3 3 - 3 6 ; Englis h transl . i n Sovie t Math . Dokl . 6 (1965) , 1408-1411 .

[Po] A . Polombo , Nombres characteristiques des varietes riemanniennes de dimension 4 , J . Differentia l

G eo m. 13(1978) . 145-162 .

[R] R . D . Reilly , Mean curvature, the luiplacian, and soap bubbles, Amer . Math . Monthl y 8 9 (1982) ,

180-198 .

[SS] I . Sabito v an d S . Shefel , The connections between the order of smoothness of a surface and its

metric, Sibirsk . Mat . Zh . 1 7 (1976) , 916-925 ; Englis h transl . i n Siberia n Math . J . 1 7 (1976) , pp .

687-694 .

[Sa] D . H . Sattinger , Conformal metrics in R 2 with prescribed curvature, Indian a Univ . Math . J . 2 2

(1972), 1-4 .

[S] Richar d Schoen , Conformal deformation of a Riemannian metric to constant scalar curvature , J .

Diff. G e o m . 2 0 (1984) , 4 7 9 - 4 9 5 .

[SY1] R . Schoe n an d S-T . Yau , Existence of incompressible minimal surfaces and (he topology of three

dimensional manifolds with non-negative scalar curvature, Ann . of Math . (2 ) 11 0 (1979), 127-142 .

[SY2] , On the proof of the positive mass conjecture in general relativity, Comm . Math . Phys .

65 (1979) , 4 5 - 7 6 .

[SY3] , Proof of the positive mass theorem. II , Comm. Math . Phys . 7 9 (1981), 231-260 .

[SY4] , Complete three dimensional manifolds with positive Ricci curvature and scalar curva-

ture. Semina r o n Differentia l Geometr y (S . T . Yau , ed.) , Ann. o f Math . Studies , vol . 104 . Princeto n

Univ. Press , Princeton , N . J. , 1982 , pp. 209-228 .

[SP] Seminair e Palaiseau , Premiere classe dp Chern et courbure de Ricci, Asterisque , vol . 58 , Soc.

Math . France , Paris , 1978.

[Se] J . Serrin , The problem of Dirichlet for quasi/inear elliptic differential equations with many

independent variables, Philos . Trans . Roy . Soc. London Ser . A. 264 (1969), 413-496 .


REFERENCES 55

[Sk] M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, 1975. fTh] J. Thorpe, Some remarks on the Gauss-Bonnet integrand, J. Math . Mech . 1 8 (1969), 779-786. [T] A. Treibergs, Existence and convexity for hyperspheres of prescribed mean curvature (to appear). [TW] A . Treiberg s an d S . W . Wei , Embedded hyperspheres with prescribed mean curvature, J .

Differential Geom . 18(1983) . [Tr] N . S . Trudinger , Remarks concerning the conformal deformation of Riemannian structures on

compact manifolds, Ann . Scuola Norm . Siup . Pisa CI. Sci. (4) 22 (1968), 265-274. [TU] N . S . Trudinge r an d J . I . E . Urbas , The Dirichlet problem for the equation of prescribed Gauss

curvature. Bull . Austral. Math . Soc . 28 (1983), 217-231. [U] Joh n I . E . Urbas , The equation of prescribed Gauss curvature without boundary conditions , J .

DifT. Geom . 2 0 (1984) , 311-327 . [V] M . Vaugon , Equations differentielles non lineaires sur les varieties riemanniennes compactes. I , II ,

Bull. Sci . Math . (2 ) 10 6 (1982), 352-367; 10 7 (1983). 371-391. [WW] N . Wallac h an d F . W . Warner , Curvature forms for 2-manifolds, Proc . Amer . Math . Soc . 2 5

(1970), 712-713. [WZ] McKenzi e Wan g an d W . Ziller , Existence and non-existence of homogeneous Einstein

metrics, Inventione s 8 4 (1986) , 177-194 . [Wb] S . M . Webster , Pseudohermitian structures on a real hypersurface, J. Differentia l Geom . 1 3

(1978), 25-41 . [Wei] Henr y C . Wente , Large solutions to the volume constrained Plateau problem, Arch . Rationa l

Mech. Anal . 75 (1980), 59-77. [We2] , The Plateau problem for symmetric surfaces, Arch . Rationa l Mech . Anal . 6 0

(1975-76), 149-169 . [We3] , The differential equation A.v = 2H(x u A xr) with vanishing boundary values, Proc .

Amer. Math . Soc . 50 (1975), 131-137. [We4] Counterexamples to a conjecture of //. Hopf, Pacific J . Math . 12 1 (1986) , 193-243 . [W] E. Witten, A new proof of the positive energy theorem, Comm. Math . Phys . 80 (1981), 381-402. [Wu] H . Wu , The Bochner technique, Proc . 198 0 Beijin g Sympos . o n Differentia l Geometr y an d

Differential Equations , vol. 2, (S. S. Chern an d W u Wen-tsun, eds.). Science Press , China, and Gordo n & Breach , New York , 1982 , pp. 929-1072.

[Ya] H. Yamabe , On the deformation of Riemannian structures on compact manifolds, Osaka Math . J. 12(1960), 21-37 .

[Y] S . T . Yau , On the Ricci curvature of a compact K'ahler manifold and the complex Monge-Ampere equation. I , Comm. Pur e Appl . Math . 31 (1978) , 339-411 .

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