Some Nonlinear Problems in Riemannian Geometry (Springer Monographs in Mathematics) 9783642082368, 9783662130063, 364208236X

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Springer Monographs in Mathematics

Springer-Verlag Berlin Heidelberg GmbH

Thierry Aubin

Some Nonlinear Problems •

In

Riemannian Geometry

Springer

Thierry Aubin University of Paris VI Mathematiques 4, Place Jussieu, Boite 172 F-7S2S2 Paris France

llbrary of Congress Cataloglng-In-Publlcatlon Data Aubin. Thierry. Some nonlinear problens in Rlemannlan geometry I Thlerry Aubln. en. -- (Springer monographs in mathenatics) p. Ineludes blbllographlcal references and Index. ISBN 978-3-642-08236-8 ISBN 978-3-662-13006-3 (eBook) DOI 10.1007/978-3-662-13006-3

1. Geometry. Rtemanntan. II. Sertes.

2. Nonltnear theortes.

OA649.A833 1998 516.3·73--de21

1. Tttle. 98-4150

CIP

Mathematics Subject Classification (1991); 35,53,58

This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction an microfilm or in any other way, and storage in data banks. Duplication of this publicat ion or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1998 The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by A. Leinz, Karlsruhe SPIN 10518869 41/3143-543210 - Printed an acid-freepaper

Preface

This book is the union of two books: the new edition of the former one "Nonlinear Analysis on Manifolds. Monge-Ampere Equations" (Grundlehren 252 Springer 1982) mixed with a new one where one finds, among other things, up-to-date results on the problems studied in the earlier one, and new methods for solving nonlinear elliptic problems. We will give below successively the prefaces of the two books, and at the end of the volume, the two bibliographies (the references * are new). A very interesting area of nonlinear partial differential equations lies in the study of special equations arising in Geometry and Physics. This book deals with some important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved. Each problem is explained, up-to-date results are given and proofs are presented. Thus the reader is given access, for each specific problem, to its present status of solution as well as to most up-to-date methods for approaching it. The book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing up technique, geometric and topological methods. My book "Nonlinear Analysis on Manifolds. Monge-Ampere Equations" (Grundlehren 252) is self-contained, and is an introduction to research in nonlinear analysis on manifolds, a field that was almost unexplored when the book appeared. Ever since then, the field has undergone great development. This new book deals with concrete applications of the knowledge contained in the earlier one. This book is adressed to researchers and advanced graduate students specializing in the field of partial differential equations, nonlinear analysis, Riemannian geometry, functional analysis and analytic geometry. Its objectives are to deal with some basic problems in Geometry and to provide a valuable tool for the researchers. It will allow readers to apprehend not only the latest results on most topics, but also the related questions, the open problems and the new techniques that have appeared recently. Some may find the pace of presentation rather fast, but ultimately, it represents an economy of time and effort for the reader. In the space of a few pages, for instance, the ideas and methods of proof of an important result may be sketched out completely here, whereas the full details are only to be found dispersed in several very long original articles.

VI

Preface Some problems studied here are not treated in any other book. For instance: -

-

-

Very few people know if the remaining cases of the Yamabe problem are really solved. The results were announced ten years ago, but parts of the proofs appeared only recently and in different articles, some not easily available. On prescribed scalar curvature. Between the author's first article on the topics in 1976, and the second one in 1991 which poses the problem again, only a few results appeared. Ever since, a lot of results have been proved. The same thing applies to the Nirenberg problem, the Kahler manifolds with C1(M) > 0 and the problem of Einstein metrics. The last chapter of the book deals with a very broad topic, on which there are many books: it is discussed here so that the reader may obtain an idea of the subject. About the methods. There are books on the variational method or on topological methods, but is there any book where we can find so many methods together ? Of course it is of advantage, when we attack a problem, to have many methods at one's disposal, and in this book there are also new techniques.

The reader can find most of the backgroung knowledge needed in [* 1]. Some additional material is given in Chapter 1. Chapter 2 is devoted to the Yamabe Problem. Thirty years were necessary to solve it entirely. After a proof with all the details, we will find new proofs which do not use the method advocated by Yamabe (minimizing his functional). The study of the Yamabe functional is not completed. We know very little about p. = sup JL[g], where JL[g] is the inf of the Yamabe functional in the conformal class [g]. This problem is related to Einstein metrics. Chapter 3 is concerned with the problem of prescribing the scalar curvature by a conformal change of metrics. When the manifolds is the sphere (Snl go) endowed with its canonical metric, the problem is very special: we study it in Chapter 4. Chapter 5 deals with Einstein-Kahler metrics. Although there has been a great progress when C1(M) > 0, not everything is clear yet. Chapter 6 deals with Ricci curvature. A problem that remains open for the next few years is the existence (or the non-existence) of Einstein metrics on a given manifold. Lastly, Chapter 7 studies harmonics maps. We present the pioneer article of Eells-Sampson on this topics, then we mention some new results. The subject is very large and is continually developing ; several books would be necessary to cover it! There are many other interesting subjects, but it is not the ambition of this book to treat all the field of research ! To explain some methods and to apply them is our main aim. It is my pleasure and privilege to express my deep thanks to my friends Melvyn Berger, Dennis DeTurck, Jerry Kazdan, Albert Milani and Joel Spruck

Preface

VII

who agreed to read one or two chapters. They suggested some mathematical improvements, and corrected many of my errors in English. I am also extremely grateful, to Pascal Cherrier, Emmanuel Hebey and Michel Vaugon, who helped me in the preparation of the book. February 1997

Thierry Aubin

Preface to "Grundlehren 252"

This volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical studies are inadequate to solve them. Each problem has its own particular difficulties. Nevertheless there exist some standard methods which are useful and which we must know to apply them. One should not forget that our problems are motivated by geometry, and that a geometrical argument may simplify the problem under investigation. Examples of this kind are still too rare. This work is neither a systematic study of a mathematical field nor the presentation of a lot of theoretical knowledge. On the contrary, I do my best to limit the text to the essential knowledge. I define as few concepts as possible and give only basic theorems which are useful for our topic. But I hope that the reader will find this sufficient to solve other geometrical problems by analysis. The book is intended to be used as a reference and as an introduction to research. It can be divided into two parts, with each part containing four chapters. Part I is concerned with essential background knowledge. Part II develops methods which are applied in a concrete way to resolve specific problems. Chapter 1 is devoted to Riemannian geometry. The specialists in analysis who do not know differential geometry will find, in the beginning of the chapter, the definitions and the results which are indispensable. Since it is useful to know how to compute both globally and in local coordinate charts, the proofs which we will present will be a good initiation. In particular, it is important to know Theorem 1.53, estimates on the components of the metric tensor in polar geodesic coordinates in terms of the curvature. Chapter 2 studies Sobolev spaces on Riemannian manifolds. Successively, we will treat density problems, the Sobolev imbedding theorem, the Kondrakov theorem, and the study of the limiting case of the Sobolev imbedding theorem.

Preface to "Grundlehren 252"

IX

These theorems will be used constantly. Considering the importance of Sobolev's theorem and also the interest of the proofs, three proofs of the theorem are given, the original proof of Sobolev, that of Gagliardo and Nirenberg, and my own proof, which enables us to know the value of the norm of the imbedding, an introduction to the notion of best constants in Sobolev's inequalities. This new concept is crucial for solving limiting cases. In Chapter 3 we will find, usually without proof, a substantial amount of analysis. The reader is assumed to know this background material. It is stated here as a reference and summary of the versions of results we will be using. There are as few results as possible. I choose only the most useful and applicable ones so that the reader does not drown in a host of results and lose the main point. For instance, it is possible to write a whole book on the regularity of weak solution for elliptic equations without discussing the existence of solutions. Here there are six theorems on this topic. Of course, sometimes other will be needed; in those cases there are precise references. It is obvious that most of the more elementary topics in this Chapter 3 have already been needed in the earlier chapters. Although we do assume prior knowledge of these basic topics, we have included precise statements of the most important concepts and facts for reference. Of course, the elementary material in this chapter could have been collected as a separate "Chapter 0" but this would have been artificial, and probably less useful to the reader. And since we do not assume that the reader knows the material on elliptic equations in Sobolev spaces, the corresponding sections should follow the two first chapters. Chapter 4 is concerned with the Green's function of the Laplacian on compact manifolds. This will be used to obtain both some regularity results and some inequalities that are not immediate consequences of the facts in Chapter 3. Chapter 5 is devoted to the Yamabe problem concerning the scalar curvature. Here the concept of best constants in Sobolev's inequalities plays an essential role. We close the chapter with a summary of the status of related problems concerning scalar curvature such as Berger's problem, for which we also use the results from Chapter 2 concerning the limiting case of the Sobolev imbedding theorem. In Chapter 6 we will study a problem posed by Nirenberg. Chapter 7 is concerned with the complex Monge-Ampere equation on compact Kahlerian manifolds. The existence of Einstein-Kahler metrics and the Calabi conjecture are problems which are equivalent to solving such equations. Lastly, Chapter 8 studies the real Monge-Ampere equation on a bounded convex set of ]Rn. There is also a short discussion of the complex MongeAmpere equation on a bounded pseudoconvex set of en. Throughout the book I have restricted my attention to those problems whose solution involves typical application of the methods. Of course, there are many other very interesting problems. For example, we should at least mention that, curiously, the Yamabe equation appears in the study of Yang-Mills fields, while a corresponding complex version is very close to the existence of complex Einstein-Kahler metrics discussed in Chapter 7.

x

Preface to "Grundlehren 252"

It is my pleasure and privilege to express my deep thanks to my friend Jerry Kazdan who agreed to read the manuscript from the beginning to end. He suggested many mathematical improvements, and, needless to say, corrected many blunders of mine in this English version. I also have to state in this place my appreciation for the efficient and friendly help of Jiirgen Moser and Melvyn Berger for the publication of the manuscript. Pascal Cherrier and Philippe Delanoe deserve special mention for helping in the completion of the text. May 1982

Thierry Aubin

Contents

Chapter 1

Riemannian Geometry §1. Introduction to Differential Geometry

§2.

§3. §4. §5.

§6. §7. §8. §9. §10. § 11.

1.1. Tangent Space 1.2. Connection 1.3. Curvature Riemannian Manifold 2.1. Metric Space . . 2.2. Riemannian Connection 2.3. Sectional Curvature. Ricci Tensor. Scalar Curvature 2.4. Parallel Displacement. Geodesic Exponential Mapping ...... . The Hopf-Rinow Theorem . . . . . Second Variation of the Length Integral 5.1. Existence of Thbular Neighborhoods 5.2. Second Variation of the Length Integral 5.3. Myers' Theorem Jacobi Field The Index Inequality Estimates on the Components of the Metric Tensor Integration over Riemannian Manifolds Manifold with Boundary 10.1. Stokes' Formula . . . . Harmonic Forms ..... . 11.1. Oriented Volume Element 11.2. Laplacian . . . . . . . 11.3. Hodge Decomposition Theorem 11.4. Spectrum . . . . . . . . .

1

2 3 3 4

5 6 6 8

9 13 15 15 15 16

17 18

20 23 25

26 26 26 27

29 31

Chapter 2

Sobolev Spaces §1. First Definitions

..... §2. Density Problems . . . . . §3. Sobolev Imbedding Theorem §4. Sobolev's Proof .....

32 33 35

37

XII §5. §6. §7. §8. §9.

§10. §11. §12. §13. §14. §15. §16. §17. §18.

Contents Proof by Gagliardo and Nirenberg . . . . . . . . . New Proof ................. . Sobolev Imbedding Theorem for Riemannian Manifolds Optimal Inequalities . . . . . . . . . . . . . . . Sobolev's Theorem for Compact Riemannian Manifolds with Boundary . . . . . . . . . . . . . . . . The Kondrakov Theorem . . . . . . . . . . Kondrakov's Theorem for Riemannian Manifolds Examples . . . . . . . . . . Improvement of the Best Constants The Case of the Sphere .... The Exceptional Case of the Sobolev Imbedding Theorem Moser's Results ........ . The Case of the Riemannian Manifolds Problems of Traces . . . . . . . .

38 39 44 50 50 53 55

56 57 61

63 65 67 69

Chapter 3

Background Material §1. Differential Calculus

§2.

§3.

§4.

§5.

§6.

§7.

1.1. The Mean Value Theorem 1.2. Inverse Function Theorem 1.3. Cauchy's Theorem Four Basic Theorems of Functional Analysis 2.1. Hahn-Banach Theorem 2.2. Open Mapping Theorem 2.3. The Banach-Steinhaus Theorem 2.4. Ascoli' s Theorem . . . . . . Weak Convergence. Compact Operators 3.1. Banach's Theorem 3.2. The Leray-Schauder Theorem 3.3. The Fredholm Theorem The Lebesgue Integral 4.1. Dominated Convergence Theorem ..... . 4.2. Fatou's Theorem 4.3. The Second Lebesgue Theorem 4.4. Rademacher's Theorem 4.5. Fubini's Theorem The Lp Spaces 5.1. Regularization 5.2. Radon's Theorem Elliptic Differential Operators 6.1. Weak Solution 6.2. Regularity Theorems . . 6.3. The Schauder Interior Estimates Inequalities ..... . 7.1. Holder's Inequality 7.2. Clarkson's Inequalities 7.3. Convolution Product .

70 71 72 72 73 73 73 73 74 74 74 74 75 75

76 77

77 77

78 78 80 81 83

84 85 88 88

88 89 89

Contents 7.4. The Calderon-Zygmund Inequality 7.5. Kom-Lichtenstein Theorem 7.6. Interpolation Inequalities §8. Maximum Principle 8.1. Hopf's Maximum Principle 8.2. Uniqueness Theorem 8.3. Maximum Principle for Nonlinear Elliptic Operator of Order Two . . . . . . . . . 8.4. Generalized Maximum Principle §9. Best Constants . . . . . . . . . 9.1. Applications to Sobolev's Spaces

XIII 90 90 93

96 96 96 97 98 99 100

Chapter 4

Complementary Material §1. Linear Elliptic Equations

§2.

§3.

§4.

§5. §6.

. . . . . . . . . . 1.1. First Nonzero Eigenvalue>. of ~ 1.2. Existence Theorem for the Equation ~tp = f Green's Function of the Laplacian 2.1. Parametrix . . . . . . . . . . . . 2.2. Green's Formula 2.3. Green's Function for Compact Manifolds 2.4. Green's Function for Compact Manifolds with Boundary Riemannian Geometry . . . . . . . 3.1. The First Eigenvalue . . . . . . . 3.2. Locally Conformally Flat Manifolds 3.3. The Green Function of the Laplacian 3.4. Some Theorems Partial Differential Equations 4.1. Elliptic Equations 4.2. Parabolic Equations The Methods . . . The Best Constants

101 101 104 106 106

107 108 112 115 115 117 119 123

125 125 129 134

139

Chapter 5

The Yamabe Problem §1. The Yamabe Problem 1.1. Yamabe's Method 1.2. Yamabe's Functional 1.3. Yamabe's Theorem §2. The Positive Case . . §3. The First Results §4. The Remaining Cases 4.1. The Compact Locally Conformally Flat Manifolds 4.2. Schoen's Article 4.3. The Dimension 3, 4 and 5 .... §5. The Positive Mass 5.1. Positive Mass Theorem, the Low Dimensions

145 146

150 150 152 157

160 160 161 162 164

166

XIV

Contents

5.2. Schoen and Yau's Article 5.3. The Positive Energy . . . . . . . §6. New Proofs for the Positive Case (J.L > 0) 6.1. Lee and Parker's Article 6.2. Hebey and Vaugon's Article 6.3. Topological Methods 6.4. Other Methods . . . . . §7. On the Number of Solutions 7.1. Some Cases of Uniqueness 7.2. Particular Cases . . . . . 7.3. About Uniqueness . . . . 7.4. Hebey-Vaugon's Approach 7.5. The Structure of the Set of Minimizers of J §8. Other Problems . . . . . . . . . . . . . . 8.1. Topological Meaning of the Scalar Curvature 8.2. The Cherrier Problem ........ . 8.3. The Yam abe Problem on CR Manifolds 8.4. The Yam abe Problem on Non-compact Manifolds 8.5. The Yamabe Problem on Domains in IRn 8.6. The Equivariant Yamabe Problem 8.7. An Hard Open Problem 8.8. Berger's Problem . . . . . .

166 169 171 171 171 172 175 175 175 176 178 178 179

179 179 180 182 183 185 187 188 191

Chapter 6

Prescribed Scalar Curvature §1. The Problem . . . . . . .

§2. §3. §4. §5.

§6.

§7. §8.

§9.

§10.

1.1. The General Problem 1.2. The Problem with Conformal Change of Metric The Negative Case when M is Compact The Zero Case when M is Compact The Positive Case when M is Compact The Method of Isometry-Concentration 5.1. The Problem . . . . . . . 5.2. Study of the Sequence {cpqj} 5.3. The Points of Concentration . ..... . 5.4. Consequences 5.5. Blow-up at a Point of Concentration The Problem on Other Manifolds 6.1. On Complete Non-compact Manifolds 6.2. On Compact Manifolds with Boundary The Nirenberg Problem First Results . . . . . . . . . . 8.1. Moser's Result . . . . . . . 8.2. Kazdan and Warner Obstructions 8.3. A Nonlinear Fredholm Theorem G-invariant Functions f ..... The General Case . . . . . . . . 10.1. Functions f Close to a Constant

194 194 196

197 204 209 214 214 216 218 221

224 227 227 229 230 231 232 233 235 238

241 241

Contents

xv

10.2. Dimension Two 10.3. Dimension n ~ 3 10.4. Rotationally Symmetric Functions §11. Related Problems 11.1. Multiplicity 11.2. Density 11.3. The Problem on the Half Sphere

243 245 247 247 247 248 249

Chapter 7

Einstein-Kahler Metrics §1. Kahler Manifolds §2.

§3.

§4.

§5. §6. §7. §8. §9. §1O. §11. §12. §13. §14.

§15.

§16.

§17. §18.

. . . 1.1. First Chern Class 1.2. Change of Kahler Metrics. Admissible Functions The Problems ..... 2.1. Einstein-Kahler Metric 2.2. Calabi's Conjecture The Method . . . . . . 3.1. Reducing the Problem to Equations 3.2. The First Results . . . . . Complex Monge-Ampere Equation 4.1. About Regularity . . . . . 4.2. About Uniqueness . . . . . Theorem of Existence (the Negative Case) Existence of Einstein-Kahler Metric Theorem of Existence (the Null Case) Proof of Calabi's Conjecture The Positive Case . . . . . . . . A Priori Estimate for ~cp A Priori Estimate for the Third Derivatives of Mixed TYpe The Method of Lower and Upper Solutions A Method for the Positive Case The Obstructions When C1(M) > 0 14.1. The First Obstruction 14.2. Futaki's Obstruction 14.3. A Further Obstruction The CO-Estimate 15.1. Definition of the Functionals J(cp) and J(cp) 15.2. Some Inequalities ...... . 15.3. The CO-Estimate . . . . . . . . . . 15.4. Inequalities for the Dimension m = 1 15.5. Inequalities for the Exponential Function ....... . Some Results On Uniqueness . . . . . . . . On Non-compact Kahler Manifolds

251 252 253 254 254 255 255 255 256 257 257 258 258 259 260 263 263 263 266 267 269 271 271 272 272 273 273 274 276 277 278 281 285 288

XVI

Contents

Chapter 8

Monge-Ampere Equations §1. Monge-Ampere Equations on Bounded Domains of JRn 1.1. The Fundamental Hypothesis 1.2. Extra Hypothesis 1.3. Theorem of Existence §2. The Estimates .... 2.1. The First Estimates 2.2. C2 -Estimate 2.3. C3 -Estimate §3. The Radon Measure Jf(ep) §4. The Functional J(ep) 4.1. Properties of J( ep) §5. Variational Problem . §6. The Complex Monge-Ampere Equation 6.1. Bedford's and Taylor's Results 6.2. The Measure m(ep) 6.3. The Function J(ep) .... 6.4. Some Properties of J(ep) §7. The Case of Radially Symmetric Fucntions 7.1. Variational Problem 7.2. An Open Problem §8. A New Method . . . .

289 289 290 291 292 292 293 296 301 306 306 311 314 314 31S 31S 31S

316 317 318

318

Chapter 9

The Ricci Curvature §1. About the Different Types of Curvature 1.1. The Sectional Curvature 1.2. The Scalar Curvature 1.3. The Ricci Curvature . . 1.4. Two Dimension . . . . §2. Prescribing the Ricci Curvature 2.1. DeTurck's Result 2.2. Some Computations 2.3. DeTurck's Equations 2.4. Global Results §3. The Hamilton Evolution Equation 3.1. The Equation . . . . . 3.2. Solution for a Short Time 3.3. Some Useful Results 3.4. Hamilton's Evolution Equations §4. The Consequences of Hamilton's Work ..... 4.1. Hamilton's Theorems 4.2. Pinched Theorems on the Concircular Curvature §S. Recent Results . . . . . . . . 5.1. On the Ricci Curvature . . . S.2. On the Concircular Curvature

321 321 322 322

323 323 323 324

324 325 326 326 327 330 333 343

343 344 345

345 346

Contents

XVII

Chapter 10

Harmonic Maps §1. Definitions and First Results §2. Existence Problems 2.1. The Problem 2.2. Some Basic Results 2.3. Existence Results §3. Problems of Regularity 3.1. Sobolev Spaces 3.2. The Results §4. The Case of aM f 0 4.1. General Results 4.2. Relaxed Energies 4.3. The Ginzburg-Landau Functional

348 351 351 352 354 356 356 357 359 359 360 361

Bibliography

365

Bibliography·

375

Subject Index

389

Notation

393

Chapter 1

Riemannian Geometry

§1. Introduction to Differential Geometry 1.1 A manifold M", of dimension n, is a Haussdorff topological space such that each point of M" has a neighborhood homeomorphic to IR". Thus a manifold is locally compact and locally connected. Hence a connected manifold is pathwise connected. 1.2 A local chart on M" is a pair (0, cp), where a is an open set of M" and cP a

homeomorphism of a onto an open set of IR". A collection (ai' CPi)i e I of local charts such that Ui e I OJ = M" is called an atlas. The coordinates of Pea, related to cP, are the coordinates of the point cp(P) of IR".

1.3 An atlas of class Ck (respectively, c ~, CW ) on Mil is an atlas for which all changes of coordinates are C k (respectively, CX, C W ). That is to say, if (O:Z' CP2) and (all' CPII) are two local charts with O2 11 Op :1= 0, then the map CP7. s cpi 1 of CPII(fJ.:z 11 all) onto cp:z(O:z 11 Oil) is a diffeomorphism of class Ck (respectively, Coo, CW). 1.4 Two atlases of class Ck on Mil (Uj, CPi)iel and (~, "':Z)lI.eA are said to be

equivalent if their union is an atlas of class Ck. By definition, a differentiable manifold of class Ck (respectively, Coo, CW) is a manifold together with an equivalence class of Ck atlases, (respectively, Coo, CW). 1.5 A mapping f of a differentiable manifold Ck : Wp into another M", is called differentiable C (r ::;; k) at P e U c Wp if'" fo qJ -1 is differentiable C at cp(P), and we define the rank ofJ at P to be the rank of", J cp - 1 at qJ(P). Here (U, cp) is a local chart of Wp and (0. "') a local chart of Mil with f(P)eO. 0

0

0

A C differentiable mappingf is an immersion if the rank ofJ is equal to p for every point P of Wp. It is an imbedding ifJis an injective immersion such that J is a homeomorphism of Wp onto JCWp) with the topology induced from Mil.

2

I. Riemannian Geometry

1.1. Tangent Space 1.6 Let (0, qJ) be a local chart and f a differentiable real-valued function defined on a neighborhood of P E O. We say that f is flat at P if d(f 0 qJ - 1) is zero at qJ(P). A tangent vector at P EM" is a map X:f -+ X(f) E IR defined on the set of functions differentiable in a neighborhood of P, where X satisfies: (a) (b) (c)

If A, j.l E IR, X(Af + j.lg) = AX(f) + j.lX(g). X(f) = 0, iffis flat. X(fg) = f(P)X(g) + g(P)X(f); this follows from (a) and (b).

1.7 The tangent space Tp(M) at P E Mn is the set of tangent vectors at P. It has a natural vector space structure. In a coordinate system {Xi} at P, the vectors (O;OXi)p defined by (O;OXi)p (f) = [c(fo qJ-l)/OXi] 0 if X =I

o.

Hereafter, unless otherwise stated, a Riemannian manifold Mil is a connected Coo Riemannian manifold of dimension n. 1.16 Theorem. On a paracompact Coo differentiable manifold, there exists a Coo Riemannian metric g.

Proof Let (OJ, lfJJiel be an atlas and {tXJ a COO partition of unity subordinate to the covering {OJ Such {tXJ exists since the manifold Mil is paracompact. Set 8 = (8 ik ) be the Euclidean metric on IR" (in an orthonormal basis Gjk = £5~, Kronecker's symbol). Then 9 = Lief tXjlfJr(tff) is a Riemannian • metric on Mil' as one can easily verify. For an alternate proof of Theorem (1.16) one can also use Whitney's theorem and give M /I the imbedded metric. Whitney's theorem asserts that every differentiable manifold M /I has an immersion in 1R2/1 and an imbedding

5

§2. Riemannian Manifold

in 1R 2n + 1. So let be an imbedding of Mil in 1R211 + 1. On Mil define the Riemannian metric g by g(X, Y) = 8(. X, . Y). This is the metric induced by the imbedding. Let {xil, (i = 1,2, ... , n), be a local coordinate system at a point P E Mn and {il} (Ot: = 1,2, ... , 2n + 1) the coordinates of (P) E 1R 2 n+ 1. The components of 9 can be expressed as follows: 2n+ 1

gij

= (1~1

oil. oy% ox i oxi"

By definition, gii are the components of the inverse matrix of the metric matrix «gi): gijgik

=

b,.

2.1. Metric Space 1.17 Definition. Let C be a differentiable curve in a Riemannian manifold 3 t - C(t) E M" with C differentiable (namely the restriction of a differentiable mapping of a neighborhood of [a, bJ into Mil). Define the arc length of C by:

(M", g): IR ~ [a, bJ

(2)

L(C) =

f

gC(1)

dC dC) ( dt' dt dt =

fba

where Ci(t) are the coordinates of C(t) in a local chart, and dCi/de the components of the tangent vector at C: dC/de = C.(c/ct), c/ec being the unit vector of IR. One verifies easily that the definition of L( C) makes sense; the integral depends neither on the local chart, nor on a change of parametrization s = set) with ds/dt =I 0. Henceforth we suppose that the manifold is connected. This implies that it is pathwise connected. Two points P and Q of M" are the endpoints of a differentiable curve. Indeed, a continuous curve from P to Q is covered by a finite number of open sets Q i homeomorphic to IR", and in each Q i one replaces the continuous curve by a differentiable one. Set d(P, Q) = inf L(C) for all differentiable curves from P to Q. 1.18 Theorem. d(P, Q) defines a distance on MII' and the topology determined by d is equivalent to the topology of M" as a manifold.

Proof. Clearly d(P, Q) = d(Q, P) and d(P, Q) ~ d(P, R) + d(R, Q). Since d(P, P.) = 0, the only point remaining to be proved is that d(P, Q) = => P = Q. Assume that P =I Q and let (Q, 4') be a local chart with (a neighborhood of zero, where the 11Ulpping is defined) onto a neighborhood of P. By definition expp(O) = P, and the identification of rR.n with Tp(M) is made by means of qJ*: X = (qJ;l)pX, (qJ is introduced in 1.30).

Proof exp P(X) is a Coo map of a neighborhood of 0 E rR." into Mil' This follows from 1.30 (/3 may be chosen greater than 1). At P the Jacobian matrix of this map is the unit matrix; then, according to the inverse function theorem 3.10, the exponential mapping is locally a diffeomorphism: ~1, ~2, ..• , ~" can be • expressed as functions of C 1, C 2 , ••• , C. 1.32 Corollary. There exists a neighborhood

n of P,

such that every point

Q E Q can be joined to P by a unique geodesic entirely included in Q. (n, expp 1)

is a local chart and the corresponding coordinate system is called a normal geodesic coordinate system.

10

I. Riemannian Geomelry

Proof Let {~i} be the coordinates ofa point QEQ, and C(t) = {Ci(t)} the geodesic from P to Q lying in n. Ci(t) = t~i, for 1 E [0,1]. Since the arc length s = IIXllt,

I

II

(8)

giJ{Q)~iej =

(e i )2 =

IIXI12.

i= 1

The length of the geodesic from P to Q is I X II. Since C(t) is a geodesic, by (1.28) we conclude that

Letting t - 0, we have qk(p)~jek = 0 for all {~i}. Thus qk(P) = 0; all Christoffel symbols are zero at P. •

Lp

1.33 Proposition. Every geodesic through P is perpendicular to (r), the 1 (~i)2 = r2, with r small enough (~i subset of the points Q En satisfying are geodesic coordinates of Q).

D'=

Ip

Proof Let Q E (r) c n. Choose an orthonormal frame of [R" such that the geodesic coordinates of Q are ~1 = r and ~2 = ~3 = ... = en = O. We are going to prove that 9li(Q) = bit for all i; thus the desired result will be established, because a vector in Q tangent to (r) has a zero first component; (if 7(U) is a differentiable curve in (r) through Q, 1 /(u) x (d"/(ll)/du) = 0, and that implies d/(u)/du = 0 at Q). Clearly, by (8), g 11 (Q) = 1. Differentiation of (8) with respect to ~k yields:

Lp

Lp

Ii'=

Hence, at Q, if k i: 1:

where gij(r) are the components of g at the point with coordinates ~i = 0 for i > 1. Moreover, r:~{r)~i~j = 0 for all It leads to

e= 1

r,

Thusgtk(r) + rOrg\k(r) = o,(c;ar) [rg lk(r)] = 0, and rglk(r) is constant along the geodesic from P to Q, so

•

*3. Exponential Mapping

II

1.34 Definition. C is called a minimi::ing curve from P to See 1.17 for the definition of L( C) and d(P, Q).

Q if L( C) = d(P, Q).

1.35 Proposition. A minimizing differentiable curve C from P to Qis a geodesic.

Proof Consider C parametrized by arc length s ([0, rJ 3 s -+ C(s) E C), and suppose that C(s) is of class C 2 and lies in a chart (n, b) and c). Let P E M and a geodesic C(s) through P be defined for 0 ~ s < L, where s is the canonical parameter of arc length. Consider sp, an increasing sequence converging to L, and set xp = C(sp). We have d(xp, x q) ~ Isp - sql. Hence {xp} is a Cauchy sequence in M, and it converges to a point, say Q, which does not depend on the sequence {sp}. Applying Theorem 1.31 at Q, we prove that the geodesic can be extended for all values of s such that L ~ s < L + e for some e > O.

Proof b) => d) and Theorem 1.38. Denote by Ep(r) the subset of the points QE Sp(r), such that there exists a minimizing geodesic from P to Q. Recall Sp(r) = {Q EM, d(P, Q) ~ r}. We are going to prove that E(r) = Ep(r) is compact and is the same as S(r) = Sp(r).

14

I. Riemannian Geometry

Let {QJ be a sequence of points in E(r),Xj(with IIXjl1 = 1 (recallX j = fP*X;) the corresponding tangent vectors at P to the minimizing geodesic (or one of them) from P to Qj, and Si = d(P, Qi)' Since the sphere §n-l(1) is compact and the sequence {Si} bounded, there exists a subsequence {QJ of {Q;} such that {X) converges to a unit vector X 0 E §n-l(1) and Sj - So. Assuming b), Qo = expp soXo exists. It follows that Qj - Qo and d(P, Qo) = So :$ r. Hence E(r) is compact. Indeed, expp is continuous: We have only to consider a finite covering of the geodesic, from P to Qo by open balls, where we can apply Proposition 1.29. According to Theorem 1.36, E(r) = S(r) for 0 < r < o(P). Suppose E(r) = S(r) for 0 < r < ro and let us prove first, that equality occurs for r = ro, then for r > ro. Let Q E S(ro) and {Q;} be a sequence, which converges to Q, such that d(P, Qi) < ro. Such a sequence exists because P and Q can be joined by a differentiable curve whose length is as close as one wants to roo Qi E E(ro), which is compact; hence E(ro) = S(ro). By Theorem 1.36, b(Q) is continuous. It follows that there exists a 00 > 0 such that b(Q) ~ 00 when Q E E(ro), since E(ro) is compact. Let us prove that E(ro + bo) = S(ro + bo). Pick Q E S(ro + bo), Q ¢ S(ro). For every kEN, there exists Ck , a differentiable curve from P to Q, whose length is smaller than d(P, Q) + 11k. Denote by 4 the last point on Ck , which belongs to E(ro). After possibly passing to a subsequence, since E(ro) is compact, 4 converges to a point T. Clearly, d(P, T) = ro, d(T, Q) :$ 00 :$ beT), and d(P, T)

+ d(T, Q) = d(P, Q),

since d(P, 4)

+ d(4, Q) < d(P, Q) + 11k.

There exists a minimizing geodesic from P to T and another from T to Q. The union of these two geodesics is a piecewise differentiable curve from P to Q, whose length is d(P, Q). Hence it is a minimizing geodesic from P to Q. This proves d) and Theorem 1.38, any bounded subset of M being included in S(r) for r large enough, and S(r) = E(r) being compact. • Finally, d) => a), obviously.

1.39 Definition. Cut-locus of a point P on a complete Riemannian manifold. According to Theorem 1.37, expp(r X) with IIXII = 1 is defined for all r E IR and X E §n-l(I). Moreover the exponential mapping is differentiable. Consider the following map §n-l(I)3X - ,u(X)E]O, +00], ,u(X) being the upper bound of the set of the r, such that the geodesic [0, r] 3 S - C(s) = expp sX is minimizing. It is obvious that, for 0 < r :$ ,u(X), the geodesic C(s) is minimizing. The set of the points expp[jt(X) X], when X varies over §n _ t (1), is called the cut-locus of P. It is possible to show that ,u(X) is a continuous function on §n-t(l) with value in ]0,00] (Bishop and Crittenden [53]). Thus the cut-locus is a closed

15

§5. Second Variation of the Length Integral

subset of M. So when M is complete, expp, which is defined and differentiable on the whole IR", is a diffeomorphism of

E> = {rX E IR"IO

~

r < Jl(X)}

onto Q

= expp E>.

M is the union of the two disjoint sets: Q and the cut-locus of P.

1.40 Definition. Let J1.(X) be as above and J p = inf J1.(X), X E §"-1(1). Jp is called the injectivity radius at P. Clearly Jp > 0. The injectivity radius J of a manifold M is the greatest real number such that b ~ Jp for all P E M. Clearly J may be zero. But according to Theorem 1.36, J is strictly positive if the manifold is compact.

§5. Second Variation of the Length Integral 5.1. Existence of Tubular Neighborhoods 1.41 Let C(s) be an imbedded geodesic [a, b] 3 S -...+ C(s) E M. At P = C(a), fix an orthonormal frame ofTP(M), {eJ, (i = 1,2, ... ,n)with el = (dC/ds)s=a,

being the parameter of arc length. Consider ej(s), the parallel translate vector of ej from P to C(s) (see Definition 1.27). {eb)} forms an orthonormal frame of TC(s)(M) with el(s) = dC(s)/ds, since gc(slej(s), eis» is constant along C. Consider the following map f defined on an open subset of IR":IR x IR"-l 3 (s, ~) -...+ expC(sl~' To define f, associate to ~ E IR"- 1 the vector ~ E IR", whose first component ( is zero. According to Cauchy's theorem (see Proposition 1.29), f is differentiable. Moreover, by 1.30, the differential of f at each point C(s) is the identity map of IR" if we identify the tangent space with IR"; thus r is locally invertible in a neighborhood of C, by the inverse function theorem, 3.10. For J1. > 0, define Til = {the set of the [(s,~) with s E [a, b] and II~II < J1.}. ~ is called a tubular neighborhood of C. The restriction f Il of f to [a, b] x Bil C IR" is a diffeomorphism onto Til' provided J1. is small enough. Indeed. it is sufficient to show that for Jl small enough r Il is one-to-one. Suppose the contrary: there exists a sequence {QJ of points belonging to T1 / j , such that Qi = f(sj, Xi) = f(uj, fi) with (Sj, Xi) ::1= (Ui' fi) and IIXdl ~ II fill < 1/i. After possibly passing to a subsequence Qi' whenj -...+ "XJ, Qj converges to a point of C, say C(so). Accordingly, Sj -...+ So and Uj -...+ So. This yields the desired contradiction, since r is locally invertible at C(so), as proved above.

S

5.2. Second Variation of the Length Integral

1.42 Let C be a geodesic from P to Q, [0, r] 3 S - C(s) E M being injective. Choose J1. small enough so that f Il is injective (for the definition of r Il see 1.41).

16

I. Riemannian Geometry

On ~, the tubular neighborhood of C, (s, ~) forms a coordinate system (called Fermi coordinates), which is normal at each point of C, as it is possible to show. We are going to compute the second variation of arc length in this chart (Til' r; 1). Set Xl = s and Xi = i , for i > 1. Let {C A} be a family of curves close to C, defined by the C 2 differentiable mappings: [0, r] x ] -e, +e[ 3 (s,,t) - Xi(S, ,t), the coordinates of the point Q(s, ,t) E CA' In addition, suppose that Q(s, 0) = C(s), X1(S,,t) = s, and that e > 0 is chosen small enough so that CA is included in ~ for all ,t E ] - e, + e[. The first variation of the length integral

e

L(,t) = is zero at ,t leads to

I

(10)

I:

= 0, since Co = C is a geodesic.

02L(,t») ),=0 = Jo r ["i~2 = ( all

A straightforward calculation

• (dyi) 2 .] ds - R1ilJ{C(S»Y'(s)y-'(s) ds,

where yi(S) = [OXi(S, A.)/OA.]A=O' Indeed, by 1.13, R lilj = -!Oijgll on C. Recall that on C, gij = t5{ and o"gij = o. 5.3. Myers' Theorem

1.43 A connected complete Riemannian manifold M" with Ricci curvature ~ (n - 1)k 2 > 0 is compact and its diameter is ::;; Tt/k. Proof Let P and Q be two points of M" and let C be the (or a) minimizing geodesic from P to Q, r its length. Consider the second variation I j U ~ 2) related to the family C). defined by xj(s, A.) = A. sin(Tts/r) and Xi(S, A.) = 0 for all i > 1, i :f. j. According to (10): Ij

=

i'[ Tt

2TtS

2'2 cos -

orr

. 2TtS - R 1j1J{S) sm - ] ds. r

Adding these equations and using the hypothesis Rll ~ (n - 1)k 2 , it follows that

" = i' [

LI

j=2

j

0

2 Tt 2 cos 2 -TtS - R 11 (s) sin 2 -Tts] ds::;; (n - 1) -2r (Tt (n - 1) 2' 2' - k 2 ) • r r r r

If r > Tt/k, this expression will be negative and at least one of the I j must be negative. It follows that C is not minimizing, since there exists a curve from P to Q with length smaller than r. Hence d(P, Q) ::;; Tt/k for all pair of points P and Q. By Theorem 1.37, M is compact. •

17

§6. Jacobi Field

§6. Jacobi Field 1.44 Definition. A vector field Z(s), along a geodesic C, is a Jacobi field if its components ~i(S) satisfy the equations: (11)

in a Fermi coordinate system (see 1.42). The set of the Jacobi fields along C forms a vector space of dimension 2n, because by Cauchy's Theorem, 3.11, there is a unique Jacobi field which satisfies Z(so) = Zo and Z'(so) = Yo. So E [0, rJ. when Zo and Yo belong to Tqso)(M). The subset of the Jacobi fields which vanish at a fixed So forms a vector subspace of dimension n. Those, which are in addition. orthogonal to C, form a vector subspace of dimension (n - 1). Indeed, if ~l(SO) = and (~l)'(SO) = 0, ~l(S) = for all SE[O, r], since (~l)"(S) = 0, for all S (by definition 1.44).

°

°

1.45 Definition. If there exists a non-identically-zero Jacobi field which vanishes at P and Q. two points of C, then Q is called a conjugate point to P. 1.46 Theorem. expp X is singular at X 0 jugate point to P.

if and only if Q =

expp X 0 is a con-

Proof expp X is singular at X 0 if and only if there exists a vector Y '# orthogonal to X 0 such that ( 0 expp(X 0 OA

(12)

+ ,l Y»)

=

°

o.

' o. •

§7. The Index Inequality 1.49 Proposition. Let Y and Z be two lacobifields along (C), as in 1.44. Then g(Y, Z') - g(Y', Z) is constant along (C). In particular, if Y and Z vanish at P, then g(Y, Z') = g(Y', Z). Indeed, [L7= 1 (yiz'i - y'iZi)], = O. 1.50 Definition (The Index Form). Let Z be a differentiable (or piecewise differentiable) vector field along a geodesic (C): [0, r] 3 t -+ CCt) E M. For Z orthogonal to dC/dt, the index form is

(14)

I(Z)

=

I

{g(Z'(t), Z'(t»

+ g[R(~~,

z) ~~,

Z]} dt.

19

§7. The Index Inequality

1.51 Theorem (The Index Inequality). Let P and Q be two points ofM n , and let (C) be a geodesic from P to Q: [0, r] 3 s - C(s) E M such that P admits no conjugate point along (C). Given a differentiable (or piecewise differentiable) vector field Z along (C), orthogonal to dC/dt and vanishing at P, consider the lacobifield Y along (C)such that YeO) = oand Y(r) = Z(r). ThenJ(Y) S J(Z). Equality occurs if and only if Z = Y. Proof First of all, such a Jacobi field exists. Indeed, by 1.44, the Jacobi fields V, vanishing at P and orthogonal to dC/dt, form a vector space "I/" of dimension n - 1. Since P has no conjugate point on (C), the map V'(O) - VCr) is one-to-one, from the orthogonal complement of dC/dt in Tp(M) to that of dC/dt in TQ(M). Thus this map is onto. And given Z(r), Yexists. Let {V;} (i = 2, 3, "', n) be a basis of "1/". For the same reason as above, {V;(s)} (2 SiS n) and dC/ds form a basis of Tc(s)(M). Hence there exist differentiable (or piecewise differentiable) functions };(s), such that Z(s) = =2 };(s) V;(s). Furthermore, set W(s) = I/=2 f;(s)V;(s) and el = dC/ds. Then by (11), g[R(e l , Z)e l , Z] = II=z};g[R(e\, V;)e\, ZJ = I;'=2J;g(V;', Z). Thus:

I/

J(Z) =

f

o

[g(W, W)

+ ~ g(}; V;,fj Vi) + ~ g(}; Vi,fi J1j) I,J

I,J

+ ~ gU; V;,fj Vj) + ~ g(}; Vi',fj J1j)] dS. I,)

I,)

By virtue of Proposition 1.49, g(V;, Vj) = g(V;, J1j). Thus, integrating the last term of J(Z) by parts gives

J(Z) =

5:g(W, W) ds + g[Y'(r), Y(r)],

because yes) = I/=2};(r)V;(s) and Y'(s) If}; are constant for all i, we find: (15)

= I/=2};(r)Vi(s).

J(Y) = g[Y'(r), Y(r)].

Hence J(Z) ~ J( Y) and equality occurs if and only if W lent to fi = 0 for all i, that is to say, if Y = Z.

= 0, which is equiva•

1.52 Proposition. Let b2 be an upper bound for the sectional curvature of M and 15 its injectivity radius. Then the ball Sp(r) is convex, if r satisfies r < 15/2 and r S n/4b.

20

I. Riemannian Geometry

Proof Let Q E Sp(r) with d(P, Q) = r, and (C) the minimizing geodesic from P to Q. In a tubular neighborhood of (C), we consider a Fermi coordinate system, (see 1.42). Given a geodesic r through Q orthogonal to (C) at Q, so that] - e, + e[ 3.-1. --+ '1(.-1.) E M, with '1(0) = Q, set Yo = (dy/d.-1.);.=o. The first coordinate of '1(.-1.) is equal to r, for all A.. By (10), the second variation of d(P, '1(.-1.» at .-1. = 0 is I(Y), where Y is the Jacobi field along (C) satisfying Y(P) = 0, Y(Q) = Yo. But I(Y)

~

{[g(Y', Y') - blg(Y, Y)] ds

= Ib(Y);

Ib(Y) is the index form (14) on a manifold with constant sectional curvature bl . On such a manifold, the solutions of (11) vanishing at s = 0 are of the type ei = pi sin bs, for i ~ 2, where pi are some constants. If br < n, a solution does not vanish for some s E ]0, r], without being identically zero. In that case, according to Theorem 1.51, and by (15):

Ib(Y)

~

sin bs ) Ib ( -'-b- Yo = b cot br g(Yo , Yo)· sm r

If r < n/2b, then I(Y) > 0 and for e small enough, the points of 'I, except Q, lie outside Sp{r). Henceforth suppose r < b/2 and r ~ n/4b. Consider Ql and Ql' two points of Sp(r), andy a minimizing geodesic from Ql to Q2 (see Theorem 1.38). Since d(Ql' Ql) ~ 2r < b, 'I is unique and included in Sp(2r). Let T be the (or a) point of y, whose distance to P is maximum. Since d(P, T) < 2r ~ n/2b, T is one end point of y. Indeed, if T is not Q1 or Q2' I is orthogonal at T to the geodesic from P to T and by virtue of the above result, 'I is not included in Sp(d(P, T» and that contradicts the • definition of T.

§8. Estimates on the Components of the Metric Tensor 1.53 Theorem. Let M" be a Riemannian manifold whose sectional curvature K satisfies the bounds - a 2 ~ K ~ b2 , the Ricci curvature being greater than a' = (n - 1)(%2. Let Sp(ro) be a ball of M with center P and radius ro < bp the injectivity radius at P. Consider (Sp(ro), expp 1), a normal geodesic coordinate system. Denote the coordinates of a point Q = (r, 0) E [0, ro] x §,,-1(1), locally by 0 = {Oi}, (i = 1,2, ... , n - 1). The metric tensor g can be expressed by

ds 2

= (dr) 2 + rl g8 8J(r, 0) dO i dOj . j

21

§8. Estimates on the Components of the Metric Tensor

For convenience let g99 be one of the components g8lfJi and Igl Then g88 and Igl satisfy the following inequalities:

= det«gfJifJJ».

(IX)

a/or log J g(}fJ(r, 0) ~ a/or log[sin(br)/r], g98(r, 0) ~ [sin(br)/br]2 when br < 7t;

(p)

a/or log J 9fJfJ(r, 0) :s; Ojor log[sinh(ar)/r], g(}(}(r, 0) :S [sinh(ar)/ar]2;

(y)

a/or log Jlg(r, 0)1 :S (n - l)(il/ilr)log[sin(lXr)/r] :s; -a'r/3,

(16)

sin(lXr)]n-l

Jlg(r, O)I:S [ -;;:-

( 0 does not occur, even if a. 2 > 0, since r < 0 and IA II B I = 1. Moreover, '7 does not vanish. 1.68 Definition (Adjoint operator *). Let Mn be a Riemannian oriented manifold and 1] its oriented volume element. We associate to ap-form 0, a (n-p)-form *0, called the adjoint of 0, defined as follows: In a chart (!l, lfJ) E sI, the components of *(X are (19)

We can verify that: (20)

where {3 is a p-form, and (et, {3) denotes the scalar product of CJ. and ( 0 , (3)

= .!..o p! AI A2, ... ,A

p

/3:

(3AI A2, ... ,A P •

Note that the adjoint operator is an isomorphism between the spaces lV(M) and A"-P(M).

11.2. Laplacian 1.69 Definition. (Co-differential 0, then ;'1 ~ nk/(n - 1).

Proof. Let f be an eigenfunction: ~f = A.f with A. > O. Multiplying formula (31), with (X = df, by ViJ, and integrating over Mn lead to:

As (Vi Vjf + (l/n)~fgij)(ViViJ + (l/n)llfgij) ~ 0, it follows that ~ (1/n)(~f)2, hence ,1.(1 - lin) ~ k.

Vi VjfViVij'

•

Chapter 2

Sobolev Spaces

§1. First Definitions 2.1 We are going to define Sobolev spaces of integer order on a Riemannian manifold. First we shall be concerned with density problems. Then we shall prove the Sobolev imbedding theorem and the Kondrakov theorem. After that we shall introduce the notion of best constant in the Sobolev imbedding theorem. Finally, we shall study the exceptional case of this theorem (i.e., H'i on n-dimensional manifolds). For Sobolev spaces on the open sets in n-dimensional, real Euclidean space [R", we recommend the very complete book of Adams [1]. 2.2 Definitions. Let (M", g) be a smooth Riemannian manifold of dimension n (smooth means COO). For a real function cp belonging to Ck(M,,) (k ~ 0 an integer), we define:

In particular, IVocpl = Icpl, IVlcpl2 = kth covariant derivative of cpo

IVcpl2

= V"cpVvCP. Vkcp will mean any

Let us consider the vector space (f~ of C7) functions cp, such that ~ t ~ k, where k and t are integers and p ~ 1 is a real number.

IV(cpl E LiM,,), for all t with 0

2.3 Definitions. The Sobolev space respect to the norm

IlcpllH/:

H~(M II)

is the completion of

(f~

with

k

=

L IlVlcpllp. (=0

Ht(M II) is the closure of P)(M II) in Ht(M II). P)(MII) is the space of COO functions with compact support in M" and Hg = Lp.

33

§2. Density Problems

It is possible to consider some other norms which are equivalent; for instance, we could use

H;

When p = 2, is a Hilbert space, and this norm comes from the inner product. For simplicity we will write Hi; for the Hilbert space H;.

§2. Density Problems 2.4 Theorem.

~(IR")

is dense in Ht(IR").

Proof Letf(t) be a Coo decreasing function on IR, such thatf(t) = 1 for t ~ 0 andf(t) = 0 for t ~ 1. It is sufficient to prove that a function rp E Coo(IR") n Hr(IR") can be approxi-

mated in Ht(IR") by functions of ~(IR"). We claim that the sequence of functions rp}{x) = rp(x).f(llxll - j), of ~(IR"), converges to qI(x) in H~(IR"). Let us verify this for the functions and the first derivatives, that is, in the case of H1(IR"). Whenj -+ :C, rp}{x) -+ qI(x) everywhere and IqI}{x) I ~ IqI(x)l, which belongs to L". So by the Lebesgue dominated convergence theorem IIrpj - qllI" -+ O. Moreover, whenj -+ 00, IVrp}{x)l-+ IVrp(x) I everywhere, and IVrp}{x) I ~ IVqI(x) I + IqI(x) I SUPte[O.1l I f'(t) I which belongs to L". Thus

IIV(rpj - rp)lI" -+ O.

This proves the density assertion for Leibnitz's formula.

H~(IR").

For k > 1, we have to use •

2.S Remark. The preceding theorem is no~ true for a bounded open set 0 in Euclidean space. Indeed, let us verify that Hf(O) is strictly included in Hf(O). For this purpose consider the inner product

For

y, E Coo(O) n

Hf(O) and rp (qI,

If Y, ¥:- 0 satisfies

y, , ilUn).

E ~(O),

i t aiiy,)rp

y,> = (y, -

y, = ~j= I auY"

n

1=1

dx.

then for all rp E ~(O), (rp,

.

y,> = 0, so that

Such a function y, exists on a bounded open set 0; for instance, '" = sinh Xl (X I the first coordinate of x), fn Isinh xII" dx, and fn Icosh xII" dx are finite.

34

2. Sobolev Spaces

For this reason we only try to prove the following theorem for complete Riemannian manifolds.

2.6 Theorem. For a complete Riemannian manifold HHMn) = HHMn)·

Proof It is not useful to consider a function f

E C rJJ on IR, as in the proof of the preceding theorem, because for a Riemannian manifold [d(P, Q)]2 is only a Lipschitz function in Q E M n , P being a fixed point of Mn. So let us consider the function f(t) on IR, defined by f(t) = 1 for t :::; O,f(t) = 1 - t for 0 < t < 1, and f(t) = 0 for t > 1. Let J be a finite covering of K such that E>i is homeomorphic to the open unit ball B of IRn, (E>j, !/Ii) being the corresponding chart. Let {~J be a partition of unity of K subordinate to the covering {E>J. We approximate each function ~i O.

Proof Let r be an integer and let t/I E C+ 1. Then (1)

To establish this inequality, it is sufficient to develop (Vl' VCll '" VClrY'.lsV"fit ... V{J r'l'.Is

-

VV,,1 V" ... V"",...".lsV%1 ... V~r'Y.Is)

We find 4IV,+1t/l12IV't/l1 2 -IVIV't/l1 2 12 ::;; O. Since H'l°(Mn) is imbedded in LPo(M n}, there exists a constant A, such that for all q> E Hio(M n):

Let us apply this inequality withq> = IV't/I I, assuming q> belongs to Hio: IIV't/lII PO (2)

::;;

A(IIVIV't/llll qo + I V't/IIIqo)'

::;; A(IIV'+ 1t/1llqo + I V't/I II qo)'

Now let rjJ E HZ(M n) Ii C'Xl(Mn). Applying inequalities (1) and (2) with q = q 0 and r = k - 1, k - 2, ... , we find: IIVk-lt/lIlPlc_1 ::;; A(II Vk t/lllq

+ II Vk - 1 rjJll q),

IIVk- 2 rjJII PIc _, ::;; A(IIVk-1rjJll q + IIV k - 2 t/1ll q), IIrjJIlPIc-, ::;; A(II VrjJll q + IIrjJll q);

37

§4. Sobolev's Proof

thus

Therefore a Cauchy sequence in HZ of crt) functions is a Cauchy sequence in H~"--l' , and the preceding inequality holds for all '" E HZ. Similarly, one proves the following imbeddings: HZ c Hr-=-l c H~,,-=-~ c· ..

C

H~t.

§4. Sobolev's Proof 2.12 Sobolev's lemma. Let p' > I and q' > 1 two real numbers. Define A by lip' + 1/q' + Aln = 2. If A satisfies 0 < A < n, there exists a constant K(p', q', n), such that for all f E Lq,([R") and g E L,,,.([R"):

f f II

f(x)g(y)

(3)

API An

X -

, ,

II). dx dy

~ K(p, q,

Y

n)llfllq·llgll p •

Ilxll being the Euclidean norm.

The proof of this lemma is difficult (Sobolev [255J), we assume it.

Corollary. Let A be a real number, 0 < A < n, and q' > 1. If r, defined by 11r = Aln + 1/q' - 1, satisfies r > 1, then

h(y)

=

f

A"

I

f(x) X -

II). dx belongs to L r , when f

Y

E Lq.(~n).

Moreover, there exists a constant C(A, q', n) such that for all f

Ilhll,

~ C(A, q',

Proof For all g E Lp.(IR"), with llr

:::

L;.

and

Lq.([RII)

n)llfll q••

+ lip' = 1:

therefore hE L r

E

Ilhllr

~ K(p', q',

n)11 f Ilq"

•

Now we will prove the existence of a constant C(n, q) such that all q> E ~(~II) satisfy: (4)

with lip

= l/q

- lin and 1 < q < n.

38

2. Sobolev Spaces

Since ~(IR") is dense in Hl(lR") Theorem 2.4 the first part of the Sobolev imbedding theorem will be proved, according to Proposition 2.11. Let x and y be points in IR", and write r = Ilx - YII. Let 0 E §,,-l (1), the sphere of dimension n - 1 and radius 1. Introduce spherical polar coordinates (r, 0), with origin at x. Obviously, because ep E ~(IR"):

( ) =epx

0) dr = - foo II X-Y 111-" aep(r, 0) ,-"-1 dr fooo aep(r, ar 0 ar

and

Integrating over §,,-1(1), we obtain: 1

lep(x)I:sW,,-l

i

1Ji"

IV'ep(y) I II x _ YII"-ldy,

where W,,-l is the volume of §,,-1(l). According to Corollary 2.12 with A. = n - 1, inequality (4) holds.

•

§5. Proof by Gagliardo and Nirenberg (1958) 2.13 Gagliardo [118J and Nirenberg [220J proved that for all ep (5)

E ~(IR"):

2"lJ I axiaep III

Ilepll,,/(,,-l) :s 1 II

ill 1 •

It is easy to see that the Sobolev imbedding theorem follows from this inequality. First lacp/axil:S lV'epl; therefore Ilepll,,!(II-1):S tllV'eplll' Then setting Icpl = UP(,,-l) n and applying H6lder's inequality, we obtain:

where l/q + l/q' = 1 and p' = p(n - I)/n - 1. But p'q' = p since l/p l/q - l/n; hence:

=

39

§6. New Proof

We now prove inequality (5). For simplicity we treat only the case n = 3; but the proof for n ". 3 is similar. Let P be a point of 1R3 , (x, y, z) the coordinates in 1R 3 , (xo, Yo, zo) those of P, and Dx (respectively, Dy , Dz) the straight line through P parallel to the x-axis, (respectively, y-, z-axis). Since cP E ~(IR"), cp(P) =

i

f

ocp -0 (x, Yo, zo) dx = -00 x xo

+ (P)1 ~ t fox IOxcpldx. Likewise for Dy and Dz:

G)'''[L.IO.", Idx L.IO,. ",Idy L.IO,

(x) = f(llxl!), f being a positive Lipschitzian function decreasing on [0, x J and equal to zero at infinity. Third step, Proposition 2.18: the proof of inequality (6) for these functions.

2.15 Proposition (Milnor [200J p. 37. This is actually due to Morse). Let M be a Riemannian manifold. Any bounded smooth function f: M - IR can be uniformly approximated by a smoothfunction g which has no degenerate critical points. Furthermore, g can be chosen so that the ith derivatives ~fg on the compact set K uniformly approximate the corresponding derivatives of f for is k. We recall that a point P E M is called a critical point of f if IVf(P) I = 0, the real number f(P) is a critical value of f. A critical point P is called nondegenerate if and only if the matrix Vi Vjf(P) is nonsingular. Nondegenerate critical points are isolated.

°

2.16 Proposition. Let f ¥:. be a coo function on M" with compact support K.

f can be approximated in H1(M") by a sequence of continuous functions fp with compact support Kp c K (the boundary of Kp being a sub-manifold of dimension n - 1); moreover /PE Coo(Kp) and has only nondegenerate critical points on K p.

41

§6. New Proof

Since they are isolated, the number of critical points of fp on K p is finite. Proof According to Proposition 2.15 there is a sequence of Crt) functions gp which have no degenerate critical points and which satisfy If - gpl < lip on M,. and 1V(f - gp)1 < lip on K. Choose a real number (Xp satisfying lip < (Xp < 21p, such that neither (Xp nor -(Xp is a critical value of gpo Then g; l«(Xp) and g; l( -(Xp) are sub-manifolds of dimension n - 1, unless they are empty. LetAp = {x E M,.lgp(x) ~ (Xp} andA_ p = {x E MlIlgp(x) ~ -(Xp}. Define fp by: fix) = [gp(x) - (Xp]X.lp(X)

+ [gp(x) + (XP]xl_ p(X),

where x£ is the characteristic function of the set E. The support Kp = Ap U A_p of fp is included in K because for x E K p' Igix) I > lip; thus I f(x) I > O. fp E Crt)(K p) and fp is Lipschitzian, hence f p EHl(M,.). Since If(x) - fp(x) I ~ (3Ip)xK(x), Ilf - fpllq-+O when p-+x. Moreover, at a point x where f(x) i= 0, we have IV[f(x) - fp(x)] I -+ 0, because x E U;"= 1 Kp. But the set of the points where, simultaneously, f(x) = 0 and IVf(x)1 i= 0, has zero measure; consequently IV[f(x) - fix)]l-+ almost everywhere. Therefore IIV(f - fp)lIq -+ 0, according to Lebesgue's theorem, • since IV(f - .f~)1 ~ (suplVfl + l/p)Xl('

°

°

2.17 Proposition. Let f ~ be a continuous function on r (r denoting the sphere S,., the euclidean or hyperbolic space), which is Crt) on its compact support K, whose boundary (if it is nonempty) is a submanifold of dimension n - 1, and assume f has only nondegenerate critical points. Pick P a point ofr, and define g(r), a decreasing function on [0, xL by Jl{Qlg[d(P, Q)]

~

a} = Jl{Qlf(Q)

~

a} = I{!(a).

Then

Proof Let d(JJ, Q) be the distance between P and Q on r, and let a > 0 be a real number; Jl denote the measure defined by the metric, and write g( Q) = g[d(P, Q)]. Let Qi(i = 1, ... , k) be the critical points of fin K. Consider the set rd = f-l(a) and note that, ifQ E rd is not one ofthe points Qj, then IVf(Q) I i= O. Ifda(Q) denotes the area element on r d , then we may write

42

2. Sobolev Spaces

J!a Ita

Furthermore, when a is not a critical value of J, cp(a) = l'Vf 1- 1 da exists. cp(a) is continuous and locally admits - tjJ(a) as primitive. l'Vf Iq - 1 da has a We consider cp(a) = - tjJ'(a) as given. Therefore minimum, in the case q > 1, when l'Vf I is constant on r a , according to Holder's inequality:

But ra is the boundary of a set, whose measure tjJ(a) is given. Hence hada is greater than or equal to the area of the boundary of the ball of volume tjJ(a) (by A. Dinghas [110]). This completes the proof. • Furthermore, one verifies that g(r) is absolutely continuous and even Lipschitzian on [0, 00[. 2.18 Proposition. Let g(r) be a decreasing function absolutely continuous on

[0,

.X; [,

and equal to zero at infinity. Then:

where K(n, q) is from Theorem 2.14.

Proof Let us consider the following variational problem, when q > 1: Maximize /(g) = Jo Ig(r)IPrn - 1 dr, when )(g) = Jo Ig'(r)lqrn - 1 dr is a

given positive constant. The Euler equation is (8)

( I9 ' Iq - 1rn -

- k9 P - 1r n - 1,

1)' _

where k is a constant. It is obvious that we have only to consider decreasing functions. One verifies that the functions y = (1. + ~/(q-l)l-nlq are solutions of (8), A. > 0 being a real number:

According to Bliss, Lemma 2.19, the corresponding value of the integral I(y) is an absolute maximum.

43

§6. New Proof

The value of K(n, q), the best constant, is

Letting q -+ 1, we establish the inequality (7) for q = 1: K(n, 1) = lim K(n, q). 4"'1

Let us compute K(n, q). f CXl 1y'1 4r"-1 dr

Jo

= (n - q)4 fCXl[A. + rII/(4-1)r"r"+1/(4- 1) dr. q- 1

Jo

Setting A. = 1 and r = t(4-1)/4, we obtain:

Furthermore, B/A

= (n -

q)/n(q - 1), because

= f IX>(1 + t)-lt"- 14 dt =

A

o

---=- fIX> n q 1 n-l q

n

q- 1

n-l

q

0

= - - - - (A

(1

+ t)1-lIt"-l-1I/4dt

+ B).

Hence: K(n, q)

= (W II _1)-l f ll ( ~1 B )1

/4 -1 /11

q

(B)

_ q- 1 1/4(q - 1 )K(n, q) - - - - - BWII-l n- q A q

A -1/4 ~1 ( -q- )1/4,

n-q q-l

lilt

•

_

,wIth B -

r(n/q)r(n - n/q) r( )

n

.

•

2.19 Lemma. Let h(x) ~ 0 a measurable, real-valued function defined on IR, such that J = SO' h4(x) dx is finite and given. Set g(x) = So h(t) dt. Then I = SO' gP(x)x Pdx attains its maximum value for the functions h(x) = (A.XIl + 1)-(11+ 1)/ 11, with p and q two constants satisfying p > q > 1, rt = (P/q) - 1 and A. > 0 a real number. ll

-

44

2. Sobolev Spaces

This is proved in Bliss [55]. The change of variable x = r(q-II}!(q-I) now yields the result used in the Proposition 2.18, above. Recall that here lip = (l/q) - (lIn) and so we have (X = pin, (ax/ar)l- q = ,"-1 and Xl +2- P = r".

§7. Sobolev Imbedding Theorem for Riemannian Manifolds 2.20 Theorem. For compact manifolds the Sobolev imbedding theorem holds. Moreover HI does not depend on the Riemannian metric.

Proof We are going to give the usual proof of the first part of the theorem, because it is easy for compact manifolds. But for a more precise result and a more complete proof see Theorem 2.21. Let {nj} be a finite covering of M, (i = 1, 2, ... , N), and (OJ, cPj) the corresponding charts. Consider {(XJ a C 0,

where K(n, q) is the smallest constant having this property. According to Proposition 2.11 and Theorem 2.6, to prove the first part of the Sobolev imbedding theorem, it is sufficient to establish inequality (10) for the functions of ~(Mn). The proof will be given at the end of 2.27 using Lemmas 2.24 and 2.25. First we will establish the second part of the Sobolev imbedding theorem.

2.22 Lemma. Let Mn be a complete Riemannian manifold with injectivity radius 0 and sectional curvature K, satisfying the bound K :S b2. There exists a constant C(q) such that for all cP E ~(Mn): (11)

suplcpl

:s

C(q)llcpIIH"1 if q > n.

Proof Let f(t) be a Coo decreasing function on IR, which is equal to 1 in a neighbourhood of zero, and to zero for t ~ b ( 0. If the curvature is constant or if the dimension is two and the curvature bounded, then AiO) exists and every 1'1' E H1(Mn) satisfies

For IRn and D-!l n the hyperbolic space, the inequality holds with AiO) = 0. For the proof, see Aubin [13J pp. 595 and 597. 2.29 Theorem. There exists a constant A(q) such that every satisfies:

1'1' E

H1(§n)

Let Mn(n ~ 3) be a Riemannian manifold, with constant curvature and injectivity radius 15 0 > 0. There exists a constant A, such that every 1'1' E Hi(M n) satisfies:

For the sphere of volume 1, the inequality holds with A = 1. See Aubin [13] pp. 588 and 598. For the proof of the last part of the theorem see Aubin [14] p. 293.

§9. Soholev's Theorem for Compact Riemannian Manifolds with Boundary 2.30 Theorem. For the compact manifolds Wn with C -boundary, (r ~ 1), the Sobolev imbedding theorem holds. More precisely: First part. The imbedding H~(W) c Hr(W) is continuous with lip = l/q (k - t)ln > 0. Moreover, for any e > 0, there exists a constant Aie) such

§9. Sobolev's Theorem for Compact Riemannian Manifolds with Boundary

51

that every qJ E H1(w,.) satisfies inequality (10) and such that every qJ E HHw,.) satisfies: o

Second part. Thefollowing imbeddings are continuous: (a) H%(W) c C~W), if k - n/q > s ~ 0, s being an integer, (b) HZ O. Proof. Recall that E is the half-space of IRn and D = B n E, where B is the open ball with center 0 and radius 1. Consider ~ defined, for x E E, by ~(x) = I/I(x) and ~(x) = I/I(x), when x = (-x .. X2'"'' XII)' (Xl> X2'"'' XII) being the coordinates of x. ~ is a Lipschitzian function with compact support, thus ~ E H1(IR") and according to Theorem 2.l4:

The lemma follows, since

2.32 The proof that every qJ E RHw,.) satisfies inequality (10) is similar to that in 2.27. But here it is easier because the covering is finite. Using Lemma 2.31, we can prove that all qJ E H1(w,.) satisfy (15); for a complete proof see Cherrier [97].

Proof of the second part of Theorem 2.30. a) There exist constants Cj(i[) such that for allf E Hi(W) n COO(W) (16)

52

2. Sobolev Spaces

Set hex) = 0 for x ¢ D, and hex) = (ad) 0 it is a Cauchy sequence in Lp:

by HOlder's inequality, with J.l = [(rip) - 1]/(r - 1). b) Proof of the second part of Theorem 2.33. Let A. satisfy (X < A. < inf(l, k - nlq). Then by the Sobolev imbedding theorem 2.30, HnO) is included in Cl(O), and there exists a constant A such that IIflle;. S;; AllfllHr. Let .91 be a bounded subset of H%(n); iff E.9I, IIfllHI S;; C, a constant, and IIflle;. S;; AC. Thus we can apply Ascoli's Theorem, 3.15 . .91 is a bounded subset of equicon~ tinuous functions of CO(O), and 0 is compact. So d is precompact in CO(O).

55

*11. Kondrakov's Theorem for Riemannian Manifolds

Then, since

If(x) - f(y)lIlx - yll-Z = (If(x) - f(y)lIlx - yll-A),Z/Alf(x) - f(y)11-z/A if a sequencef", E sI converges tofin

CoCO), l!flle"" :$

AC and

Thus sI is precompact in CZ(O). c) q) (Q) is included in q) (IR"), so we can apply the theorem of Sobolev, o 2.10, to the space H%(Q). A proof similar to those of a) and b) gives the desired result.

§11. Kondrakov's Theorem for Riemannian Manifolds 2.34 Theorem. The Kondrakovtheorem,2.33,holdsfor the compact Riemannian manifolds Mit, and the compact Riemannian manifolds w:a with CI-boundary. Namely, the following imbeddings are compact:

H%(M,.) c LiM,.) and H%(W,.) c Lp(w:a), with 1 ~ lip> l/q kin> O. (b) HZ(M,.) c CZ(M,.) and HZ(w:a) c CZ(W,.), if k - r:x. > nlq, with 0:$ r:x. < 1. (a)

Proof Let (Qj, 0, but the Kondrakov theorem does not. Consider the sequence offunctionsh defined on IRn (n > 4) by: 1)(n-2'/4(1 )1-nI2 h(x) = ( k k + IIxl12 . Let us verify fie E Hi{lRn). Now

is finite and so is IIVhll~ =

[00 (n - 2)2 wn_Ikl-nI2 10

= wn-l(n -

2)2

(1k + r2 )-n,n+l dr

LOO(l + t 2)-n t "+l dt = A.

Also,h belongs to L N , with N = 2n/(n - 2), because

Lf~,n-l dr oo

= =

(~r/2

L(~ + r2) oo

L +( oo

(1

2 )-lIrn- 1

-n ,n-l

dr

dt = C
t

i'

If(t)-f(r)l:s r1f'ldt:S

(fOOr 1f'lqdt ) I/q(t-t)l-I/q.

.

Since we can choose r so that J~ I f' Iq dt is as small as one wants, t l q- If(t) -+ 0 as t -+ 00. Hence Plf(t)lq/(q-l)t- 1 -+ 0 as t -+ ,X) and the integral in (29) ~~.

2.49 Proposition. Let g be as in Proposition 2.48. Then there are constants C and A. such that (30)

the inf of A. such that C exists is equal to A.q = «q - l)/n)q-I q-q. Proof It is easy to verify that all real numbers u satisfy

thus according to Proposition 2.48,

• Corollary 2.49. Let

n be a bounded open set of IR" and set p" =

n l - 2"w;_11' Then all cp E H~(n) satisfy

(n - 1)"-1

(31)

where C depends only on n. Proof After symmetrization we use Proposition 2.49 with q = n and we get p" = ~w;~ I' This result may also be obtained from (28) by using the inequality: uv:S il"lul"/("-l) + ,ulllvl" with v = IIVcpll" and u = cpIIVcpll;l. •

§17. The Case of the Riemannian Manifolds 2.50 Return to Theorem 2.46. Set &", the sup of

such that 1I] is integrable and Ji", the inf of p, such that C and \' exp[il( Icp III cp I Hl t exist in inequality (24). Two questions arise. Does Ji" depend on the manifold? l (" -

(1,

68

2. Sobolev Spaces

Is fi,. attained? (I.e., is J-L = fi,. allowed.) The answers were first found in Cherrier [95]. In Cherrier [96] there are similar results when qJ E H'k'" or ll"qJ E L,./u. He proved the following: Theorem 2.50. For Riemannian manifolds M,. with bounded curvature and global injectivity radius (in particular this is true if M is compact), the best constants il,. and~,. in Theorem (2.46) depend only on n. They are equal to

For compact Riemannian manifolds with C 1 boundary the best constants are equal, respectively, to 2J-L,. and 2- 1/(,.-1)CX". J-L,. is attained for the sphere §,. and J-L2 is attained for compact Riemannian manifolds of dimension 2. 2.51 The case of the sphere. We have seen that we have the best possible o inequality (31) for qJ E Hi(n) when n is a bounded open set of [Rn: J.l = J-L" and v = 0 in (24). This is also the case for the sphere §". The following was proved by Moser [209] when n = 2, and by Aubin [21], p. 156. Theorem 2.51. All qJ E Hi{§,,) with integral equal zero (Js.. qJ dV = 0) satisfy (32)

i e'" dV =:;; C exp(J.l" II VqJ II:), Js "

where C depends only on nand J-L,. = (n J-L2 = 1/16n.

1),,-1 n 1-2n(l);_1 1 ;

in particular

2.52 As in other inequalities concerning Sobolev-spaces, the best constants can be lowered when the functions qJ also satisfy some natural orthogonality conditions. Theorems similar to those in 2.40 and 2.42 are proved in Aubin [21], p. 157. The sequence ofhest constants is {J-LJm}me"". For the sphere §" the following is proved. Theorem 2.52. Let A be the eigenspace corresponding to the first nonzero eigenvalue. The functions cp E H~(§,,) satisfving Js" ~e'l' dV = 0 for all ~ E A and Js"qJ dV = 0, satisfy the inequality (33)

f.

s.

e'" dV =:;; C(J-L) exp(J-LIIVqJll:),

where it is possible to choose J-L > J-LJ2 as close to J-LJ2 as one wants. C{J.t) is a constant which depends on J-L and n.

69

§18. Problems of Traces

2.53 The case of the real projective space IPII • Theorem. For any e > 0 there is a constant C(e) which depends only on n such that alil/! E H~(lPn) with integral zero (Jp"l/!dV = 0) satisfy

i . e til dV S; C(e) exp[(Jl,. + e)IIVt/lU:J. Jp

(34)

Proof. p: S,. -+ /P,., the universal covering of /P,. has two sheets. We associate to t/I E Hi(/P n) the function rp on S,. defined by cp(Q) = t/I(p(Q» for Q E SII' The function rp so obtained satisfies the hypotheses of Theorem 2.52. Js"rp dV = 2 Jp"t/I dV = 0 and e" is orthogonal to A. In~eed, if Q and Qare antipodaUy symmetric on S,., ~(Q)e 0 there is C(e) which depends only on n, such that all t/I E Hi (/P,.) satisfy

2

i

p..

etildV

=

f.s.. elPdV

S;

C(8)ex P[(Jl,./2

= C(e) exp [ (JL" + 28) Thus we get (34) with C(e)

= C(e/2)/2.

+ e)

f.

Sn

IVCPlndV]

t. 1 1Vt/I I" dV

•

§18. Problems of Traces 2.54 Let M be a Riemannian manifold and let V c M be a Riemannian su b-manifold. Iff is a C k function on M, we can consider Jthe restriction of f to V,J E Ck( V). Now iff E Hf(M), it is often possible to define the trace Joff on V by a density argument and there are imbedding theorems similar to those of Sobolev. Adams [1] discusses the case of Euclidean space. In Cherrier [97] the problem of traces is studied for Riemannian manifolds; he also considers the exceptional case. The same problems arise for a Riemannian manifold W with boundary oW. We can try to define the trace on oW of a function belonging to Hf(W). The results are useful for problems with prescribed boundary conditions.

Chapter 3

Background Material

§1. Differential Calculus 3.1 Definition. A normed space is a vector space ~over C or R), which is provided with a norm. A norm, denoted by I II, is a real-valued functional on

fj, which satisfies:

(a) fj 3 X -+ IIxll ~ 0, with equality if and only if x (b) II Ax II = IAlllxll for every x E fj and AE C, (c) I/x + yll ~ Ilxll + Ilyll for every x, y E fj.

= 0,

A Banach space ~ is a complete normed space: every Cauchy sequence in ~ converges to a limit in ~. A Hilbert space f> is a Banach space where the norm comes from an inner product:

f>2 3 (x, y) ~

-+

(x, y) E C, so

is an inner product provided that

~

IIxll2

= (x, x).

is linear in x, that it satisfies (x, y)

(y, x), and that (x, x) = 0 if and only if x

= o.

=

3.2 Definition. Let fj and 6> be two normed spaces. We denote by 9'(~, 6» the space of the continuous linear mappings u from fj to 6>. 9'(fj, 6» has the

natural structure of a normed space. Its norm is lIuli = supllu(x)1I

for all x E fj with Ilxll ::; 1.

~*, the dual space of~, is 9'(~, C) or g'(~, R), according to whether fjis a vector space on C or IR. ~ is said to be reflexive if the natural imbedding ~ 3 X -+ X E fj**, defined by x(u) = u(x) for u E F*, is surjective.

3.3 Proposition. A linear mapping u from fj to 6> (where fj and 6> are two normed spaces) is continuous if and only if there exists a real number M such

that Ilu(x)1I ~ Mllxll for all x E~. If~ is a Banach space, then 9'(~,

~)

is a Banach space.

71

§l. Differential Calculus

a,

3.4 Defiaidon. ICn is an open subset of (ij and 0, there exists a number ~ > 0 and a compact set ReO, such that for every fEd: c

L_ltlf(X)I'dX < e

and

11

f(x

+ y) -

f(x) I' dx
1, and f E Lp(Q), p > 1. Suppose that for all functions v E C nt(n) n H~(a),

f

uA(v) dV =

Then u E H~nt(a) n

o

H~(a)

f

fv dV.

and

where C is a constant depending only on a, A, n, and p. Moreover, if p > n/(m + 1) then u E C,"-I(O) and u is a solution of the Dirichlet problem A*u

=f

in n, V'u

=0

on

aa, 0 s I

;5; m -

1

in the strong Lp sense.

3.58 Theorem (Gilbarg and Trudinger [125] p. 177). Let a be a bounded open set of IR" with Ck+ 2 boundary (k ;;::: 0) and A a linear elliptic operator of order two, such that a2 E Ck+ l(a) and at> ao E C"(a). Suppose u E HI(n) is a weak solution of A(u) = f, with f E H,,(n). Then u E H,,+ in) and (19)

where the constant C is independent of u and f. Thus, if the coefficients and f belong to Coo(a) and if the boundary is Coo, then UE Coo(a).

3.59 Theorem (Gilbarg and Triidinger [125] p. 106). Let n be a bounded open set oflR" with Ck+ 2. '" boundary and let A be a linear elliptic operator oforder two, with coefficients belonging to C",2(n) (k ;;::: 0 an integer and 0 < ex < 1). Suppose u E CO(a) n C 2(n) is a solution of the Dirichlet problem A(u) = fin n, u = von an, with f E Ck. "'(a) and v E ck+ 2. 2(a). Then u E Ck+ 2. 2(a). Now let us prove a result which will be used in Chapter 8.

Proposition 3.59. Let 0 be a bounded open set oflR" with Coo boundary and let A(u) = F(x, u, Vu, V 2u) be a differential operator of order two, defined on n, F being a Coo differentiable function of its arguments on a. Suppose that A is uniformly elliptic on!l at Uo E C 2 ,2(!l), with 0 < ex < 1. Ifuoieo E Coo(aO) and if A(uo) E c 0 and J§,,_dl)W(X) da = 0, then 11K, * f - Ko * flip - 0 when e - 0 and (25) holds. In addition, if WE C 1(1R" - {O}), K, * f - Ko * f a.e. (see Dunford and Schwartz [111 ]). 7.5. Korn-Lichtenstein Theorem

3.66 Theorem. If w(x) is a function with the properties described in 3.65 and Ko is defined as above, there eXists,for any IX(O < IX < 1), a constant A(IX) such that IIKo * fllc .. ::;; A(IX)llfllc", for allf E C'(IR") with compact support.

91

§7. Inequalities

3.67 Theorem. Let Mn be a compact Riemannian manifold and p, q, and r real numbers satisfying l/p = l/q - l/n, 1 ~ q < nand r > n. Define .PI = {cp E LtiJ cp dV = O}. Then there exists a constant k, such that, for all (X ~ 1, any function cp E .91 with IVlcplO!I E Lq, satisfies (26)

If cp E .91 with IVcpl E L" then suplcpl llcp E Lq (in the distributional sense), then

If cp E Ll and llcp

E

~

Const x IIVcpllr' If cp E Ll and

L r , then IVcpl is bounded and suplVcpl

~

Const x II11CPlir.

The constants do not depend on cp, of course. Let Mn be a compact Riemannian manifolds with boundary. Then the theorem holds for functions cp E ~(M). Proof. First of all, we are going to establish (26) for (X = 1. Let G(P, Q) be the Green's function of the Laplacian. As Jcp dV = 0, in the distributional sense (Proposition 4.14): cp(P)

(27)

=

f

G(P, Q)llcp(Q) dV(Q),

whence: (28)

and according to Proposition 3.64, we find Ilcpllq

~

IlVlPllq sup PeM"

f,v QG(P, Q)I dV(Q)

for all cp E .91, such that IVlPl E L q • Using the Sobolev imbedding theorem 2.21, we obtain (29)

with ko

=K +

A

sUPPeM

JIVQG(P, Q)I dV(Q).

92

3. Background Material

Let us now prove (26) for oc > 1. Since the set of the COO functions which have no degenerate critical points is dense in the spaces H~ (Proposition 2.16), we need only establish (26) for these functions. Let cp ::f: 0 be such a function, with Jcp dV = O. Set if> = sup( cp, 0) and

0 be such that the measure of Q"

= {xeMJlcp(x)1

~

a} is equal toe:/l(Q,,)

= e.

We have ae ~ !!CP!!I' Since 1q>11
n.

= I(tx),

with 0 < t < oc. Applying (44) to qJ

Choosing t = (II I IIvIIVI II; I)'"'~+ I -",r)-l, we find: (45)

with d- 1 = 1 + v(I/n - l/r). If q > n, we can choose \' = q, and the result follows for p Ifq ~ n,since 1I/IIv ~ II II I:Iv(supl/l>l-q,., (45) gives: (sup II 1>1-(I-d)O-qIV) which is the result for p a-I

~

= + oc, (d = a).

2c(v)IIV111~11/11~I-d)q/v,

= + oc, with

= q/v d + 1 -

q/v

= 1 + q(l/n -

l/r).

«5) The case j = 1, m = 2. We have established (Theorem 3.69) inequality (38) for a = j/m = 1/2. If r < n, inequality (38) for a = 1 is just the Sobolev imbedding theorem (Corollary 2.12, Theorem 3.67). By interpolation (22), we find the inequality for! < a < 1. If r ~ n, according to (38) with j = 0, m = 1, applied to the function IVI I:

(46) with l/p = 1/5 + b(I/r - I/n - 1/5) > 0 and 0 ~ b ~ 1. Using (35) in (46) yields the desired inequality. Indeed, IIV/II; ~ Const x IIV 2/11rll/llq with 2/5 = I/r + I/q. Thus we find inequality (38) wherej = 1, m = 2 and a = (I + b)/2.

96

3. Background Material

We can verify that a S al = [1 + (lIn - l/r)/(l/n b S ao = (1 + sin - sir). Thus (38) holds.

+ llq)r 1

implies •

§8. Maximum Principle 8.1. Hopf's Maximum Principle 2

3.71 Let a be an open connected set of II(" and L(u) a linear uniformly elliptic differential operator in a of" order 2: L(u)

02U i 0 j x x

= L ai;{x) 0 i,j

+

OU L" b.{x) ~ + h(x)u

i= 1

OX

with bounded coefficients and h S O. Suppose u E C2(0) satisfies L(u) ~ O. If u attains its maximum M ~ 0 in 0, then u is constant equal to M on n. Otherwise if at Xo E 00, u is continuous and u(xo) = M ~ 0, then the outer normal derivative at xo, if it exists, satisfies ou/ov(xo) > 0, provided Xo belongs to the boundary of a ball included in O. Moreover, if h == 0, the same conclusions hold for a maximum M < O.

Remark 3.71. We can state a maximum principle for weak solution (see Gilbarg and TrUdinger [125] p. 168). Let Lu = oi(aij Oju) + bi 0iU + hu be an elliptic operator in divergence form defined on an open set of IR", where the coefficients aij, bi and hare assumed to be measurable and locally bounded. u E H1(0) is said to satisfy Lu ~ 0 weakly if for all qJ E 9'(0), qJ ~ 0:

a

In this case, if h S 0 then supn u S sUPcnmax(u, 0). The last term is defined in the following way: we say that v E H1(0) satisfies vlaO s kif max(v - k,O) E Hl(O). 8.2. Uniqueness Theorem

3.72 Let W be a compact Riemannian manifold with boundary and L(u) a linear uniformly elliptic differential operator on W:

with bounded coefficients and h S O. l

Protter and Weinberger [239].

97

§8. Maximum Principle

Then, the Dirichlet problem L(u) = f, uloW = 9 (f and 9 given) has at most one solution. Proof Suppose it and u are solutions ofthe Dirichlet problem. Then 0 = it - U satisfies LD = 0 in Wand DloW = O. According to the maximum principle D ~ 0 on W. But the same result holds for - D. Thus D = 0 in W. •

3.73 Theorem. Let W and L(u) be as above. If WE C2 (W) (\ CO(W) is a subsolution of the above Dirichlet problem, i.e. w satisfies:

Lw

~

f in W, w/aw

~

g,

then w :$; u everywhere, if u is the solution oj the Dirichlet problem. Likewise, if D is a supersolution, i.e. D satisfies Lu :$; J in Wand o/aw ~ g then u :$; D everywhere.

8.3. Maximum Principle for Nonlinear Elliptic Operator of Order Two 3.74 Let W be a Riemannian compact manifold with boundary, and A(u) = f(x, u, V'u, V' 2u) a differential operator of order two defined over W, where f is supposed to be a differentiable function of its arguments. Suppose D, WE C 2 (W) satisfy A(u) = 0 and A(w) ~ O. Define Dr by [0 + t(w - D)].

Theorem 3.74. Let A(u) be uniformly elliptic with respect to Or ,for all t E JO, 1[. Then cP = ro - cannot achieve a nonnegative maximum M ~ 0 in W, unless it is a constant, if af(x. ur • Vu r • V 2 ur )/cu :$; 0 on W. Moreover suppose 0, WE CO(W) and w ~ on the boundary, then ro :$; everywhere provided the derivatives oj f(x, Or' V'D r • V 20r) are bounded (in the local charts of a finite atlas)for all t E JO, 1[. If in addition at Xo E oW, cp(xo) = 0 and ocp/ov(xo) exists, then ocp/av(xo) > 0, unless cp is a constant, provided the boundary is C 2 .

°

°

°

Proof Consider yet) = f(x, or}. For some B E JO, 1[ the mean value theorem shows that 0 ~ A(ro) - A(D) = y(1) - yeO) = y'(B) with '(B) = af(x, D8 ) y av··u .)

.

v.. + af(x, Do) v. + Of(x, 0 8 )

av.u'Cp

,Jcp

Thus cp = ro - D satisfies L( cp) present statements.

~

ou

cp

= L(

cp

)

O. Applying the above theorems yields the •

3.7S As an application of the maximum principle we are going to establish the following lemma, which will be useful to solve Yam abe's problem.

98

3. Background Material

Proposition 3.75. Let Mn be a compact Riemannian manifold. If a function t/I ~ 0, belonging to C 2 (M), satisfies an inequality of the type ~t/I ~ t/lf(P, t/I), where f(P, t) is a continuous numerical function on M x IR, then either t/I is strictly positive, or t/I is identically zero. Proof. According to Kazdan. Since M is compact and since t/I is a fixed nonnegative continuous function, there is a constant a > 0 such that ~t/I + at/l ~ o. By the maximum principle 3.71, the result follows: u = -t/I cannot • have a local maximum ~O unless u == O. Here L = -~ - a. 8.4. Generalized Maximum Principle 3.76 There is a generalized maximum principle on complete noncom pact manifolds Cheng and Yau (90). Namely: Theorem. Let (M, g) be a complete Riemannian manifold. Suppose thatfor any x E M there is a C 2 non-negativefunction 0

1/1 °

in W}. Obviously ({Jo - Pl/lo ~ 0 in W. But we have more: there is a point PEW where the function 0 in W. According to the maximum principle 3.71, (olav) ( 0 such that ({Jo - (P + 6)1/10 > 0 in W. Hence our initial supposition is false and P does exist. Applying the maximum principle yields ({Jo - Pl/lo = 0 everywhere, since (5) implies that ({Jo - PI/!o cannot achieve its minimum in W, unless it is constant. • 4.5 Remark. If l'Y.t is only of class Cle, the preceding proof is valid, except for the regularity on the boundary. When k > n12, we can prove that CPo E C(W) with r < k - n12. The proof is similar to the previous one, except that now the atlas fn i, I/!Jie~ is of class Cle. The functions I; satisfy elliptic equations on 15 with coefficients belonging to CIe - 2(D); then by Theorem 3.5S,I; E HIe(.D). Applying Theorem 2.30, we have

I;EC(D). 4.6 Coronary. Let W be a compact Riemannian manifold with boundary of class C", k ~ 1, or at least Lipschitzian. There exists Al > 0, such that Ilcplii :S All IIVcpll~ for all ({J E A1(W). Thus IIVcpll2 is an equivalent norm for R 1(W).

Proof. Since the Sobolev imbedding theorem 2.30 and the Kondrakov theorem 2.34 hold. the proof of Theorem (4.4) is valid, except for the regularity at the boundary. Thus J1. is attained in d and consequently J1. = ..1.1 is strictly positive. •

1.2. Existence Theorem for the Equation ~cp =

f

4.7 Theorem. Let (Mn, g) be a compact CIXl Riemannian manifold. There exists a weak solution cpE HI 0/(1) ifand only ifJf(x)dV = O. The solution cp is unique up to a constant. Iff E C+2, (r ~ 0 an integer or r = + Xl, 0 < ~ < I).

then cp E cr+2+2.

Proof. IX) if cp is a weak solution of (1) in H10 by Definition 3.53, Jaii VicpVil/! dV = I/If dV for all 1/1 E H1· Choosing 1/1 = 1, we find Sf dV = O. This con-

J

dition is necessary.

§1. Linear Elliptic Equations

105

P) Uniqueness up to a constant. Let (/)1 and (/)Z be weak solutions of (1) in HI' Set fp = (/)Z - (/)1' For all '" e HI' Jaij Vi",Vifp dV = O. Choosing", = fp

J

leads to aij VifpVifp dV = O. Thus fp = Constant.

y) Existence of (/). Iff == 0, the solutions of (1) are (/) = Constant. Henceforth suppose f ~ O. Let us consider the functional J«(/» = SaiiVi(/)Vi(/) dV. Define Jl = infl«(/» for all cpe:1l, with:1l = {cpeHl satisfying f cpdV= 0 and (/)f dV = 1}. Jl is a nonnegative real number, 0 ~ Jl :::;; IUllflll Z), Let {cpJieN be a minimizing sequence in :11: J(lfJi) -+ Jl. Since aiJ{x) are the components of a Riemannian metric, there exists (X > 0 such that I«(/)J ~ (XIIVCPill~. Thus the set {I V bo and a"iix)~i~j ~ A.lel 2 for some bo > 0 and A. > 0 independent of k. By the first part of the proof, E" has a COO solution qJ". And these solutions (k = 1,2, ... ) are uniformly bounded. Indeed, considering the maximum and then the minimum of cp", we get

Now by the Schauder interior estimates 3.61, the sequence {cp,,} is bounded in C 2 • 2 • To apply the estimates we consider a finite atlas {OJ, I/Ii} and compact sets K j C OJ such that M = Ui K i • As {cp,,} is bounded in C2 .:I, by Ascoli's theorem 3.15, there exist qJ E C 2 and a subsequence {cpj} of {qJ,,} such that cpj _ cP in C2 • Thus cP E C 2 .:I and satisfies (23). Lastly, according to Theorem 3.55, the solution cp belongs to'C+2+1% and is unique (uniqueness does not use the smoothness of the coefficien ts).

Remark. For die proof of Theorem 4.18, we can also minimize over HI the functional

We considered a similar functional in the proof of Theorem 4.8.

§3. Riemannian Geometry

115

§3. Riemannian Geometry 3.1. The First Eigenvalue

4.19 Let A\ be the first non-zero eigenvalue of the Laplacian on a compact Riemannian smooth manifold (Mn • g) of dimension n ~ 2. Lichnerowicz's Theorem 4.19 [185]. If the Ricci curvature of the compact manifold (Mn, g) satisfies Ricci ~ a > 0, then A\ ~ nn~\.

Proof We start with the equality (24) valid for any f E C 3(M). Multiplying (24) by "if and integrating leads (after integrating by parts twice) to

Choosing as f an eigenfunction of the Laplacian Ll = Llf = At!, we obtain at once

AT

J

f 2 dV

~

a!

l"fl 2 dV

=aAI

"i"

i

related to A\ :

J

Thus Al ~ a, but we have beUer, because for any

f 2 dV.

f

E C2:

(26) This inequality is obtained expanding 1 ) ( ( "i"jf + -:;;,Llfgij

,,',,3

. , f + -:;;,Llfg'3 1 "") ~ O.

When f satisfies Llf = At!, (25) and (26) imply the inequality A\ ~ nn~1 of Theorem 4.19. After this basic result, a lot of positive bounds from below and from above for A\ have been obtained. 4.20 For a Kahler manifold the Laplacian Ll is one half of the real Laplacian Ll. In Chapter 7 we will write the complex Laplacian without the bar, but in this section we must have another symbol than that for the real Laplacian.

A = 1,2, ... , m, where m is the complex dimension (n

=2m).

116

4. Complementary Material

For a compact Kahler manifold the first non-zero eigenvalue of the Laplacian is equal to >1} /2 (in Chapter 7, we write the first non-zero eigenvalue of the complex Laplacian without bar).

5.)

Theorem 4.20 (Aubin [20] p. 81). If the Ricci curvature of the compact Kahler manifold (M2m , g) satisfies Ricci ~ a > 0, then 5.) ~ a. Proof The complex version of (24) is (27) since 'Vii 'V IL 'V vf = 'V IL 'Vv'V vf. Multiplying (27) by 'Vp. f and integrating yield

J

(15.f) 2 dV -

J

'Vv'Vp. f'V v'V p.f dV =

J

Rp.v 'VIL f'Vv f dV.

Thus, for any f E C 2, J(15.f)2 dV ~ J Rp.v 'VIL f'V v f dV. The inequality of Theorem 4.20 follows. This inequality will be the key for solving the problem of Einstein-Kahler metrics when C)(M) > 0 (see 7.26).

Corollary 4.20. Thefirst non-zero eigenvalue 5.) of the Laplacian on a compact Einstein-Kahler manifold satisfies 5.) ~ Rim, where R is the scalar curvature of (M2m , g), that is one half the real scalar curvature R:

R- -- 9 p.VR p.V -- 9 ji.vR{J.v -- RI2 . We verify that 5.) = Rim for the complex projective space Pm(C). But there are other Kahler manifolds having this property. There is no complex version of Obata' s theorem [*260] for the sphere. S2 x S2 or more generally Pm(C) x Pm(C) have this property: 5.) = Rim (see Aubin [20]).

4.21 The preceding results concern the case of positive Ricci curvature. Without this assumption we have the Theorem 4.21 (Berard. Besson and Gallot [*36]). Let (Mn , g) be a compact Riemannian manifold satisfying Ricci ~ (n-l)€a 2 D- 2, where D is the diameter -1,0 or 1. Then A) ~ nD- 2 a2 (n, €, a). and €

=

For the value of a(n, €, a). see Theorem 1.10.

§3. Riemannian Geometry

117

3.2. Locally Conformally Flat Manifolds 4.22 Definition. The Riemannian manifold (Mn , g) is locally conformally flat if any point P E M has a neighbourhood where there exists a conformal metric 9 (9 = ef 9 for some function f) which is flat. When (Mn' g) is locally conformally fiat, there exists an atlas (n i , 3. Set 9 = e f g. A computation gives (30)

=

-!

with Tij \1 i \1j l \1d\1jl+~ \1 k l\1klgij; thus, we have \1 kSij - \1jSki 0, in particular when n = 3. Indeed, if 9 = £, ~jkl == 0 and

==

2~jkl \1j 1= \1 kl\1 i \1z/ - \1z/\1i \1 kl + \1j I (gil \1 k \1 j l - gik \11 \1 j/)· Thus \1kTij = \1jTi k. Since Sij == 0, we have \1kSij - \1jSik particular when n = 3.

== 0, in

118

4. Complementary Material

We verify that

V'jWijkl

= n-:;3(V'kSi/ -

Thus

V'ISik).

== 0 implies

Wijkl

V' kSil - V'ISik == 0 when n > 3. The condition is also sufficient. Assume there exists a I-form ponents Wi satisfying in a local chart {xk}:

W

with com-

(31) with Aij

k 11k = rijWk + 2WiWj - 4"w wkgij

1=

=

- Sij.

=

Since Sij Sji and r i rj~' OiWj OjWi. Thus, locally, there exists a function f such that W = df. According to (30) and (31), for the corresponding metric g, Sij = O. Th~s implies, R = (n - I)Sijg i j =0 and then "Hij =O. So 9 is fiat since Wijkl = 0 (by assumption when n > 3, in any case when n = 3). The local integrability conditions of system (31) are OkAij

OAij

OAkj

UWI

UWI

+ ~Akl = OiAkj + ~Ail'

A computation shows that they are equivalent to the conditions Wi) klWj

= 'hSil -

which are satisfied by hypothesis (when n V'kSil - V'ISik

== 0).

V'ISik

> 3,

we saw that

W ijkl

== 0 implies

4.25 Proposition (Hebey). Let (Mn, g) be a locally conformally jiat manifold (n ~ 3) and let P be a point of M. Then there exists in a neighbourhood of P a metric 9 conformal to g, which isjiat and invariant by any isometry 0' of(Mn , g) such that O'(P) = P. Proof Let us go back to the proof of Theorem 4.24. If we fix df(P) = 0 and f(P) = 0, the solution of (31) with W = df is unique. Now WOO' satisfies (31) and the conditions at P. Thus f 0 0' f.

=

4.26 Examples. The Riemannian manifolds of constant sectional curvature are locally conformally fiat. The Riemannian product of two manifolds (M), g)) and (M2, g2) is locally conformally fiat if one of them is of constant sectional curvature k and the other of dimension I, or of constant sectional curvature - k.

We also have the Theorem 4.26 (Gil-Medrano [*142]). The connected sum of two locally conformally jiat manifolds admits conformally flat structure.

§3. Riemannian Geometry

119

3.3. The Green Function of the Laplacian

4.27 Gromov [135] found a new kind of isoperimetric inequalities, which concern the compact Riemannian manifolds (Mn , g) of positive Ricci curvature. By an homothety, we can suppose that the Ricci curvature is greater than or equal to n - 1 which is the Ricci curvature of the sphere (Sn, go) of radius 1 (endowed with the standard metric). Let n c M be an open set which has a boundary an. Gromov considers a ball B C Sn such that Vol B/Vol Sn = Vol n/Vol M.

(32)

The Gromov inequality is Vol(an)/ Vol M ~ Vol(aB)/ Vol Sn.

(33)

With such inequality, we can for instance obtain an estimate of the constants in the Sobolev imbedding theorem, or a positive bound from below for the first non-zero eigenvalue .AI of the Laplacian, see Berard-Gallot [*37], Berard-Meyer [*38] and Gallot [* 133]. However these results concerned only compact manifolds with positive Ricci curvature. This extra hypothesis has been removed.

4.28 Let (Mn, g) be a compact Riemannian manifold. Berard, Besson and Gallot defined the isoperimetric function h({3) of M as follows:

h«(3) = inf [Vol(an)/ Vol M]

(34)

n

n/

for all c M such that Vol Vol M = (3 with (3 E]O, 1[ of course. Changing in M \ proves that h( 1 - (3) = h({3). The properties of h«(3) are studied in Gallot [* 133] (regularity, underadditivity). We denote by Is«(3) the isoperimetric function of (Sn, go) of radius 1. Let D be an upper bound for the diameter of (M,g) and let r be the inf of the Ricci curvature of (M, g).

n

n

Theorem 4.28 (Berard, Besson and Gallot [*36], see also Gallot [* 133]). Assume (35)

r D2 ~ c(n - 1)0:2 with c E { -1,0, +1}

Then,for any (3 E]O, 1[, (36)

with a(n, 0, 0:)

Dh«(3)

~ a(n, c, 0:) Is«(3),

= (1 + nwn/Wn-I) lin -

1,

and 0: E R+.

120

4. Complementary Material a(n, +1, a)

=a [wn/wn-d lin (21OoI\cos t)n-Idt) -lin (in this case a

~

71')

and a( n, -1, a) = ac( a) where c( a) is the unique positive solution x of the equation x J;(cht + xsht)n-I dt = Wn/Wn-I. This solution c(a) satisfies c(a) ~ a) = inf(k, kIln) with

ben,

I dt _ fOoJ0 (sin tr(1)12

k -

1l"

dt

Jo (ch 2t) n-

_

-

( (n-I)Oo

(n - l)wn/wn-I e

In dimension 2, we can choose a(2, +1, a) a(2, -1,a) = a/sh(a/2).

) - 1 .

=a / sin(a/2), a(2, 0, a) = 2 and

4.29 Let G(x, y) be the Green function of the Laplacian on (M, g) satisfying

!

G(x, y)dV(y) = O.

In this section, we want to find a lower bound of G(x, y) in terms of n, r, V and D, that is, resp., the dimension, the inf of the Ricci curvature, the volume and the diameter of the compact manifold (Mn , g). In [*31] Bando and Mabushi gave such a lower bound (37)

,en,

n

where a) is a positive constant depending only on and a ~ 0 a constant such that rD2 ~ l)a 2. With the result of Theorem 4.28, independently Gallot found an explicit lower bound for G(x, y). His proof is unpublished, we give it below.

-en -

Proposition 4.29 (Gallot). For any x, y, G(x,y)

(38)

~

-V-Ill ,B(l-,8)h- 2(,8)d,B,

where V is the volume of (M, g), h is defined by (34). Proof Note that the integral at the right side converges since h(,8) C,BI-I/n when ,B --+ 0 and h(1 - ,8) G(1 - ,8)1-lln when ,B --+ 1. Fix x E M and set fey) G(x, y). Let us define the function a: IR --+ IR by ('oJ

('oJ

=

a(J-L) = V-I Vol{y/f(y)

> J-L}

and the function J of [0,1] in IR by J(,8) = inf{J-L/a(J-L) < ,B}. Since f is harmonic on M - {x}, Vol{y/f(y) = J-L} = 0 and J-L --+ a(J-L) is continuous. As J.L --+ a(J.L) decreases, is the inverse function of a.

J

§3. Riemannian Geometry

121

According to Gallot [*133] (Lemma 5.7, p.60), (i) for any regular value J-L of I, j 0 a(J-L) = J-L and

r

Va'(J-L) = VI j' [a(J-L)] = -

(ii)

JU=JL}

1\711- 1 dO", {I = J-L}.

where dO" is the (n - I)-measure on the manifold For any continuous function u: JR -+ JR,

J

u0

We have

r

J{f=JL}

1\7 II dO" = =

I

dV = V

r

J{f>JL}

11

j(f3) df3.

U 0

L1ldV

r (8 JU>JL}

-V- I )dV=1-a(J-L).

x

Moreover, using (i) and the Cauchy-Schwarz inequality, we have

(VolU = J-L})2 ~

(39)

r

J{f=JL}

1\7 /1- 1 dO"

r

J{f=JL}

1\7 II dO"

=- V [I - a(J-L)] a' (J-L). Thus, by the very definition of h,

Vh 2[a(J-L)] ~ -[1 - a(J-L)]a'(J-L).

We can rewrite this inequality in the form

Integrating yields j(f3)

(40)

~ j(1) + V-I

hi

(1 - s)h- 2(s)ds.

Using (ii) with u(x) = x gives V-I

JI 11 dV =

j(f3)df3

~ j(1) + V-I

J

Since I dV = 0 and j(1) = inf I(y) integrating by parts the last integral.

4.30 Let

H be a C l

positive function on h*(f3)

11 hi (1 -

s)h- 2(s) ds df3.

= infy G(x, y),

we get (38) after

[0,1/2]. We define the function

=f3 1- I / n H(f3)

h* by

122

4. Complementary Material

for (3 E [0, 1/2] and h*«(3) = h*(1 - (3)

for (3 E [1/2,1]. . rl/ 2 Let us consIder the f unction S «(3) = Jj3 A: [0, L] ~ [0,1/2] where L = S(O). Definition 4.30. M* = [- L, L] x parameter family of metrics

where

gS,,-1

Sn-I

is the canonical metric of

ds

hOes)

and its inverse function

is the manifold endowed with the one-

Sn-I (1).

We identify all the points of {+ L} X Sn-I to a pole noted Xo (resp. all the points of { - L} X Sn-I to a pole noted Xl) of the Riemannian manife>ld (M*, gt). B(xo,r) being the geodesic ball of (M*,gt) centered at Xo of radius r, by

construction, Vol [8B(xo, r») I Vol M* = h*[VoIB(xo,r)/VoIM*],

(41)

where the volumes are related to the metric gt. We denote by C;o C*(xo,') the Green function of the Laplacian on (M* ,gt) with pole xo, and V* = Vol(M*, gt).

=

4.31 Proposition (Gallot). For any compact Riemannian manifold (M, g) whose isoperimetric function h satisfies h ~ h* on [0,1],

(42) X,

C(X, y)

2

(V* /V)C*(xo, XI)

Y being two points of M.

Proof IV'C;ol is constant on each hypersurface {C;o = I-"}, so that the CauchySchwarz inequality used in (39) is an equality for C;o' Thus, according to (41), the same proof as that of Proposition 4.29 leads to (40) with equality.

C;o«(3) = C*(Xo, XI) + (V*)-I

(43)

where V*

=Vol(M* ,gt) and

(44)

infC*(xo,Y) = G*(xo,xd

y

h\1 -

= _(V*)-I iot

s)[h*(s)r 2ds

s(1- s)[h*(x)r2ds.

(38) together with (44) imply (42). If the manifold (Mn , g) has its Ricci curvature bounded from below by I)K 2 , according to Theorem 4.28,

-en -

§3. Riemannian Geometry

123

h({3) ~ ,(KD,n) [inf({3, 1 -

(3)r- Iln ,

where D is the diameter of (M, g) and , an universal function. I-lin Set then h*({3) = ,(K D, n) [inf({3, 1 - (3)] . For a suitable choice of t, (M*, gt) is Bn(R)#Bn(R) the union of two euclidean balls of radius R = R(K D, n) glued on their boundaries by the identity. We obtain the Corollary 4.31. Assume Ricci(Mn,g) ~ -en - I)K 2, then

C(x, y) ~ [2wn-J/nV] RnCBn(R)#Bn(R) (xo, Xl), where R = R(KD, n) and where Xo and Xl are the centers of the two balls. 4.32 Theorem (Gallot). Assume Ricci(Mn , g) ~ -en - I)K 2, then (45)

with R = R(n, K, D) = K-lb-l(n, KD), GSn(R) being the Green junction of the sphere Sn(R) with Xo and XI their two poles. ben, K D) comes from Theorem 4.28. Proof If we choose h*«(3) = Kb(n, K D)Is ({3), for a suitable choice of t, (M*, gt) is a canonical sphere with radius R K- I b- l (n, K D). Moreover according to (13), h({3) ~ h*({3). Then (42) implies (45).

=

3.4. Some Theorems 4.33 The Sard Theorem [*279] (see also Sternberg [*294]). Let Mn and Mp be two C k differentiable manifolds of dimension nand p. If f is a map of class k of Minto £1, then the set of the critical values of f has measure zero provided that k - 1 ~ max(n - p, 0).

c

P E M is a critical point of I if the rank of I at P is not p. All others points of M are called regular. Q E £1 is a critical value of I, if 1-1 (Q) contains at least one critical point. All other points of £1 are called regular values. Since our manifolds have countable bases, a subset A c £1 has measure zero if for every local chart (0, 1/J) of £1, 1/J(A 0) c RP has measure zero.

n

4.34 The Nash imbedding Theorem [*252]. Any Riemannian C k manifold of dimension n, (3 ::; k ::; 00) has a C k isometric imbedding in (RP, £) when p = (n + 1)(3n + II)n /2, in fact in any small portion of this space. If the manifold is compact, the result holds with p = (3n + II)n/2. Previously Nash [251] had solved the C l isometric imbedding problem. If in the sequence of successive approximations, we keep under control only the

124

4. Complementary Material

first derivatives, Nash does not need more dimensions than Whitney (see 1.16). So for k = 1, the theorem holds with p =2n + 1 and with p =2n in the compact case. 4.35 The Cheeger Theorem [*86]. Let (Mn, g) be a Riemannian manifold, and let d, V and H be three given real numbers, d and V positive. There exists a positive constant Cn(H, d, V) such that if the diameter d(M) < d, the volume v(M) > V and the sectional curvature K of M is greater than H, then every closed geodesic on M has length greater than Cn(H, d, V). Thus we have a positive lower bound for the injectivity radius. Proof Let P be a point of the simply connected space MH of constant curvature H, and v a non-zero vector of Rn. We define the angle 0 < < IT /2, by

e,

e

Volexpp[ad,o(v)] = V/2 where ad,o(v) denotes the set of vectors in R n of length ::; d making an angle of or more with both v and - v. Then we define r by

e

Volexpp [Br(O) - ar,o(v)] = V/2. Since

e < 7r /2,

there exists a constant Cn(H, d, V)

>

0 such that, if a,

r are geodesics in M H, a(O) = r(O), (a l (0), r' (0») ::; 0, then the distance dMH(a(r),r(t») < r for 0 < t ::; Cn(H,d, V). Suppose now there exists on (Mn , g) a closed geodesic, of length l < Cn(H, d, V), and let us prove then

that v(M) ::; V, which is a contradiction. By the Rauch comparison Theorem (see 1.53), since K v [ exp,,(O) ad,O (,' (0»)]

~

H,

::; V /2

and

v { exp,,(O) [Br(O) - ar,o (,' (0») ]} ::; V /2. These inequalities imply v(M) ::; V since

Me exp,,(O) {ad,o(,I(O») U[Br(O) - ar,o(,I(O»)]}. Indeed, let a be a geodesic with a(O) = ,(0) and (a l (0), " (0») ::; 0; then dM(a(r), ,(l)) < r since l < Cn(H, d, V). But ,(l) = ,(0), thus a is not minimal between a(O) ,(0) and a(r).

=

From this result, Cheeger proved his finiteness Theorem (see [*86]), which asserts that there are only finitely many diffeomorphism classes of compact ndimensional manifolds admitting a metric for which an expression involving d(M), v(M) and S(M) a bound for the sectional curvature IKI (or for the norm of the covariant derivative of the curvature tensor) is bounded.

125

§4. Partial Differential Equations

4.36 The Gromov compactness Theorem [*147] asserts that the space m(S, V, D) of compact Riemannian n-manifolds of sectional curvature IKI S S, v(M) 2:: V > 0 and d(M) S D, is precompact in the el,o topology. The following theorem has the same purpose. Theorem 4.36 (Anderson [*3]). The space m(A, io, D) of compact RiemanA, d(M) ~ D and injectivity radius nian n-manifolds such that IRicci I 2:: io > 0, is compact in the el,o topology. More precisely, given any sequence (Mi' gi) E m(A, io, D), there are diffeomorphisms Ii of Mi such that a subsequence of (Mi,ftgi) converges, in the Cl,o topology, to a Cl,o Riemannian

s

manifold (M, g).

§4. Partial Differential Equations 4.1 Elliptic Equations 4.37 Let E and F be two smooth vector bundles over a Coo manifold M. We consider the vector spaces of the Coo sections of E and F: eoo(E) and Coo(F). Let ({lj, 0 such that n

aij(x)eie i

i=l

for x Efland q

~ I-" l:(e i )2 i=l

> n such that

(51)

are bounded by I-" for all 1 ::; i ::; n.

Theorem 4.40 (Ladyzenskaja-Ural'ceva [*206]). On any open bounded subset e c fl such that the distance dee, 8fl) ~ 8 for some 8 > 0, a weak solution in HI (fl) of (49) is bounded and belongs to Gel on e for some Q > 0, if we suppose conditions (50) and (51) satisfied. Moreover IluIlCO(6) ::; M a constant which depends only on n, v, 1-", q, 8 and Ilullv(.(]). Furthermore Q and k an upper bound for lIullc"'(6) depend only on n, v, 1-", q, 8 and M. We have a uniform estimate of max l\lul on depending on the same quantities ifin addition 118kaijllq, 118k bi ll q, Ilall q, IIfllq and 118kfillq are bounded by I-" for all i, j, k. According to the first part of the theorem, we have then a uniform estimate of Ilullcl,13(6) for some (3 > 0, this estimate and (3 depending on the same quantities.

e

Indeed, differentiating (49) with respect to xk, of the following form: n

n

V

= 8 ku satisfies an equation

n

l : l : 8i (aij 8j v) = F(x) + l:8i Fi (x). i=l j=l i=l

4.41 Let A be an elliptic linear differential operator of order two on set of lRn : (52)

Au =

n an open

L aij(x)8ij u + L b (x)8i u + c(x)u n

n

i,j=l

i=1

i

such that ai/X) satisfy (53)

for x E n and any e E lRn of norm 1(lel be bounded, IIb(x)11 2 + Ic(x)1 ~ k for x E

= 1), bi(x) and c(x)

n.

are supposed to

Theorem 4.41 Harnack inequality (see Krylov [*204] and Safonov). Let u E Hr(fl) be a non-negative function (u ~ 0) which satisfies Au = f in Br C fl. Then, for 0 < Q < 1, (54)

sup u(x)::;

xEB",r

~ ( inf u(x) + rllfIILn(Br») 1xEB",r Q

4. Complementary Material

128

where C depends on n, at a given point Xo E D.

AI A and kr 21 A.

Bp is the ball of radius p with center

From (54) we can deduce uniform estimates on osc u in Bp(p < r) and on the Holder continuity of u (see Moser [*244]). In [*244], Moser gave a proof of (54) in case u ~ 0 satisfies n

(55)

Au

==

L 8 [aij(x)8j u] =0

on

i

il.

i,j=l

His conclusion is: in any compact set KeD, max u(x) xEK

~

c min u(:t), xEK

where c depends on K, D, A and A only. The proof of (54), as that of (55), is given in two parts corresponding to the following two propositions. 4.42 Proposition. Let u E Hf(D) satisfy Au B 2r C il and p > 0,

~

I,

with

I

E

Ln(il). Then for

where C depends on A, A, n, k and p. u+ = sup(u, 0) and 1- = sup( -

1,0).

For the proof we use the following Alexandrov-Bakelman-Pucci inequality.

n

Theorem 4.42. Let u E CO(n) Hf,loc(D) satisfy Au ~ I, where A is given we assume c(x) ~ 0 in il, Ibl/o by (52) and (53) holds. Setting det(aij») = and 110 belonging to Ln(D). Then

(57)

on,

sup u(x) ~ sup u+(x) + CII/- IOIIL n({1),

xED

xE8D

where C depends on n, diam il and

II IblO- l II Ln({1) only.

4.43 Proposition. Let u ~ 0 satisfy Au ~ that

where C depends on

I

in Q2. Then there exists p

>., A, nand k only.

Qh denotes the cubes

Ixil < hl2(i = 1,2, ... , n).

>0

so

§4. Partial Differential Equations

129

For Moser [*244], who studied equation (55), the two propositions are: (i) If u is a positive subsolution of (55) in Q4, then for P > 1

maxu(x)~CI(~1)2( f

(ii)

i Q2

P-

XEQI

uPdX)I/P.

If u is a positive supersolution of (55) in Q4, then

(1

Q3

uP dx

)

I/P

~

C

2 )2 min u(x)

(Po - P

XEQI

for 0 < P < Po = n/(n - 2), where C 1 and C 2 denote constants which depend on n, ). and A only. 4.2. Parabolic Equations

4.44 The heat operator L. On a compact Riemannian manifold (Mn' g), we consider the operator L=f1+8/8t

on C 2-functions u:

M x [O,oo[--IR.

K(P,Q,t)

= (2J7f)-nc n / 2exp[-p2(p,Q)/4t]

is a parametrix for L with p smooth, pep, Q) = d(P, Q) when d(P, Q) and pcP, Q) = 0 when d(P, Q) > 8 the injectivity radius. We define N 1(P, Q, t) = -LpK(P, Q, t) and Nk(P,Q,t) =

< 8/2

lot dT 1M Nk_I(P,R,t-T)N (R,Q,T)dV(R). 1

The fundamental solution of the heat operator L is (58)

H(P, Q, t) = K(P, Q, t)

+

t dT 1Mr K(P,R,t-T)fNk(R,Q,T)dV(R)

~

k=1

(see Milgram-Rosenbloom [*235], Pogorzelski [*265]). H(P, Q, t) is Coo except for P = Q, t = 0; it is positive and symmetric in P, Q. In the sense of functions, it satisfies LpH(P, Q, t) O. Any function u(P, t) on M x [O,oo[ which is C 2 in P and C I in t satisfies for t > to

=

(59)

u(P,t)=

t ito +

dT

f

iM

H(P,Q,t-T)Lu(Q,T)dV(Q)

1M H(P,Q,t - to)u(Q,to)dV(Q).

4. Complementary Material

130

The spectral decomposition of H(P, Q, t) is

l: exp( -Ait) n+2, u Q and 8i u Q are continuous. Using the regularity properties for a single equation (65), 0: fixed, by induction we show that u is smooth for t :> O. Let

t/J

=~

i)u

Q

)2.

Q=I

t/J satisfies (66)

Lt/J

=-

n

L Q=l

\7 i u Q\7 i UQ +

n

L uQ(a3iaiUf3 + b3 u(3 ). Q=I

Then, for an appropriate constant G, we have that the right side of (66) is smaller than Gt/J: Lt/J ::; Ct/J. Since t/J(P,O) == 0, the maximum principle 4.46 shows that t/J O. Thus u == O. Let us prove now the existence. Denote by fIf (M x [0, to]) the subspace of the functions of Hf(M x [0, to]) which vanish for t = O. According to Theorem 4.45, u --+ Lu defines an isomorphism of fIf (M x [0, to]) onto Lp (M x [0, to]) Let Ku = {a3i8iu.B + b3u.B}. The map K: fIf --+ Lp is compact, since the inclusion fIr c Lp is compact.

=

4. Complementary Material

132

By the theory of Fredholm mappings, the map fIf --t Lp given by u --t (Lu - K u) has finite dimensional kernel and cokemel. Moreover its index is zero, since the index is invariant under compact perturbations. Since we saw that its kernel is zero, this map is an isomorphism.

4.48 Definition. A strictly parabolic equation is an equation of the type

81Jtt

at

(67)

= AttPt,

where t --t IJtt belongs to C t ([0, oo[ COO (E) ) and [0, oo[.3t --t At is a smooth family of strongly elliptic operator ofC OO(E) into COO (E), see Definition 4.39.

4.49 We now prove local existence of solutions for the non linear parabolic equation of Eells and Sampson (see 10.16). Let u = {u Q} be k unknown functions on M smooth functions on M (a = 1,2, ... , k).

Theorem 4.49. There exists e solving the equation {

(68)

>

°and u

E

x [0, T], and

Hf(M x [O,e]) with p > n + 2

LuQ - F!f_'/u(x, t»)gii(x)8iUf3ai U'Y = uQ(x,O) = fQ(x),

fQ be k given

a = 1,2, ... , k.

°

Moreover, u is unique and smooth on (M x [0, e]). r!f'Y are smooth functions on IRk, u(x, t) being the point ofJRk whose coordinates are uQ(x, t).

°

°

Proof (Hamilton [* 149]). We will find u as a sum uQ(x, t) = fQ(x) + vQ(x, t) and write (45) as P(J + v) = with vex, t) = when t =0. The linearized equation Auh of (45) at u E H2 has the form (Auh)Q = LhQ - a~i(u)8ihf3 - b~(u)hf3, hex, 0) = 0,

(69)

with at(u) and bfi(u) continuous since p

° n ° =

>

n

fIf( M

+ 2. So

V --t

P(J

+ v) defines

x [0, T]) into Lp (M x [0, T]). Its is A/:fIf(M x [O,T]) --t Lp(M x [O,T]) which is an

a continuously differentiable map of

derivative at v isomorphism according to Theorem 4.47. Therefore by the inverse function theorem the set of all P(J + v) for v in a of E fIf (M x [0, T]) covers a neighbourhood of P(J) in neighbourhood

°

e

Lp(M x [O,T]).

°

If c > is small enough, the function equal to for t ::; c and equal P(J) for c < t ::; T belongs to e. Thus there exists W E fIf(M x [0, T]) which satisfies P(J +w)

on M x [O,e].

=°

§4. Partial Differential Equations

133

4.50 Corollary. Let (Mn, g) and (Mm , g) be Coo compact Riemannian manifolds and 10 a smooth map M ~ M. Then there exists c > and a map I: Mx[O,c] 3 (x, t) - !t(x) E M belonging to Hf(M x [0, e], M) satisfying the parabolic equation

°

with 10 as initial value. Moreover I is unique and smooth on M x [0, c]. {Xi} (1 ~ i ~ n} denote local coordinates of x in a neighbourhood of a point P E M and Y'\1 ~ A ~ m) local coordinates of y in a neighbourhood e of I(P) E M. The parabolic equation is written in these systems of coordinates, are the Christoffel symbols in e.

f;v

Proof (Hamilton [* 149]). Hamilton embeds M in ]Rk, k large enough. He considers a tubular neighbourhood T of M in ]Rk and extends the metric 9 on M smoothly to a metric on T. There is an involution i: T - T corresponding to multiplication by -1 in the fibres, i(Q) = Q for Q E M. We can choose the extension 9 of 9 to T so that i is an isometry of (T, g). Finally we extend 9 smoothly to all of ]Rk. Now we apply Theorem 4.49 with the Christoffel symbols of (]Rk, g) and u(x,O) = 10. We have 10(M) c M. If u(x, t) does not always remain in M, we can suppose e small enough so that u(x, t) E T for any x E M and t E [0, c] . Since i is an isometry, i 0 u would be another solution of (68). which is in contradiction with the uniqueness of the solution. For more details see [* 149].

rs-y

4.51 Theorem. Let E be a bundle of tensors over a smooth compact Riemannian manifold (M, g). We seek a smooth family [0, T[ - Ut of smooth tensor fields on M (Ut E COO (E) ) which satisfies the equation (71)

"o, 0)] (U() ~ R(Lo), then (>"0, 0) is a simple bifurcation point for f. More precisely, let Z be any closed complementary subspace of Uo in BI, (BI = Z ® Ker L o), then there is a 8 > 0 and a C l - curve ] - 8,8[3S ---t (>..(s),"(0) = 0, ..(s), s( un + "0,0) such that any zero of I either lies on this curve or is of the form (>",0). This Theorem was used by Vazquez-Veron [*312] to solve the problem of prescribing the scalar curvature in the negative case (see 6.12). 4.57 The method of Moving Planes. This method uses the maximum principle in an essential way. To understand how the method works, let us give the original proof of the

Theorem 5.57 (Gidas-Ni-Nirenberg [140)). Let il c lRn by a bounded open set symmetric about xl = 0, convex in the Xl direction and with smooth boundary ail. Suppose U E C 2 (ti) is a positive solution of Llu = I(x, u) in il satisfying u = 0 on ail. Assume I and aul are continuous on ti, and I is symmetric in Xl with I decreasing in Xl for Xl > O. Then u is symmetric in Xl and al u < 0 for X l > O. Proof Set >"0 = maxxE.II xl and let Xo E ail with xA = >"0. Since u > 0 in il and u(xo) = 0, al u(xo) ~ O. First we prove that for x E il close to xo, l u(x) < O. If 81u(xo) < 0, this is obvious by continuity. If 8 1u(xo) 0, the proof is by contradiction. Assume there is a sequence { x j} c il converging to Xo such that a1u(Xj) ~ O. Consequently 8 11 u(xo) = 0 and hence Llu(xo) 0 (since u 0 on 8il). Thus we must have l(xo,O) = O. In that case u > 0 in il satisfies an equation of the type Llu + h(x)u ~ 0 for some function hex). By the version 1.43 of the maximum principle al u(xo) < 0, thus the contradiction. Now we start with the method of moving planes.

=

=

a

=

137

§S. The Methods

=

We denote by T>., the plane Xl A. For A < AO, A close to AO, we consider the cap E(A) {x E njA < Xl < AO}, the set of the points in n between T>., and T>.,o' For any x in n, we use x>., to denote its reftexion in the plane T>.,. When A ~ 0, x>., is defined on E(A) since n is convex in the Xl direction and symmetric about xl =O. At the beginning, when A decreases from AO, since u(x) is strictly decreasing for x close to Xo, w>.,(x) = u(x>.,) - u(x) > 0 in E(A). For x E aE(A) with Xl > A, w>.,(x) > 0 and, for x E T>., naE(A), w>.,(x) = 0 and alw>.,(x) > O. Decrease A until a critical value I-' is reached, beyond which this result no longer holds: at a point yETI-' n n, l w(y) = 0 (we drop the subscript I-' in wl-'). But w satisfies in E(I-'), when I-' ~ 0

=

a

.1w

=I (XI-"

U(XI-'») - I(x, u(x») ~ I(x, u(XI-'») - I(x, u(x»).

We can write this inequality in the form .1w ~ h(x)w.

(74)

Moreover w satisfies w ~ 0 in E(I-'). Thus, according to Proposition 4.61, w == 0 in E(I-') since w(y) =0 and al w(y) =O. The result follows and I-' =O. We must have I-' ~ 0, since otherwise we start with the reftexions from A = Al = infxEii Xl and we increase A. 4.58 Corollary (Gidas-Ni-Nirenberg [140]). In the ball u E C 2(ii) be a positive solution in of

n

(75)

.1u = I(u) with u

=0

n: Ixl
0 in

n.

Proof We can start at once with the method of moving planes. Since I is Lipschitz, W>., satisfies (74) in a narrow band E(A) when A is close to AQ.

4. Complementary Material

138

Moreover w>. ~ 0 on oE()"), thus w>. > 0 in E()") (according to Proposition 4.60) and, on T>. n n where w>. = 0, we must have OIW>. > 0, otherwise the function vanishes.

4.59 The method of moving planes may be used also for unbounded domains. To start with the process, we need an assumption on the asymptotic expansion of u near infinity. Using this method in [*69], Caffarelli and Spruck proved uniform estimates for solutions of some elliptic equations. In [*39], Berestycki and Nirenberg use with the method of moving planes a new one, the sliding method introduced by them. They compare translations of the function.

4.60 The Maximum Principle (see 3.71). It concerns second order elliptic operators A in a bounded domain n c ]Rn. Let gij(X) be a Riemannian metric on nand e(x) be a vector field on n. Set (76)

A is supposed to be uniformly elliptic (a- I I7J12 coefficients to be bounded by b in n. The maximum principle holds for A in n, if

Au

~

0

in nand

:s gij 7Ji7Jj :s al7J12), and its

limsupu(x):S 0 x-+an

imply u(x) :S 0 in n . . The usual condition for this to hold is h(x) :S 0 (see 3.71).

Proposition 4.60. The maximum principle holds if there exists a positive function f E Hzo(n) n oo(.!?) satisfying Af :S 0, or if n lies in a narrow band a < Xl < a + € with € small, or (Bakelman-Varadhan) if the measure Inl is small enough (Inl < 15)More precisely, assume diam n :S d, there exists 8 > 0 depends only on n, d, a and b. Proof. (i)

(ii)

Consider v = uf

-I,

v satisifies

Since lim sUPx~an vex) :S 0, the usual maximum principle implies v :S 0 in n ,thus the same is true for u. If n lies in a narrow band, we construct a function f as above.

§6. The Best Constant

(iii)

139

We use the following theorem of Alexandroff, Bakelman and Pucci [*268] (see 4.42). If hex) ~ 0 and if u satisfies Au ~ / and lim sUPx--+an u(x) ~ 0, then sUPxEn u(x) ~ Gil/lin where G depends only on n, d, a and b. u satisfies [A - h+(x)]u ~ -h+(x)u+. Thus

sup u+ ~ G(sup h+)(supu+) Inl l / n .

n

n

Choose 8 = (Cb) -n, then u ~ O.

n

4.61 The Maximum Principle (Second part). Suppose there is a ball B in n with a point P E annaB and suppose u is continuous at P and,u(P) = O.lfu ¢. oin nand ifu admits an outward normal derivative at P, then ~~(P) > O. More generally, if Q approaches P in B along lines, then lim infQ--+p >0 otherwise u == 0 in n. This holds for u E C 2(n) satisfying (76) if hex) ~ O..

U(02='t}f)

Proposition 4.61 (Gidas-Ni-Nirenberg [* 140]). If u E C 2(n), u ~ 0 satisfies Au ~ 0, the maximum principle holds. That is, ifu vanishes at some point in n, or if u vanishes at some point P E an with ~~ (P) =0, then u == 0 in .n. Proof Set A = A - h+. u ~ 0 satisfies Au ~ -h+u ~ O. Since -h- ~ 0, the usual maximum principle holds.

§6. The Best Constant 4.62 Theorem (Aubin [13], [17]). Let (Vn, g) be a complete Riemannian manifold with positive injectivity radius and bounded sectional curvature, n ~ 2 the dimension. Let q be a real number satisfying 1 ~ q < n: then for all € > 0 there exists a constant A£(q) such that any function cp belonging to the Sobolev space H?(Vn ) satisfies (77)

with lip = l/q - lin. The best constant K(n, q) depends only on nand q, its value is in 2.14. Remark. Recently, tanks to sharp estimates on the harmonic radius obtained by Anderson and Cheeger [5], Hebey [165A] was able to prove that Theorem 4.62 still holds if one replaces the bound on the sectional curvature by a lower bound on the Ricci curvature. We can ask the question: does A£(q) tend to 00 when c -+ O? In [13] we made the conjecture that the best constant K(n, q) is achieved (Ao(q») exists. The conjecture is proved when n = 2 and when n ~ 3 if the manifold has constant sectional curvature.

4. Complementary Material

140

This result is obtained by choosing a nice partition of unity and by using the isoperimetric inequality (which holds when the curvature is constant). Later Hebey and Vaugon extended this result to the locally conformally flat manifolds by a similar argument. But recently by new methods they proved 4.63 Theorem (Hebey and Vaugon [*171], [*172]). For any complete Riemannian manifold with positive injectvity radius, bounded sectional curvature, bounded covariant derivative of the curvature tensor and dimension n ~ 3, the best constant K(n, 2) is achieved. The statement of Hebey and Vaugon is more precise. We still sketch the proof when the manifold (M, g) is Coo compact, because it is very interesting and a good illustration of new technics; but before to read it, the reader must see chapter 6 (Note that the assumption of Theorem 4.63 are obviously satisfied by compact manifolds). Assume Proposition 4.64 below. Let (fli ,1/Ji) (i 1,2, ... , m) be a finite We can choose the atlas atlas such that 1/Ji({li) = B the unit closed ball of such that B is convex for (1/Ji l )*g (1 ~ i ~ m), since any point has a convex neighbourhood. Let us consider {7]i} a Coo partition of unity subordinated to the covering (li such that y'rii and IV' y'riil belong to cl(iii)' Setting Ui = y'riiu for some U E COO(M), we have by using (79)

= an.

m

m

i=1

~ K2

i=1

m

m

i=1

i=1

L lIV'uill~ + CL lIuill~·

Since 2::1 J IV' U iI 2dV = J lV'ul 2dV + 2::1 Ju21V'7]iI 2dV (indeed the additional term 42::1 J V'j 7]; V' jU 2 dV = 0), there is a constant C such that any u E HI (M) satisfies (78)

K = K(n, 2) is achieved. When the manifold has constant curvature, we use the isoperimetric inequality to prove Proposition 4.64, but in the general case the proof is harder. 4.64 Proposition. Let B c lR n be the closed ball of radius 1, and 9 be a Coo Riemannian metric on a neighbourhood of B such that B is convex for g. Then there exists a constant C such that any

K2(IIV'uall~ + alluallD. Thus Th~

Since Aa < K- 2 , the minimum is achieved. The proof is that of the basic theorem 5.11 on the Yamabe problem. As a consequence, there exists 'Pa E ilJ(B), with lI'PaliN = 1, which satisfies the equation and 'Pa

(80)

>0

10

B.

Hence 'Pa E C oo (8), 'Pa/ BB = 0 and (81)

'Pqi

Therefore lima-+oo II 'Pa 112 = 0 and there exists a sequence qi ~ ~ 0 a.e .. By interpolation

(82)

for

00

such that

2:S p < N.

Lemma 4.64. There exists a sequence {qi} such that {'Pqi} has a unique simple point of concentration. Moreover Aqi ~ K-2 and qill'Pqi II~ ~ 0 when qi ~ 00. According to Theorem 6.53, there is only one point of concentration. Indeed here f(P) = 1 and J-L/ J-Ls = Aq , K2 :S 1. Moreover since the energy of a point of concentration is at least K- 2 (see 6.52, formula 64), Xo is a simple point of concentration. Consequently Aqi ~ K-2 and qill'Pq.ll; ~ O. Remark that lim

(83)

(

qi-+ oo JBp(6)

'P~ dV = 1.

since P is the unique point of concentration.

4.65 Proof of Proposition 4.64 (continued).

. ( Aq . ) (n-2)/4 'Pqi' For convemence set Ui = ~

Ui

. fi satls es

(84)

Xi

Denote by Xi a point where Ui is maximum, Ui(Xi) = mi = sUPB Ui ~ ~ Xo and Ui ~ 0 uniformly on any compact set K c B - {xo}. Let J-Li (mi)-2/(n-2). Now we study the speed of convergence of Xi to BB (if any).

=

a) lim infi-+oo d(x~~B) = O.

00,

142

4. Complementary Material

a

This implies Xo E B. After passing to a subsequence if necessary we can suppose that the limit exists. We do a blow-up at Xo. Define the maps 1/Ji of IRn in IR n : Y ~ 1/;i(Y) = J.iiY + Xo; Bi

=1/J;I(B) =B-XO/J.li (:J

and

) _ Ui(J.iiY + xo) _ (n-2)/2 ( Vi (Y) - J.ii Ui J.iiY + Xo .

mi

=J.i;21/Ji g. On B i , vi

Let us consider the metrics hi and

satisfies 0 ::; Vi ::;

(85)

On any compact set of an, hi ~ £ uniformly in C2 (we are able to do so that g(xo) = £(xo»). A similar proof of that of Corollary 8.36 of Gilbarg-Trudinger [143] shows that the Vi are uniformly bounded in CIon a neighourhood of 0 E B i . . But hypothesis a) implies limi-+oo d( 1/J; 1(Xi), aBi ) = 0 which is in contradiction with Vi (1/;;1 (Xi») = 1 and Vi =0 on aBi . So a) is impossible. (3) lim infi-+oo d(X~~B) = l

> O.

This implies also Xo E aBo As previously we suppose that the limit exists. Since O(n) acts on B, we can suppose without loss of generality that all points Xi and Xo are on the same ray (the n - 1 first components of Xi and Xo are zero). We denote by gi the metric corresponding to 9 after the action of the element of O(n). Now we do the same blow-up as previously (in a). Then U::;I i is the half space

B

Since the sequence {Vi} is equicontinuous, a subsequence converges uniformly on any compact set K c E to a function V which satisfies Ll£v

(86)

viaE =0 and v(z)

=n(n -

= 1 where z =(0,0, ... , -l) E E,

Indeed qiJ.i7 ~ O. Let K QiJ.i;

2)v(n+2)f(n-2)

c

in E (Yi

= Xi~X()

~ z).

E be a compact set. When i is large enough

r v2 dV£ :::; 2QiJ.i; lKr v; dVhi ::; 2QiJ.i; lBir v; dVhi = 2QiJ.i 2- n r vi dVhi(J.i;)n/2 = 2Qi r u; dV lBi lB

lK

gi

~0

according to Lemma 4.64. Now such a positive function v cannot exist by Pohozahev's identity. First by the inverse of a stereographic projection, we get a function on a half sphere

143

§6. The Best Constant

of pole Q. Then a stereographic projection of pole Q (opposite to Q) yields a function v satisfying equation (86) on a ball 13 and also v(O) = 1 and v / aB =O. So (3) is impossible. ,) limi-+oo d(x~~B) = +00. In this case Xo may be on 8B or inside B. We do a blow-up at Xi. We define the maps 1/Ji by [li :3 Y !i expx, (J-tiY) with [li = 1/J;l(B). 1/Ji is well defined since [li is star-shaped according to the convexity of B for

g.

We consider on [2i the metric hi =J.L;21/J;g. hi(O) = £(0) according to the properties of the exponential mapping. hi ---t £ uniformly in C 2 on every compact set K. The function Vi on [2i defined by

satisfies and (87)

The sequence {Vi} is uniformly bounded in C t on every compact set K. A subsequence converges, uniformly on any K, to a function V satisfying 0 ~ V ~ I, v(O) = 1 and equation (86) on lRn, since qiJ.Lr ---t 0 (see (3). We know such function v, V = (I + Ily11 2) l-n I 2. In order to exclude the third case ,), and to establish a contradiction to the existence for any a 2:: 1 of a function CPa satisfying (80), we need to use the Pohozahev identity (88)

l,

'\Jk r 2'\J k Vi.:1£vi dV

= -~

r

k~

'\Jkr2'\Jkvi8vVi da - (n - 2)

1 ~

Vi.:1£Vi dV.

In (88) the metric is the euclidean metric, 8v is the outside normal derivative and r = liyli. Since [li is star-shaped, X =

1 n,

= --21 Define (89)

y=

tl ~

'\Jk r 2'\J kVi.:1£Vi dV£ + (n - 2)

r

Jan,

1 n.

Vi.:1£Vi dV£

8 vr218vvil 2 da ::; O.

8kr28kvi.:1h,vidv£+(n-2)1 vi.:1h,vi dV£

~

= X +Y - X

~

~

Y - X.

4. Complementary Material

144

In; v; dYe.

Using (87), we get Y = 2qiJ.L; Now some computations lead to (90)

Y - X

~ CIJ.L~ [In; vi dYe + In; r vf' dVe] 2

for some constant C 1 independant of i. Moreover there exists a constant C 2 such that for any i, Vi ~ C2 v. This result is proved by different authors; it holds when the point of concentration is simple. According to the value of Y, (89) and (90) yield (91) But

Vi

---t

V

uniformly on any K, hence for i large enough

f vi dYe ~! f v2 dYe > o.

in;

So we get a contradiction:

2

Vi

iK

cannot exist for i large enough.

Chapter 5

The Yamabe Problem

Yamabe wanted to solve the Poincare conjecture (see 9.14). For this he thought, as a first step, to exhibit a metric with constant scalar curvature. He considered conformal metrics (the simplest change of metric is a conformal one), and gave a proof of the following statement "On a compact Riemannian manifold (M, g), there exists a metric g' conformal to g, such that the corresponding scalar curvature R' is constant". The Yamabe problem was born, since there is a gap in Yamabe's proof. Now there are many proofs of this statement. We will consider some of them, but if the reader wants to see one proof, he has to read only sections 5.11, 5.21, 5.29 and 5.30.

§1. The Yamabe Problem 5.1 Let us recall the question. Let (Mn, g) be a compact Coo Riemannian manifold of dimension n :2: 3, R its scalar curvature. The problem is: Does there exist a metric g', conformal to 9 , such that the scalar curvature R' of the metric g' is constant? In fact Yamabe [269] said that such metric always exists, but there is a gap in his proof which is impossible to overcome in general. He said that the set {'f(x)c.pN-I,

(2)

where hex) and f(x) are Coo functions on M n , with f(x) everywhere strictly positive and N = 2n/(n - 2). The problem is to prove the existence of a real number >. and of a Coo function c.p, everywhere strictly positive, satisfying (1). 1.1. Yamabe's Method 5.4 Yamabe considered, for 2 < q :$ N, the functional

where c.p =t 0 is a nonnegative function belonging to HI, the first Sobolev space. The denominator of Iq(c.p) makes sense since, according to Theorem 2.21, HI C LN C L q • Define

/-Lq = infIq(c.p)

for all c.p E HI, c.p

~

0, c.p

== o.

It is impossible to prove directly that /-LN is attained and thus to solve Equation (2). (We shall soon see Why.) This is the reason why Yamabe considered the approximate equations for q < N:

(4)

/lc.p + h(x)c.p =>'f(X)c.pq-l

and proved (Theorem B of Yamabe [269]):

§l. The Yamabe Problem

147

5.5 Theorem. For 2 < q < N, there exists a Coo strictly positive function rpq satisfying Equation (4) with ..\ = J-tq and Iq(rpq) = J-tq.

Proof a) For 2

< q ~ N, J-tq is finite. Indeed

(5)

Iq(rp) ?: [inf (0, heX))] [sup I(X)] -2/q xEM

IIrpll~llrpll;2

xEM

and

(6) with V = f M dV. On the other hand,

J-tq :S Iq(l) =

[1M hex) dV] [1M I(x) dV] -2/q

(b) Let {rpi} be a minimizing sequence such that

rpi

E

HI,rpi?: 0, lim Iq(rpi) ~-+oo

fM I(x)rpr dV = 1:

=J-tq.

First we prove that the set of the rpi is bounded in HI,

Since we can suppose that Iq(rpi)

< J-tq + 1, then

and

c) If 2 < q < N, there exists a nonnegative function rpq E HI, satisfying

Iq(rpq)

= /-Lq

and

1M I(x)rp~ dV = 1.

Indeed, for 2 < q < N, the imbedding HI C Lq is compact by Kondrakov's theorem 2.34 and, since the bounded closed sets in HI are weakly compact (Theorem 3.18), there exists {rpj} a subsequence of {rpi}, and a function rpq E HI such that: (a) (/3)

(")')

rpj rpj rpj

-t -t -t

rpq in Lq, rpq weakly in HI. rpq almost everywhere.

148

5. The Yamabe Problem

The last assertion is true by Proposition 3.43. (a) ==> iM f(x)'PZ dV = 1; ~ 0, and (3) implies

,) ==> 'Pq

(Theorem 3.17). Hence Iq('Pq) ~ limj-+oo Iq('Pj) = j..£q because 'Pj --.. 'Pq in L 2, according to (a) since q ~ 2. Therefore, by definition of j..£q' Iq('Pq) = j..£q. d) 'Pq satisfies Equation (4~ weakly in HI. We compute Euler's equation. Set 'P = 'Pq + //'I/J with 'I/J E HI and // a small real number. An asymptotic expansion gives:

Iq('P) = Iq('Pq)

[1 + //q 1M f(x)'Prl'I/J dV] -2/q

+ 2// [1M ,\i'V 'Pq \I v'I/J dV + 1M h(x)'Pq'I/J dV] + 0(//). Thus 'Pq satisfies for all 'I/J E HI: (7)

1M \IV 'Pq \I v'I/J dV + 1M h(x)'Pq'I/J dV

=j..£q 1M f(x)'Prl'I/J dV.

To check that the preceding computation is correct, we note that since is dense in HI and 'P ~ 0, then

!i} (M)

I ('P) = 1(1 'P I) when 'P E Coo because the set of the point P where simultaneously 'P(P) = 0 and I'V'P(P) I :f 0 has zero measure (or we can use Proposition 3.49 directly). e) 'Pq E Coo for 2 ~ q < Nand the functions 'Pq are uniformly boundedfor 2 ~ q ~ qo < N. Let G(P, Q) be the Green's function (see 4.13). 'Pq satisfies the integral equation (see 4.14) (8)

'Pq(P) = V-liM 'Pq(Q) dV(Q)

+ 1M G(P, Q)[j..£qf(Q)'Pr l - h(Q)'Pq] dV(Q). We know that 'Pq E Lro with TO = N. Since, by Theorem 4.13c there exists a constant B such that IG(P, Q)I ~ B[d(P, Q)]2-n, then according to Sobolev's lemma 2.12 and its corollary, 'Pq E Lrl' for 2 < q ~ qo with ~= n-2+qo-l_1=qo-l_~ TI

n

TO

TO

n

§1. The Yamabe Problem

149

By induction we see that 0

and

n- 2]

+ c: 4(n _ 1) J-Lq

since J-Lq ~ J-L when q ~ N. In this case, (15) and (16) imply lim infIIcpqll2 2: Const q~N

> O.

Because 'Pqi converges strongly to 'Po in L 2 , lI'Poll2 =f O. Thus 'Po =1= 0 and 'Po > O. Picking 'lj; 'Po in (14) gives J( 0 and L= ~ + h such that any

6

and

J('l1 k ) = 30W!/3 [1

=

for n 6 with a 2 enough.

-

a2 k-2(1og k)/80 + O(k- 2 )]

= IWijkl(P)12/12n. Thus J('¥k) < n(n -

l)w~n for k large

5.22 Remarks. For any compact manifold Mn(n ~ 3), J('¥k) tends to n(n - l)w~n when k ---7 00. This implies the first part of Theorem 5.11.

160

5. The Yamabe Problem

In dimension 3 to 5, there are integrals on the manifold in the limited expansion of J(Wk) instead of a coefficient like a2, and it is not possible to conclude a priori, but see 5.50. For locally conforrnally flat manifolds, it is obvious that local test functions cannot work since for the sphere JL = n(n - l)w!ln (Theorem 5.58). 5.23 Theorem ([14] p.291). For a compact locally conformally flat manifold Mn, (n ~ 3), which has a non trivial finite Poincare's group, JL < n(n-l)w!ln. For the proof, we consider Un the universal covering of Mn. Un is compact, locally conforrnally flat and simply connected. Kuiper's theorem [172] then implies that Un is conforrnally equivah~nt to the sphere Sn. Hence Equation (1) has a solution with R' =JL

5.24 Proposition. When the minimum JL is achieved, let J(cpo) = JL. In the corresponding metric 90 whose scalar curvature Ro is constant, the first nonzero eigenvalue of the Laplacian

>'1

~

Ro/(n - 1).

For the proof one computes the second variation of J(cp) (see Aubin [14] p.292).

§4. The Remaining Cases 4.1. The Compact Locally Conforrnally Flat Manifolds 5.25 The effect of §4 is to prove the validity of conjecture 5.11. The results of the preceding paragraph do not concern the locally conforrnally flat manifolds with infinite fundamental group for which v 2/n-1 RdV > n(n - 1)w!ln. The known manifolds of this type are

J

a)

(3)

'Y)

some products Sn-l X C and Sp x Hn - p where C is the circle and Sq (resp. Hq) are compact manifolds of dimension q with constant sectional curvature p > 0 (resp. -p < 0). some fibre bundles with basis one of the manifolds with constant sectional curvature mentioned previously and for fibre Sq or Hq according to the situation. the connected sums Vi # V2 of two locally conformally flat manifolds (Vi,9}), (Vi,92).

Most of these manifolds are endowed with a metric of constant scalar curvature by definition. But for them, according to the conjecture 5.11, the problem is to prove that the infimum of J(cp) is achieved, and thus we shall prove

§4. The Remaining Cases

161

5.26 Theorem (Gil-Medrano [*142]). The manifolds a), (3), and,) satisfy /-L

< n(n -

l)w~n.

Proof. It consists to exhibit a test function u such that J(u) < n(n - l)w~n. By an homothetic change of metric, we can suppose that p = 1. Let II be the projection Sn-l xC -+ C.

On Sn-l X C the function u will be u(P) = (ch r)l-n/2 where r is the distance on C from II(P) to a fixed point Yo E C. On Sp x fI n - p, the same function u(P) works, but here II is the projection Sp x fIn - p -+ fI n - p and r is the distance on fIn - p from II(P) to a fixed point Yo E fIn-po The proof is similar for the fibre bundles. For the connected sums we have first to study the conformal class of the locally conform ally flat metric go constructed on the connected sum Va = Vi # Vi. Then Gil-Medrano proved that /-La ~ inf(/-Ll, 1-'2). where I-'i(i = 0, 1,2) is the I-' of (Vi, gil. 4.2. Schoen's Article [*280] 5.27 As I-' is a conformal invariant, it is possible to do the computation of J(lfl), for some test function lfl, in a particular conformal metric (as in 5.20). When the manifold is locally conformally flat. after a suitable change of conformal metric, the metric is flat in a ball B6 of radius fJ and center Xo. We saw above that locally test functions yield nothing for these manifolds. The idea of Schoen is to extend the test functions used in 5.21 by a multiple of the Green function GL of the operator L = b.. + (n - 2)R/4(n - 1). We are in the positive case (I-' > 0), L is invertible and G L precisely let p < fJ /2 and r = d(xo, x). For e > 0 set (27)

(e + r 2/e)l-n/2 { lfl(x) = co [G(x) - h(x)a(x)] eoG(x)

> O. More

for r ~ p, for p < r ~ 2p, for r > 2p.

G(x) is the multiple of GL(XO, x) the expansion of which is the following in B6:

(28)

G(x) = r 2- n

+ a(x)

where a(x) is an harmonic function in B 6 • h(x) is a Coo function of r which satisfies h(x)=1 for r r > 2p and IVhl ~ 2/p.

~ p,

h(x) = 0 for

162

5. The Yamabe Problem

t:o = (p 2 - n + A)-l(t: + p2 jt:)I-n/2 with A = a(xo) in order the function W is continuous hence lipschitzian, p will be chosen small, t:o infinitely small with respect to p, then t: is well defined and t: rv t:~/(n-2) when t:o --t O. Indeed the function t --t [t + p2 jtf- n / 2 is increasing for t E ]0, p] and goes from 0 to (2p)l-n/2. 5.28 Proposition (Schoen [*280] 1984). IfG(x) is of the form (28)forany n ~ 3 with a(xo) = A > 0 then p, < n(n - l)w~n. The proof is easily understood. By an integration by parts, all the computations can be carried out in B 2p . They yield (29) where C

J(w)

>0

'S

n(n -1)w~n - CAt:5+0(Pt:5)

is a constant which depends on n. The result follows.

5.29 Theorem (Schoen and Yau [*289] 1988). If (Mn , g) is a compact locally conformally flat Riemannian manifold ofdimension n ~ 3 which is not conformal to (Sn,gO), then A > O. Hence conjecture (5.11) is valid and there exists a conformal metric 9 with R = p, V- 2/ n, V being the volume of the manifold (Mn,g). The result follows from Proposition 5.28 combined with 5.37 for n = 3 and Theorem 5.48 for n ~ 4. 4.3. The Dimension 3, 4 and 5 5.30 Theorem (Schoen [*280]). If(Mn , g) is any compact Riemannian manifold of dimension 3 to 5, which is not conformal to (Sn,gO), then p, < n(n - l)w~n. Hence according to theorem 5.11, there exists a conformal metric 9 with R = p,V- 2/ n , V being the volume of the manifold (Mn,g).

Proof The result follows from Proposition 5.28 combined with the fact A > 0 to be established below §4. When n = 3, the Green function G p of L at P E M has for limited expansion in a neighbourhood of P:

where A is a real number and r = d(P, x). This expression is the same as (28). So the method of 5.27 works. For the dimensions 4 and 5, Schoen [*280] replaces in a small ball Bp(p) the metric 9 by a flat metric. He considers a Coo metric which is euclidean in Bp(p) and equal to 9 outside the ball B p (2p). Thus he can use his method, but the approximation is too complicated. It is simpler to use the following fact which is one of the hypotheses of Proposition 5.28.

§4. The Remaining Cases

163

5.31 Proposition. Let (Mn, g') be a compact Riemannian manifold of dimension 4 or 5, belonging to the positive case (J-t > 0). Pick P E M n , there exists a metric g conformal to g' such that the Green function G p of L at P has, in a neighbourhood of P, the following limited expansion

e

Gp(X) = (r 2 - n

where A is a real number and r lipschitzian for n = 5.

+ A)/(n - 2)Wn _1 + a(x)

=d(P, x). a(P) =0, a E c l for n =4 and a is

With this proposition, the method of 5.27 works and Proposition 5.28 implies theorem 5.30.

Proof of 5.31. We consider a conformal metric g to g' which has at P the properties (21) and (23). Thus (24) Jlg(x)1 = 1 + 0(r4) in normal coordinates. As in 4.10, consider H(P, Q) !(r)r 2- n /(n - 2)Wn_1 with r d(P, Q) and! a Coo function equal to 1 in a neighbourhood of zero and to zero for r ~ 6 > 0 (6 small enough). Recall 4.10, the singularity of D.QH(P, Q) is given by rl-nar Log v19f/Wn -1 which is in 0(r 4 - n ):

=

=

(30) According to the Green formula (4.10), any

0). We can choose 9 so that the scalar curvature R 2:: Ro > O. Moreover we suppose the dimension n 2:: 4. Consider (!VI, g) the universal Riemannian covering manifold of (M, g). Set 7r : M ----t M, 9 = 7r* g.

167

§S. The Positive Mass

(M, g) is complete, locally conformally flat and simply connected. A well known theorem of Kuiper [*205] asserts that there exists cI> a conformal immersion of (M, g) in (Sn, go) where go is the standard metric of Sn. 5.39 Theorem (Schoen-Yau [*289] 1988). cI> is injective and gives a conformal diffeomorphism of M onto cI>(M) C Sn. Moreover Sn - cp(M) has zero Newtonian capacity and the minimal Green junction of L at P E M is equal to a multiple of Icp'l n2:2 H 0 CPo Where H is the Green junction of Lo at cp(P) on (Sn,90) and Icp'l is the (9,90)-norm of CP'. Thus M is the quotient of a simply connected open subset n of Sn by some Kleinian group, Sn - n having zero Newtonian capacity. This theorem allows to prove A > 0 for manifolds of this type not conformal to (Sn, go). The proof (starting at the end of p. 59 of [*289]) must be completed at least at one point. First we will give the definitions of the new words used above and explain the existence of the minimal Green function Gp (lemma 5.44), so as the positiveness of the energy of (Mn , 9), A > 0, if the manifold is not conformal to the sphere (Sn, 90). For this we follow Vaugon (private communication) who first clearly explained the proof of 5.39.

5.40 Definition. Let (M, g) be a Riemannian manifold with scalar curvature R ~ 0 and dimension n ~ 3. A Green function G p of L at P is a function on M - P which satisfies LGp =bp. Recall L = !l. + (n - 2)R/4(n - 1).

(36)

G p is the minimal Green function if any Green function G'p satisfies G p ~ G'p. If some Green function G'p exists, the minimal Green function Gp exists and is obviously unique. Let {n i } be a sequence of open sets of M with Coo boundary and fli compact, such that for all i P E n i C fli C n i +1 and U~\ni = M. Let G i be the Green function of L at P with zero Dirichlet condition on We have G i > 0 on ni - P. At Q E niQ (Q # P), Gi(Q) is an increasing sequence for i ~ io, according to the maximum principle since L(Gi +1 - G i ) = 0 Likewise G i < G'p for all i, if we extend G i by and G +1 - G i > 0 on zero outside fl i . So when i - 00, G i tends to some positive function Gp which satisfies in the distributional sense LG p = bp on M.

ani.

i

ani.

5.41 Proposition (Vaugon). If G p is a Green junction for L at P and 9 = cp4/(n-2)g is a conformal metric then (37)

Gp(x) = Gp(x)/cp(P)cp(x)

if

168

5. The Yamabe Problem

is a Green function for the operator L related to g. In particular if G p is the minimal Green function for L at P, Gp(x)/cp(P)cp(x) is the minimal Greenfunction for L at P. Proof. For any function f E D(V),

Iv

GpL(fcp)dV = f(P)cp(P).

We have dV = cp2n/(n-2) dV and a computation gives (38)

so

tl(cpf) + n - 2 Rcpf = cp(n+2)/(n-2) (Af + n - 2 itf) 4(n - 1) 4(n - 1)

Iv

Gp(x)cp-l(x)Lf(x)dV(x) = f(P)cp(P).

Thus Gp(x)/cp(x)cp(P) is a Green function for L at P. 5.42 Definition. If 9 is an euclidean metric in a neighbourhood function G p at P is equal in (J to (39)

(J

of P a Green

G p(x) = [d(P, x)f- n /(n - 2)wn-l + a(x)

where a(x) is an harmonic function in

(J.

When G p is the minimal Green function of L at P, we call energy at P related to 9 the real number a(P). 5.43 Proposition. If 9 and 9 = cp4/(n-2)g are euclidean metrics in a neighbourhood (J of P, &(P) = a(P)cp-2(p). In particular the sign of the energy is a conformal invariant in the set of the euclidean metrics near P. By 5.42 and Proposition 5.41 for x E

(J:

dl-n(p, x)/(n - 2)Wn -l + &(x) [d2- n(P, x)/(n - 2)Wn -l + a(x)]jcp(P)cp(x).

=

Moreover we can prove that ~-n(p, x) lows.

=cp(p)cp(x)dl-n(p, x), the result fol-

5.44 Lemma. (M,g) being the covering manifold of(M,g) considered in 5.38, at each point P E M, there exists a minimal Green function for L.

9 = cI>*(go) is conformal to g, so there exists a Coo function u > 0 such that 9 = u 4/(n-2)g.

169

§5. The Positive Mass

Set W = ~-I(~(p». As L(Ha~) = EQEw 8Q, according to Proposition 5.41, there is fI a multiple of U -I H a~ such that with aQ

> 0 and

ap = 1.

Let us return to the definition and to the construction of the minimal Green function (5.40). Set Gi be the Green function of L at P with zero Dirichlet condition on 80+ Pick Oi C !VI an open set such that (W - P) n Oi C Oi with Oi small enough so that fI - Gi > 0 on 80i. We extend by zero Gi on M - i . On Oi - Oi n Oi, L(fI - Gi ) 0 and fI - Gi > 0 on 8(Oi - Oi n Oi)' Thus by the maximum principle Gi < fI and Gp the minimal Green function for L at P exists. Moreover Gp ~ fI. We have fI - Gp > 0 ifW i= {Pl.

n

=

5.3. The Positive Energy 5.45 Definition. A compact set F C Sn (n ~ 3) has zero newtonian capacity if the constant function 1 on Sn is the limit in HI of functions belonging to V(Sn - F).

We verify that the measure of F is zero. And we can prove that the minimal Green function for Lo at P E Sn - F on (Sn - F, 90) is the restriction to Sn - F of the Green function H for Lo at P on (Sn, 90)' 5.46 Remark an the proof afTheorem 5.39. Return to the proof of lemma 5.44. We have fI - Gp > 0 if W i= {P}. So if we prove that fI = Gp, the injectivity of ~ follows. For this, define v = GpfI- l • We have 0 ~ v ~ 1. After some hard computations which must be detailed, Schoen and Yau infer v = 1. Set F = Sn - ~(!VIn)' since fI = Gp, the restriction of H to Sn - F is a minimal Green function for Lo on (Sn - F, 90)' This implies F has zero Newtonian capacity. Before the main proof, one step consists in showing that 1M Gp dV < 00 when n ~ 4 and 1M G~-e dV < 00 for some € < 0 when n = 3. The last inequality holds because the Ricci curvature of (M, g) is bounded. Let us prove the other inequality. Return to the construction of the minimal Green function Gp (5.40). Let Ui the unique solution of LUi = 1, uilall; = O. We extend Ui by zero outside Oi.

1M G dV = Ui(P), i

At Q a point where Ui is maximum, LiUi(Q) ~ 0 so sup Ui ~ 4(n - 1)(inf R)-I I(n - 2) ~ 4(n - 1)/(n - 2)Ro.

170

to

5. The Yamabe Problem

Thus {G i } is an increasing sequence of non-negative functions which goes p. According to Fatou' s theorem, p is integrable and

a

a

1M a

p

dV =

i~~ 1M G i dV ~ 4(n -

l)/(n -

2)~.

5.47 Proposition. If (Mn, §) is flat near a point P, the energy of § at P is zero. Assuming R ~ ~ > 0, we have proven the existence of the minimal Green function p for L corresponding to the metric g. But the manifold (£1, g) is locally conformally flat, so there exists a Coo function u > 0 such that § = u4 /(n-2)§ is flat near P (we can choose u = 1 ouside a compact neighbourhood of P). According to Proposition 5.41. apex) = ap(x)lu(x)u(P) is the minimal Green function of L. Now the energy of the sphere is zero since H 4 /(n-2)gO is the euclidean metric on ]Rn with zero mass. So by Theorem 5.39 the energy of § is zero.

a

5.48 Theorem (Schoen-Yau [*289]). Let (Mn, g) be a compact locally conformalty flat manifold which belongs to the positive case (J.L > 0) but which is not conformal to (Sn, go). If 9 is flat in a neighbourhood of some point P then the energy of gat P is positive. Proof As the Riemannian manifold is not conformal to (Sn, go), it is not simply connected and (Mn, §) is a non trivial Riemannian covering of (Mn, g). Set II : Mn ---t M n , 9 = II* 9 is flat near each point of II-I (P), let P one of them. We know (lemma 5.44) that the minimal Green function cJp of L at P exists. In a neighbourhood f) of P. a p(x) = f2-n I(n - 2)Wn -1 + 6:(x) with f = d(P, x). 6: is an harmonic function and 6:(P) = 0 (Proposition 5.47). On the other hand Gpo II satisfies L(G p 0 II) = LQEn-'(p) oQ, G p being the Green function of L at P. Thus Gp 0 II - Gp > 0 on £1 (see the proof of lemma 5.44) because

But in () G p oII(x) = f2-n I(n - 2)Wn -l +a oII(x). So a(P) of 9 at P (see Definition 5.42) is positive.

> 0, the energy

§6. New Proofs for the Positive Case (f-L

> 0)

171

§6. New Proofs for the Positive Case (J1, > 0) 6.1. Lee and Parker's Article [*208] 5.49 In this article Lee and Parker present, among other things, an argument which unifies Aubin and Schoen's works. They transfer the Yamabe problem from (Mn,g) to (Mn - P,g) an asymptotically flat manifold, P EM. If necessary, we change 9 by a conformal metric which has the property of Proposition 5.20 and g = dj 0 small),

> 8, r = d(P, x)

O.

6.2. Hebey and Vaugon's Article [* 166] 5.50 Theorem (Hebey-Vaugon [*166]). If the compact Riemannian manifold (Mn, g) is not conformal to (Sn, go), the test functions: ut:(x) = (c + r 2)1-n/2

if r ~ 8, (8) 0 small)

ut:(x) = (c + 82)I-n/2

if r ~ 8,

in the Yamabe functional, yield the strict inequality of the fundamental theorem 5.11.

These test functions are the simplest one. In fact the following proof is in my opinion the clearest. Proof. First we choose a good conformal metric. When n ~ 6, if the Weyl tensor W ijkl is not zero at P, we choose the conformal metric 9 so that Rij(P) 0 (as in 5.19). J(ut:) has the same limited expansion than in (5.21) with r = d(P, x) 2/n and c 11k. Then J(ut:) < n(n - l)wn for c small enough. When the manifold is locally conformally flat, we choose a conformal metric 9 which is euclidean near a point P. We get

=

=

(40) J(ut:)

which C

= n(n-l)w;!n+Ccn / 2- 1

[l RdV -

4(n - 1)8n - 2wn _l + G(c)]

> 0 a constant which does not depend on c.

172

5. The Yamabe Problem

When the dimension n equals 3 to 5, we choose a conformal metric g' so that (21) and (23) hold. Then we have (24) and R'(x) = 0(r2). A limited expansion yields (40). In the remaining cases, we will have J(ue ) < n(n-l)w;!n if in a conformal RdV < 4(n-l)on-2wn _ 1 for some o. This comes metric as above, we have from the following caracterisation of the mass together with (5.37) and Theorem (5.48).

Iv

5.51 Theorem (Hebey-Vaugon [*166]). When the compact manifold is locally conformally flat at P (41)

A = lim sup [4(n - 1)(/ RdV)-1 - t 2 - n /wn-d(n - 2)-1 t-+O

where the sup is taken over the conformal metrics g = cp4/(n-2)g which are flat in Bp(t) with cp(P) = 1. When n = 3 to 5, (41) holds the sup being taken over the conformal metrics which are equal to 9 in Bp(t). Recall m(g) = 4(n-l)A and m(g) = m(g) by Proposition 5.43. The theorem holds in the low dimensions thanks to Proposition 5.31 and 5.36. 6.3. Topological Methods

5.52 In Bahri [*20] for the locally conformally flat manifolds, and in BahriBrezis [*23] for the dimensions 3 to 5, the authors, by using the original method of Bahri-Brezis-Coron (see 5.78 for a more complete discussion of this method) solved the Yamabe problem in the remaining cases without the positive mass theorem. They analyse the critical points at infinity of the Yamabe functional J and prove by contradiction the existence of a critical point which yields a solution of the Yamabe problem, but in general it is not a minimizer of J.

5.53 In Schoen [*282] a different approach is used. As we are in the positive case, the operator L= ~ + (n - 2)R/4(n - 1) is invertible, let L -I its inverse. For any A > 1 and any p E [1, (n + 2)/(n - 2)], we define Fp by (42) where

and

§6. New Proofs for the Positive Case (JL

> 0)

173

Theorem (Schoen [*282]). Let (Mn , g) be a compact locally conformally flat manifold which is not conformal to (Sn, go). There exists a large number Ao > 0 depending only on g such that F;I(O) c n~ for all p E [1, (n + 2)/(n - 2)]. Actually Schoen wrote this theorem for any Riemannian manifold not conformal to (Sn, go), but he gave a complete proof only for locally conformally flat manifolds. It is not known (to the Author at this time) that a general proof is written up. 5.54 When p = 1 it is well known that the equation F\ (u) = 0 has only one positive solution. Let AO be the first eigenvalue of L. Since we are in the positive case AO > O. By minimizing E(u) JuLudV on the set A = {u E Htillul12 = l,u ~ O} we find (as in 5.4) a positive eigenfunction cp : Lcp = AOcp.

=

Proposition. cp is the unique positive solution of Fl (u) = o. First a solution of Fp(u) = 0 satisfies

J uLFp(u)dV. So F\(cp) = 0

lIullp+-\ =

1, indeed compute

Then, it is a general result for the normal-compact operators, that the eigenspace corresponding to the first eigenvalue AO is one dimensional. To see this, let 'lI ¢. 0 be such that L'lI = Ao'll. Pick k E R so that'll +kcp -:; 0 and equals zero in some point P E M. We apply the maximum principle to (-~)('lI

+ kcp) - h('lI + kcp) = [en - 2)R/4(n - 1) - AO - h)('lI + kcp)

which is ~ 0 when h E lR is chosen large enough. If the maximum of 'lI + kcp is ~ 0 then 'lI + kcp = Const. As 'lI + kcp is zero at P, it is zero everywhere and 'lI is proportional to cpo Finally if 'lI ¢. 0 is an eigenfunction of L corresponding to an eigenvalue A # AO, 'lI changes of sign. Indeed multiplying L'lI A'lI by cp and integrating on M yield

=

A

J

cp'll dV

=

J

cpL'lI dV

=

J

'lI Lcp dV

=Ao

J

cp'll dV

since L is self-adjoint. We get J cp'll dV = O. 5.55 When 1 < p < (n + 2)/(n - 2) we can prove Theorem 5.53, for any manifold, by using the following theorem: Theorem (Gidas and Spruck [*141]). In lRn , n 1 -:; p < (n + 2)/(n - 2) has no non-negative

> 2, the equation ~v = v P with

e2 solution except vex) == O.

Following Spruck the proof is by contradiction. On the compact manifold (Mn' g), let us suppose there exists a sequence of positive 2 ,a functions Ui which satisfy:

e

(43)

SUPUi -+ 00.

174

5. The Yamabe Problem

This is equivalent to Fp(VUi) = 0 for some v since p > 1. Pick Zi a point where Ui is maximum: Ui(Zi) = mi without loss of generality, since the manifold is compact, we can suppose that Zi --+ P. We blow-up at P. In a ball Bp(c5), consider {xi} a system of normal coordinates at P with xi (P) =O. We suppose zi E Bp(c5 /2). Define for y E ]Rn with lIyll < c5mf /2 = ki . with a

=(p -

1)/2

Zi + miO:y is suppose to be the point of Bp(c5) of coordinates the sum of the coordinate of Zi in Bp(c5) and the coordinates of miO:y in ]Rn. We set

yi

= (xi - zl)mf.

The function Vi satisfies on the ball Bo(kio ) for i ki a 2Vi gi (y) a ka .

Y

where g~j(y)

=gkj(ymiO: ai(y) =(n k'

y3

i

~

io the elliptic equation

aVi

p

+ ai (Y)-a. + ai(y)vi = Vi (y) y3

Zi), ai(y)

*

=-miO:(gkjr~j)(ymiO: -

Zi) and

2)mi 20: R(ymiO: - zi)/4(n - 1). k'

Bg le ;

•

_ When i --+ 00, g/ ~ E 3, --+ 0, ai --+ 0 and ai --+ 0 uniformly on Bo(kio ). Moreover the functions Vi are uniformly bounded 0 < Vi ~ 1. The conditions of Theorem 4.40 are satisfied. So IIvillca is uniformly bounded on Bo(kio ) for some a > O. By the Ascoli Theorem, there exists a subsequence of {Vi} which converges uniformly to a continuous function v. V satisfies in the distributional sense on ]Rn the equation .1.v = vP, so V E C 2 and v(O) = 1 since Vi(O) = 1. This is in contradiction with Theorem 5.55. Thus the assumption sup Ui --+ 00 is impossible and there exists a real number such that IUil ~ k. GL(P, Q) being the Green function of L (see 5.14) we have Ui(P) =

(44)

IUil

J

GL(P,Q)uf(Q)dV(Q)

k implies lIudlc1.a < Const. and then lIuillC2.a ~ Ao some constant. Moreover (44) implies Ui(P) ~ m I uf dV. As lIuill:;/ ~ (n - 2)p,/4(n -1) > 0, I uf dV is bounded away from zero since I uri dV ~ k I uf dV. Hence there exists a constant Vo such that Ui ~ Vo > O. ~

=

Remark. For p (n + 2)/(n - 2) the preceding arguing yields no contradiction. The equation .1.v = v(n+2)/(n-2) on lRn has positive solutions. The solution, with v(O) = 1 as maximum, is

w = [ 1 + lIyIl2/n (n - 2)]

l-n/2

.

New proof by Schoen [*282]. Let us return to 5.53. The map u --+ E(u)L-lup is compact from OA into C2,0:. So Fp = 1+ compact and the Leray-Schauder degree makes sense. By

§7. On the Number of Solutions

175

Theorem 5.53, 0 f/. Fp(OnAo) for any p E [1, (n+2)j(n-2)] thus deg(Fp, nAo, 0) is constant for p E [1, (n+2)j(n - 2)]. Moreover 'P the unique positive solution of FI(U) = 0 (Proposition 5.54) is nondegenerate. Thus deg(FI' nAo' 0) ±1 and deg(F(n+2)/(n-2), nAo' 0) is odd. So the Yamabe problem has at least one solution on the compact locally conformally flat manifolds.

=

6.4. Other Methods 5.56 In [* 181] Inoue uses the steepest descent method to solve the basic theorem of the Yamabe problem. R. Ye in [*320] studies the Yamabe flow introduces by Hamilton

ogjOt

= (8 -

R)g

with

8

!

=

RdV /

!

dV.

Ye proves that the long-time existence of the solution holds on any compact Riemannian manifold. In the positive case for the scalar curvature, if the manifold is locally conformally flat, Ye shows that the solution converges smoothly to a unique limit metric of constant scalar curvature as t tends to 00. The estimates are obtained by using the Alexandrov reflection method.

§7. On the Number of Solutions 7.1. Some Cases of Uniqueness 5.57 In the negative and null cases (J..L :s: 0) two solutions of (1) with R = Const. are proportional. Let 'Po be a solution of (1) with R' = J..L. In the corresponding metric go the Yamabe equation is always of the type of Equation (1), since Yamabe's problem is conformally invariant. So let 'PI be a solution of Equation (1) with R = R' = J..L. If J..L 0, ~'PI 0 , thus 'PI Const .. If J..L < 0, at a point P where 'PI is maximum, ~'PI :s: 0 , thus ['PI(P)]N-2 :S 1, and at a point Q where 'PI is minimum, ~CPI :S 0 thus [CPI(Q)]N-2 ~ 1. Consequently CPo = 1 is the unique solution of (1) when R = R' = J..L < O. In the positive case, we do not have uniqueness generally, neversless we have below Obata's result.

=

=

=

Examples. The sphere Sn (Theorem 5.58).

Mn = T2 X Sn-2 with T2 the torus when R(J dV)2/n > n(n - l)w;!n and 6. Indeed in this case there exists on Mn at least two solutions of Equation R Const. First, CPI == 1 and second, CPo for which (Theorem (1) with R' 5.21) J(cpo) = J..L < n(n - l)w;;n < J(I).

n

~

= =

176

5. The Yamabe Problem

On the other hand, according to Obata [225], we have uniqueness for Einstein manifolds other than the sphere. Proposition (Obata [225]). Let (Mn, g) be a compact Einstein manifold not isometric to (Sn, go). Then the conformal metrics 9 to 9 with constant scalar cur-

vature are proponional to g.

Proof Let us consider the conformal metric 9 on the form 9 =u2g.

Set Tij = Rij - (R/n)9ij. We have (see 5.2):

Tij

(45) As Tijgi j

J

=Tij + (n -

=0

ugikgjlTijTkl dV

~

2)u -i['\1 i '\1 jU + (Au/n)9ij ] .

J

uTijTij dV + 2(n - 2)

J

T i j'\1ijU dV.

U-

But '\1 i T] = ~) '\1jR by the second Bianchi identity (see 1.23). If 9 has constant scalar curvature, we get f uTijTij dV = 0 since 'ii j = O. Thus if R = Canst., 9 is Einstein. According to (45), if u =t Canst. there exists a non-trivial solution of

'\1 i '\1 jU + (Au/n)gij

=O.

In that case Obata proves that (Mn, g) is isometric to (Sn, go). Remark. When 9 is Einstein, J.t the inf. of the Yamabe functional J is attained by the constant function. J(1) =J-t. So Ai ~ R/(n - 1) (see [14] p.292), which is the inequality of Theorem 1.78. But we have more for Einstein manifolds: 2

J(I)

=R ( / dV) n =J.t ~ n(n -1)w~n,

with equality only for the sphere. 7.2. Particular Cases 5.58 The case of the Sphere. Theorem 5.58 (Aubin [13] p.588 and Aubin [14] p.293). For the sphere Sn' (n ~ 3), J-t n(n - l)w~n and Equation (1) with R' = R has infinitely many solutions. In fact, the functions cp(r) = ({3 - cos or)1-n/2, with 1 < {3 a real

=

number and 0 2 =R/n(n - I), are solutions of (1) with R' =R({32 - 1). Moreover on the sphere Sn with f dV = I, all cp E Hi satisfy: (46)

§7. On the Number of Solutions

177

Proof Recall r is the distance to a fixed point P. According to Theorem 6.67,

Equation (1) has no solution on the spere when R' = 1 + € cos ar, € :f o. Indeed, F = cos ar are the spherical harmonics of degree 1: D.F = )\} F, with ..\1 = R/(n - 1) the first nonzero eigenvalue.

If J.L < n(n - l)w~n, we can choose f small en ought to apply Theorem 5.12 with h = Rand f = 1 + € cos ar since 1I ~ J.L(1 - €)-2/N. This contradicts the nonsolvability of Theorem 6.67. Writing J.L = n(n-l)w~n yields (46) when dV 1. (46) is an improvement of the inequality of Theorem 2.28. Both constants are optimal, the second, Aq(O) 2: 1, since the inequality must be satisfied by the function 'P == 1. For the unit Sphere (Sn, 90) the solutions of (1) with R = R = n(n - 1) are

J

=

(47)

with (3 E)I,oo), P E Sn and r

=d(P,Q), (see [14] p.293). All solutions are

minimizing for J: J('P{3,p) = n(n - 1)w~n. There is no other solution. To see this, let 7r be the stereographic projection at P, 7r is a conformal map from Sn \ {P} onto JR n . Consider (P,Oi) i = 1, 2, ... , (n - 1) polar coordinates in ]Rn and set g = (7r- I )* 90. As p = cotg(r /2), 9 = 4 sin4(r /2) 4(1 + p2)-2c. By virtue of (38), L('P{3,p) = n(n4-2)'P;'pl yields

=

(48) with cos r = (p2 - 1)/(p2 + 1) is a solution of n

(49)

L8

jj W+W N - I

=0

j=1

According to Gidas-Ni-Nirenberg [* 140], the positive solutions of (49) are radial symmetric. The solutions W(r) satisfy a second order equation, moreover W' (0) = o. So there is only one positive radial solution such that W(O) = k a given real positive number. This solution is W(p) = k [1 + k 4 /(n-2) p2 /n(n - 2)f- n / 2 .

It is a solution found in (48) with (3 E ]-00, -1[ U]1, 00]. It is of the kind (47) ; indeed 'P-{3,P = 'P{3,P with P the opposite point to P on the sphere. 5.59 Schoen [*281] found all solutions of the Yamabe problem for C X Sn-I the product ofthe circle of radius T with the sphere of radius 1. Set TO = (n - 2)-1/2. The result is: If T ~ TO the unique solution of (1) with R = R is 'P == 1. If T E ]TO, 2To] there are two inequivalent solutions, the constant solution and the minimizers of J which are a C-parameter family of solutions with fundamental period 27rT.

178

5. The Yamabe Problem

For T E ](k-l)To, kTo] there are k inequivalent solutions, (k-l) C-parameter families of solutions and the constant solution for which J has the greatest critical value. 7.3. About Uniqueness 5.60 In the positive case there is no uniqueness in general. It is very easy to construct manifolds for which equation (1) with R = R has more than one solution. Let us consider two compact manifolds (M., g.), (M2, g2) with dimension n. (resp. n2) volume V. (resp. "\12) and constant scalar curvature RI (resp. R2 > 0). Pick k large enough so that (R 1 +kR2)(V1"\I2)2/n > n(n-l)w;(n. In that case the functional J for the manifold (M., kgd x (M2' g2) satisfies J(1) > n(n-l)w;(n. The constant function is not a minimizer for J. Hence there are at least two solutions. Now we will discuss another method to exhibit examples with several solutions. 7.4. Hebey-Vaugon's Approach 5.61 Let us consider (Mn , g) be a compact Riemannian manifold of dimension n ;::: 3 and G be a compact subgroup of G(M, g) the group of conformal diffeomorphisms.

Theorem (Hebey-Vaugon [* 168]). The inf, on the set of G-invariant metrics g' -(n-2)/n f conformal to g, of J(g') = [f dV' ] R'dV' is achieved. The case G = {Id} is the Yam abe problem. For most cases the proof consists in two steps. First they prove that the info is attained if it is strictly less than f3 = n(n - l)w;(n(infxEM CardOa(x»)2/n (when G = {Id} this is Theorem 5.11). Then they prove that the inf is smaller than f3 (we have always f3 ~ 0). Now if, under some conditions, J(u) inf u>o,u G-invariant

> J.I. =inf J(u) > 0

the theorem above yields two solutions to the Yamabe problem. The corresponding critical values of the Yamabe functional J are not the same. A more general case is that of the Riemannian covering manifolds 7r : (M, g) --t (M, g) with 9 = 7r* g. The question is: find conditions so that

infuEH1(M),u~0 j(u 0

7r)

> infuEH1(M),u~0 j(u),

j being the Yam abe functional of (M, g). 5.62 Theorem (Hebey-Vaugon [* 167]). Let (Mo, go) be a compact Riemannian manifold not conformally equivalent to the standard sphere. We suppose there

§8. Other Problems

179

exist m Riemannian manifolds (Mi ,9i) (i = 1,2, ... , m) such that (Mo, 90) is a Riemannian covering manifold of (Mi , 9i) with bi sheets (lI'i : (Mo,90) --+ (Mi ,9i), 1 = bo < bi < ... < bm ). Iffor each i there exists k i E [0,1] such that Ck, (Mi, gil

(L.

dVo

t

n

"n(n - 2)w;'n

[(I - ki)b;/n - b;~~1/4

then on (Mo, 90) there exist at least m + 1 metrics conformal to go with same constant scalar curvature (and different critical values of J). Ck(M, g) is the smallest positive real number C such that any satisfies (1 - k)n(n - 2)w;jn2- 2

~

Jl\1ul

2

dV +

JluI

~~ __

2n /(n-2) dV

21)! Ru

2 dV

+C

!u

2

U

E HI (M)

dV

Ck(M, g) always exists when 0 < k ~ 1 (see the value of the best constant K(n,2)). It is proven that Co(M, g) exists when the manifold has constant curvature (Aubin [14]), more generally when the manifold is locally conformally fiat (Hebey-Vaugon [* 166]), and recently by the same authors for any compact manifolds (see 4.63). For applications of the Theorem above see Hebey-Vaugon [* 167].

7.5. The Structure of the Set of Minimizers of J 5.63 Theorem (Y. Li [*Thesis, Univ. of Paris VI]). In a conformal class the set of the metrics with volume 1 and constant scalar curvature JL is analytic compact offinite dimension bounded by a constant which depends only on n. Moreover for a generic conformal class, the minimal solution is unique.

§8. Other Problems 8.1. Topological Meaning of the Scalar Curvature 5.64 We have seen that JL is a conformal invariant (Proposition 5.8), but does its sign have a topological meaning? Considering a change of metric (a nonconformal one, obviously) it is possible to prove: Theorem (Aubin [11] p.388). A compact Riemannian manifold Mn (n carries a metric whose scalar curvature is a negative constant.

> 3)

Proof According to Theorem 5.9 if we are not in the negative case, there exists a metric 9 with R ~ O. Then we consider a change of metric of the kind:

180

5. The Yamabe Problem

9ij = 'l/Jgij + ai'I/J aj'I/J with 'I/J > 0 a Coo function. It is possible to determine 'I/J such that the corresponding functional j(u) is negative for some u. Hence the result follows by Theorem 5.9. •

Since on every compact manifold Mn (n ~ 3) there is some metric with f.1- < 0, there is no topological significance to having negative scalar curvature. In contrast to this, Lichnerowicz [186] has proved that there are topological obstructions to admitting a metric with f.1- > 0, that is, to positive scalar curvature. He showed that if there is a metric with nonnegative scalar curvature (not identically zero), then the Hirzebruch A-genus of M must be zero. This work was extended by Hitchin [145], who proved that certain exotic spheres do not admit metrics with positive scalar curvature-and hence certainly have no metrics with positive sectional curvature. In a related work, Kazdan and Warner [161] proved that there are also topological obstructions to admitting metrics with identically zero scalar curvature, that is, to J.L = O. Thus there are obstructions to f.1- > 0 and J.L = 0, but not to J.L < O. More recently, Gromov and Lawson [136] and [137] proved that every compact simply-connected manifold Mn (n ~ 5), which is not spin, carries a metric of positive scalar curvature. For the spin manifolds they generalize Hirzebruch's A-genus in order to obtain almost necessary and sufficient conditions for a compact manifold to carry a metric of positive scalar curvature. In particular, the tori Tn, n ~ 3, do not admit metrics of positive scalar curvature. For the details see the article in references [136] and [137] or Bourbaki [34]. 8.2. The Cherrier Problem 5.65 It concerns the Coo compact orientable Riemannian manifolds (M, g) with boundary and dimension n ~ 2. Denote by the unit vector field defined on the boundary aM, normal to aM and oriented to the outside. When n ~ 3, let h be the mean curvature of aM. h is the trace of the following endomorphism of the vector fields X on aM : X - \l xe/(n - 1). 4 If we consider as previously the change of conformal metric 9 O. There are only two compact manifolds which are involved §2 if X = 2 and the real projective space lP2 if X = 1. We suppose M is one of these manifolds. From now on R is a positive real number. More generally than previously, we will consider the equation (73) where f E Coo is a function positive at least at one point. This property of f is necessary in order that Equation (73) have a solution, since dV = fe'P dV.

HI

I

§8. Other Problems

193

Henceforth, without loss of generality, we suppose the volume equals 1. Set /J =inf J(cp) for all cp E HI satisfying Je'P dV =R, where J(ep) is the functional (71).

J

Theorem 5.93. Equation (73) has a Coo solution if R < 87r.

Proof a) /J is finite. First of all there are functions satisfying j is positive somewhere. On the other hand, according to Theorem 2.51 or 2.53

R=

(74)

J

je'P dV

~ supj

J

e'P dV

~ C(c)supjexp [(fL2+c)II\7epll~+

J j e'P dV = R since

J

epdV].

Thus

J(ep) ~ [~ - (fL2 + c)R] II\7epll~ + R 10g(R/C(c) sup f).

(75)

fL2 = 1/167r, if R < 87r we can choose c = co > 0 small enough so that 2(fL2 + co)R < 1. Therefore /J is finite, /J ~ R 10g(R/C(co) sup f).

b)

/J

is attained. Let {cp hEN be a minimizing sequence; (75) implies [ "2I

-] - (fL2 + co)R II\7CPilb2 ~ Const, thus II\7CPiI12 ~ Const.

J

J

Moreover, J(CPi) ~ Const implies epi dV ~ Const, and by (74) epi dV ~ Const. Therefore {epdiEN is bounded in HI (Corollary 4.3). As in 5.92b it follows that /J is attained. There exists rj; E HI such that l(rj;) = /J and J je'P dV = R. c) Writing Euler's equation yields

J

\7 V rj;\7 v hdV + R

J

for any h E HI. Picking h = 1 gives K and by the proof in 5.92c, rj; E Coo.

hdV = K

J

hje'P dV

= 1. rj; satisfies Equation (73) weakly •

Chapter 6

Prescribed Scalar Curvature

§1. The Problem 1.1. The General Problem

6.1 Let (Mn, g) be a Coo Riemannian manifold of dimension n ~ 2. Given 1 a smooth function on M n , the Problem is: Does there exist a metric g' on M such that the scalar curvature R' of g' is equal to I? This problem was solved entirely by Kazdan and Warner [* 195] [* 198] [*200]. Since the equations are different for n = 2 and n ~ 3, the proofs are different as are the results. When n = 2 the scalar curvature R has strong topological meaning because the sectional curvature is fully determined by R (At a point where the coordinates are chosen orthonormal R = 2R I 212). So more often than not, we will present the proofs when n ~ 3.

6.2 Theorem (Kazdan and Warner [* 198]). Let M be a Coo compact manifold of dimension n ~ 3. If 1 E COO(M) is negative somewhere, then there is a Coo Riemannian metric on M with 1 as its scalar curvature. Proof Using Aubin [11], on any M we can choose a smooth metric 9 whose scalar curvature R = -I. This shows that negative scalar curvature has no topological meaning. In a pointwise conformal metric g' u4 /(n-2)g with u > 0, we use formula (1) in 5.2 for R'. If u E Hf(M) with p > n, then u E C1(M) according to the Sobolev imbedding theorem. Set n = {u E Hf(M)/u > o}. Now we consider the map

=

n x Lp

:3 (u, K)

where N = n2~2' respect to u is

r

--S 4(n -

I)(n - 2)-1 ~u - U - KU N - 1 E

Lp,

is continuously differentiable and its partial differential with

Duf(v) = 4(n - 1)(n - 2)-1 ~v - V - (N - I)Ku N - 2v.

At (1,-1), Dur(v) = 4(n-1)(n-2)-1 ~v+(N -2)v is invertible as an operator acting on C(Hf, Lp). Hence, by the implicit function theorem, there exists c > 0

195

§1. The Problem

such that if K satisfies 11K + Il1p < € the equation r(u, K) =r(1, -1) =0 has of the constant function u == 1. We can a solution in a neighbourhood in choose € small enough so that the functions u in this neighbourhood are positive everywhere (this follows from the Sobolev theorem). Since f is negative somewhere there exist a > 0 and r.p a Coo diffeomorphism of M such that K = afor.p satisfies 11K + 111p < €. So af is the scalar curvature of the metric 9 = (r.p-l)*(u4 /(n-2)g), where u is the solution founded above of the equation r( u, K) = O. u E Coo by the bootstrap method, since u E Cl is a solution of Llu = (n - 2)(u + Ku N - 1)i4(n - I) with K E Coo. Therefore a COO metric (homothetic to g) has f as scalar curvature.

Hi

6.3 Theorem (Kazdan and Warner [* 198]). Let M be a Coo compact manifold of dimension n ~ 3 which admits a metric whose scalar curvature is positive. Then any f E COO(M) is the scalar curvature of some Coo Riemannian metric onM.

Proof We know that there are compact manifolds, such as the torus Tn which have metrics with zero scalar curvature but no metric with positive scalar curvature. Here by hypothesis there is a metric with positive scalar curvature, hence the manifold admits a metric with zero scalar curvature. Indeed we can pass continuously from a metric with positive scalar curvature to a metric with negative scalar curvature. So we get a metric which is in the zero case: J.L = 0 (J.L is defined in 5.8). Thus we have to consider only the case f positive somewhere. By the theorems which solve the Yamabe problem, there exists a metric 9 with scalar curvature equal to + I which minimizes the Yamabe functional in the conformal class [g]. Then we procede as for Theorem 6.2. We consider on n x L p , feu, K)

=4(n -

l)(n - 2)-1 Llu + u - Ku N - 1 .

At (1,1), Dur(v) = 4(n - l)[Llv - vi(n - 1)]j(n - 2) is invertible only if > 1i(n - 1) which is not always true (we can have Al(g) = 1i(n - 1), for instance on the sphere with the standard metric satisfying R = 1). If )q (g) = 1i(n - 1), we choose a metric 9 close to 9 (so that R(g) > 0) not belonging to [g]. For 9 well chosen, a minimizing metric in [9] for the Yamabe functional, with scalar curvature equal to 1, will have its Al > li(n - 1). With this metric the proof of Theorem 6.2 will work, K = afor.p satisfying 11K - Illp < €. Al(g)

Using this result, the problem of describing the set of scalar curvature functions on Mn is completely solved if n ~ 3. To see this, note that the topological obstructions mentioned above show that there are tree cases. The first case: M does not admit any metrics with J.L ~ O. Then J.L < 0 for every metric, so the scalar curvature functions are precisely those which are negative somewhere.

196

6. Prescribed Scalar Curvature

The second case: M does not admit a metric with J.L > 0, but does admit metrics with J.L = 0 and J.L < O. This is identical with the first case except that the zero function is also a scalar curvature. The third case: M has some metric 9 with J.L > O. Any function is scalar curvature.

6.4 Theorem (Kazdan and Warner [*198]). Let M be a non compact manifold of dimension n 2:: 3 diffeomorphic to an open submanifold of some compact manifold !VI. Then, every! E COO(M) is the scalar curvature of some Riemannian metric on M. Proof Without loss of generality, we can suppose that M - M contains an open set. On !VI we pick a metric 9 with scalar curvature equal to -1. Consider a diffeomorphism ep of M such that !oep E Lp(M), and an extension of locp on !VI by defining it to be identically equal to -Ion !VI - M. Therefore given c > 0 there exists a diffeomorphism 'l1 of M such that IIJo'l1 + Illp < c. Now we can apply the proof of Theorem 6.2.

J

1.2. The Problem with Conformal Change of Metric 6.S Henceforth on this chapter we will deal with the following problem: Let (Mn' g) be a Coo Riemannian manifold of dimension n 2:: 2. Given I E COO(M) does there exist a metric 9 conformal to 9 @ E [g]), such that the scalar curvature of 9 equals I? We suppose I ¢ Const., otherwise we would be in the special case of the Yamabe problem. The problem turns out to be very special when the Riemannian manifold is (8n , go) the sphere endowed with its canonical metric. This comes from the fact that (8n , go) is the unique Riemannian manifold for which the set of conformal transformations is not compact. This result was a conjecture of Lichnerowicz solved by Lelong-Ferrand [175]. Thus the problem on (8n , go) is especially hard. It was raised by Nirenberg on 8 2 in the sixties. Chapter 4 will deal with the Nirenberg Problem. In this chapter we suppose that (Mn , g) is not conformal to (Sn, go). 6.6 Recall the equations to solve. When n = 2, we write the conformal change of metric on the form The problem is equivalent to finding a Coo solution of

9 = e'P g.

(1)

where R is the scalar curvature of (Mn, g). When n 2:: 3, we consider the change of conformal metric 9 = cp4/(n-2)g. The problem is equivalent to finding a positive Coo solution of

(2)

4(n - 1)(n - 2)-I!::t.ep + Rep = !epN-l,

§2. The Negative Case when M is Compact where N becomes

= 2n/(n -

197

2). For simplicity set D.ep + Rep

(3)

R = (n -

2)R/4(n - 1). Then (2)

=fep N-1 ,ep > 0,

where we have written f for (n - 2)f /4(n - 1) without loss of generality. As the problem concerns a given conformal class of metrics, in writing equations (1) and (2) we may use in any metric in this conformal class. So when M is compact, we choose 9 the (or one of the) metric minimizing the Yamabe functional, accordingly R = Const..

§2. The Negative Case when M is Compact 6.7 In this section we consider (l)-{3) when R (or result is in Aubin [11].

R)

are negative. The first

°

Theorem 6.7 Let (Mn, g) be a compact Coo Riemannian manifold with J.L < and n ~ 2. Given a Coo function f < 0, there is a unique conformal metric with scalar curvature f. J.L is defined in 2.1. When n

Set

Vq

~

3 we consider the functional

=infI(ep) for all ep E Aq = {ep E Hdep 2: 0,

J

fepq dV = -1 }

with 2 ::; q < N. Consider a minimizing sequence {epi}. Since J ep~ dV ::; 5U~' J fepq dV = \lepi\l2 ~ Const.; and IlepillH ~ Const. because J(epi) -+ V q. The proof proceeds now as for the Yamabe problem. In the negative case, a uniform bound in CO for the minimizers epq satisfying D.ep q+Rep q = - v qf ep~ -1 is very easy to find. At a point P where ep q is maximum D.epq(P) ~ 0, thus ep~-2(P) ~ -R/vqf(P) ~ Const.. Uniqueness is proved by Proposition 6.8 and the solution 'l1 = limq-+N epq. When n = 2 we consider the functional

5:;1"

J(ep) = Set v = inf J(ep) for all

~

J

l'Vepl2dV +

I

J

RepdV.

198

6. Prescribed Scalar Curvature

v ::; J(O) = 0 since 'P == 0 is not a solution of (I) when R = Const. and Const.. Consider a minimizing sequence {'Pi}, 0 > J('Pi) --+ v.

f ¢

J

First step. I 'Pi dVI follows from

< Const..

Obviously

J

and

J

e'Pi dV

where V =

(J J(-

~ V log

'Pi dV

~ [inf( - f)r l

e'Pi dV/V)

f)e'Pi dV =

J

RdV/ sup f

JdV.

Second step. II'PiIIHI < Const.. J('Pd < 0 implies IIVill~ < -2R

II'Pill~ ~ where CPi =

J'Pi dV > O. Furthermore the result

lI'Pi -

J'Pi dV < Const. and

cpill~ + Vcp;

::; IIV'Pill~'\11 + Vcp;

J 'Pi dVIV·

Third step. v is attained by a Coo function. The map HI :3 'P --+ e'P E Lp is compact for any p (Theorem 2.46), so a subsequence {'Pj} of {'Pi} tends to a weak solution of (1) in HI. By the bootstrap method together with the regularity theorem, the solution is smooth.

Remark 6.7. Kazdan pointed out that one does not need any assumption on the exponent q > 2 in the negative case when f is negative. He proved that the equation (3b)

~u

= g(x,u)

has a solution when the continuous function g(x, t): M x R --+ R has the property that there exists numbers a < b so that if t > b then g(x, t) < 0, and if t < a then g(x, t) > O. When g(x, u) = f(x)uluI Q- 2 - Ru, we get a positive solution. Indeed we can use the method of lower and upper solutions with b > a > O. We verify that a > 0 is a lower solution of (3b) if a is small enough: g(x, a) -

2

a( -R + inf f(x)a QxEM

I

when a < [R/ infxEM f(x)] q-2. Moreover b upper solution of (3b).

2)

> 0 =~a -

1

> [R/ SUPxEM f(x)] q-2 is an

6.8 Proposition (Aubin [14], Kazdan and Warner [*198]). If f ~ 0 on M, equation (1) (resp. equation (2» has at most one solution (resp. one positive solution).

199

§2. The Negative Case when M is Compact

We suppose I t= 0 otherwise the problem has no solution. Set n = {x E M/I(x) = O} and let Wbe a solution of (1) when n = 2 (resp. a positive solution of (2) when n ~ 3). Consider 9 = eWg when n=2 (resp. 9 = 'lj;4/(n-2)g when n ~ 3). When n ~ 3, if there is another solution, equation (2) (written in the metric

g)

(4)

4(n - 1)(n - 2)-I.6.u + lu =

lu N - 1 ,

(u = t.p/'Ij;)

would have a solution not equal to the constant function u == 1. First suppose n = 0 then u == 1 is the unique solution of (4) indeed at a point P where u is maximum .6.u(P) ~ 0 thus u(P) ~ 1; and at a point Q where u is minimum .6.u(Q) ~ 0 thus u(Q) ~ 1. If n t 0, .6.u = 0 on nand u cannot reach a maximum or a minimum on Therefore, if u > 1 somewhere on M, u attains its maximum at a point P ¢ Accordingly there is a sequence {Pd c M - n which tends to P . .6.u(P) = 0 and for Pi near enough to P, .6.u(Pi ) ~ O. Thus u(Pi ) ~ 1 and u(P) ~ 1. Likewise if Q is a point where u is minimum, u(Q) ~ 1. Similarly when n = 2, we prove that u == 0 is the unique solution of equation (1) written in the metric 9

n. n.

6.9 Proposition (Kazdan and Warner [* 198]). A necessary condition for a solution of (3) to exist is that the unique solution of (5)

~u - (N - 2)(Ru - f) = 0

is positive. A necessary condition for a solution of (1) to exist is that the unique solution of

(6)

~u

- Ru+ f = 0

is positive. In both cases this implies the weaker necessary condition

JI dV < O.

Proof. If t.p > 0 satisfies (3), multiplying both members by t.pl-N and integrating yields J f dV < O. Since u > 0, integrating (5) gives J f dV = R J u dV < O. As R < 0, the operator f = ~ - (N - 2)R is invertible (in the space of COO functions for instance). We have to prove that if (3) has a solution t.p > 0 then the unique solution u of (5) is positive. For this we compute f(t.p2-N) and find

Thus _f(t.p2-N -u) ~ O. According to the maximum principle t.p2-N -u < 0 and u > 0 (we have t.p2-N - u t Const.). Similarly, when n = 2, we prove (-~ + R)(e-I.p - u) ~ O. This yields u > e-I.p which is positive.

200

6. Prescribed Scalar Curvature

Remark. With Proposition 6.9 it is easy to find functions I satisfying JI dV < 0 such that equations (1) and (3) have no solution. For instance 1= -tl.u/(N - 2) + Ru when n 2: 3, and 1= Ru - tl.u when n = 2, where u is a function changing sign and satisfying u dV > O.

J

6.10 Proposition (Kazdan and Warner [* 198]). If I E COO is the scalar curvature of a conformal metric, any h E Coo, satisfying h ~ al for some real number a > 0, is the scalar curvature of a conformal metric. More generally, if (3) has a positive solution for some I E CO, the equation tl.u + au = hU N -

(7)

1

with

R~ a < 0

will have a positive solution for any h E CO satisfying h ~ oJ with a (1) has a solution for some I E Co, equation tl.u + a

(8)

= he'U

will have a solution for any h E

with

> O. If

R~ a O.

Proof As equation (3) has a solution for af, we have to prove Proposition 6.10 when a = 1. Let

0 ~ f(u+)N-l.

f ::; -€

for some

€

> 0,

we will have

~ _€a N - 2cpN-l ~ a N - 2 fcpN-l,

§2. The Negative Case when M is Compact

203

6.13 Theorem (Rauzy [*273]). On (Mn, g) a Coo compact Riemannian manifold of dimension n ~ 3, let I be a Coo function satisfying (11) where .A is the first eigenvalue for .6. on 0 with zero Dirichlet data (as defined in Theorem 6.12). There exists a positive constant C which depends only on 1- = sup( - 1,0) such that if I satisfies sup I

(14)

o. Equation (3) has more than one positive solution when 6 ~ n < 10 if at a point P where I is maximum .6.1(P) = 0, and when n ~ 10 if in addition IIWijkl(P)11 =f. 0 and .6..6.I(P) = o. The first part of the theorem is proved by using the mini-max method. Condition (14) means that, when 1- is given, equation (3) has a solution for any on 0 satisfying (14). For the proof of the second part of Theorem 6.13, Rauzy uses the method of points of concentration.

r

Remark 6.13. We can ask how C depends on 1-. The answer is given by Aubin-Bismuth in [* 13]. Set K = {x E Mn/ I(x) ~ O}, K must satisfy >.(K) > -R. Condition (14) IS

sup I ~ C(K) inf [- I(x)]. (M-I1)

where 0 is a neigbourhood of K such that >'(0) eigenvalue of .6. on 0 with zero Dirichlet data.

> - li, >'(0) being the first

6.14 Theorem. When n = 2, if I a Coo junction on (M2' g) satisfies I ~ 0 and I ¢ 0, there is a conformal metric with scalar curvature f. Ifwe consider 1- = sup( - 1,0) ¢ 0 as given, there exists a positive constant C such that the same conclusion holds whenever sup I ~ C. Proof Assume f ~ 0 and set 0 = {x E M/ f(x) = O}. Let W be a manifold with boundary which is a neighbourhood of O. W exists since 0 =I M. On W let w be a solution of .6.w + R = 0, for instance with zero Dirichlet data. If k is large enough let us verify that w+ = , + k is an upper solution of (1) when, = w on a neighbourhood of 0, with c W. On 0, .6.(,+k)+R 0 ~ le'"Y+k. And on M -0, as I ~ -€ for some € > 0, we will have .6.(r + k) + R 2: _€e'"Y+k ~ le'"Y+k. On the other hand when k is large enough w- = -k is a lower solution of (1) satisfying w- ~ w+. Indeed .6.w- + R = R ::; le- k = le w - for large k. The method of lower and upper solutions yields a solution of (1). For the proof of the second part of Theorem 6.14, we use Theorem 6.11. According to the proof above, - 1- is the scalar curvature of a conformal metric, so there exists a neighbourhood V of - f- in CQ such that each function in V

=

°

e

204

6. Prescribed Scalar Curvature

is the scalar curvature of a conformal metric. In V there are functions h :::: - 1which are equal on n to a positive constant C if C is small enough. Now if sup I :S C, I :S h on M and by Proposition 6.10, I is the scalar curvature of some metric conformal to g. Remark. The necessary condition of Proposition 6.9 is satisfied under the hypothesis I :S 0, I ¢ O. Indeed, G R beeing the Green function of ~ - R, the solution of (6) is u(P) = - J GR(P, Q)f(Q)dV(Q). We know that G R satisfies G R :::: 6 for some 6 > O. Thus u :::: -6 JI dV >

O.

Similarly when n :::: 3, if I :S 0 and I ¢ 0, the solution of (5) is positive. In case I changes sign, if Theorem 6.12 can be apply to the function - 1- (i.e. (11) is satisfied), there is a positive constant C(f-) such that I is the scalar curvature of a conformal metric whenever sup I :S C(f-). The proof is similar to that of the second part of Theorem 6.14. It is an alternative proof to the first part of Rauzy's Theorem.

§3. The Zero Case when M is Compact 6.15 In this case, the manifold carries a metric with zero scalar curvature. In this metric equations (1) and (3) are

(15)

~cp

= leep,

when n = 2.

cp > 0 when n :::: 3.

~cp = IcpN-I,

(16)

Obviously there are two necessary conditions:

f changes sign

(17)

J

(18)

IdV

< O.

J ~cpdV = 0

implies (17). Multiplying (15) by e-ep, (16) by r.pl-N and integrating yield (18). The zero case is not different than the positive case, we can use the variational method. Whereas the negative case, when I changes sign, is very peculiar. 6.16 Theorem. When n = 2, I E Coo is the scalar curvature of a conformal metric (equation (15) has a solution) if and only if (17) and (18) hold. Proof. Define

1I

= infllV'ul12 for all u E

A = { r.p

E HI /

Jf

e'P dV = 0 } .

§3. The Zero Case when M is Compact

205

To see that A "f 0, let 'P E Coo a function which is positive in a ball n where I > 0 and which is zero outside n, leOl.'P dV = 0 for some a > 0 since

J

J IdV < O.

Consider a minimizing sequence Ui E A. Set Ui = Vi + Ui where Ui = V-I Ui dV, 1/ = limi ..... oo IIVvil12 and leVi dV =O. As i\ =0 the set {Vi} is bounded in HI. So there exist W E HI and a subsequence {Vj} such that Vj --> 'It weakly in HI. strongly in L2 and such that eVj --> e1/J in LI since the map HI 3 'P --> e'P E LI is compact (Theorem 2.46). This implies J le1/J dV = limj ..... oo J leVj dV =0, thus 'Ij; E A and IIVII2 =1/ since II V 112 ~ lim II V V j 112 1/. We cannot have 1/ 0, otherwise 'Ij; == 0 which contradicts I dV < O. Hence 'Ij; satisfies

J

J

=

J

D.'Ij;

(19)

=

=kle1/J

with k E lR .

Multiplying both members by e-1/J and integrating implies k J I dV < O. Thus k > 0 and 'P = 'Ij; + log k is a solution of (15). Regularity follows by a standard bootstrap argument.

6.17 Proposition. When n ~ 3, if (17) and (18) hold, there is a positive C= solution 'Pq of the equation D.'P = l'P q- 1 for 2 < q < N.

Proof. Define I/q = inf IIVull~ for all U

E Aq = {

U

E HI

/u ~

0, JIuq dV = I} .

Aq"f 0 (see the proof of 6.16). Consider a minimizing sequence {ud. If no subsequence of the sequence {lluiI12} is bounded, set Vi = udlluil12. The functions Vi satisfy Ilvill2 = 1, IIVVil12 --> 0 and Iv'! dV --> O. Thus {vd is bounded in HI and Vi --> V- 1/2 in HI (V is the volume). This implies a contradiction with (18) since we would have I dV =Vql2lim Iv'! dV =O. Similarly I/q "f 0, otherwise as we know now that {Ui} is bounded in HI, the constant function is a minimizer and this implies a contradiction. Consequently I/q > 0 and there exists a subsequence {Uj} which is bounded in HI. As HI C Lq is compact, we prove (as we did many times) the existence of a positive solution 'Pq E Coo satisfying

J

J

(20)

where I/q

D.'Pq

>0

=I/ql'PrI

and

Jl O. If at a point P where f attains its maximum PEW and tl,f(P) = 0 and IWijkl(P)1 :f 0, then f is the scalar curvature of a conformal metric when n = 6. When n 2: 7, the same conclusion holds if in addition l~ta~)1 is small enough. We use the test-functions (c + r2)I-n/2 of the original proof of the Yamabe problem. When n 2: 6 the first term after the constant A, in the limited expansion in r = d(P, Q), will be that with -IWij kd 2 if tl,f(P) = O. When n 2: 7 a term with 1tl, 2f(P)l/lf(P)1 is of the same order of that with -IWiJkLi2. We set tl,1 f = tl,f and tl, k f = tl,tl, k-I f for k > 1 entire. 6.33 Theorem (Hebey-Vaugon [* 168]). Let f be a Coo function satisfying sup f > 0 and tl,f(P) = 0 at a point P where f is maximum. Then f is the scalar curvature of a conformal metric when n = 4 or 5. When n 2: 6 we suppose IWijkl(P)1 = O. The same conclusion holds if 1tl,2 f(P)1 = 0, when n = 6 or 7, and when n = 8 ifin addition I\7Wij kl(P)1 :f 0 or tl,3 f(P) = O. When n > 8 the same conclusion holds if I\7Wijkl(P)1 :f 0, tl,2 f(P) = and tl,3 f(P) = 0, or when I\7Wijkl (P)1 = 0 if 1\72Wij kl(P)1 :f 0, tl,2 f(P) = 0, tl,3 f(P) = 0 and tl, 4f(P) = 0, or when all derivatives of Wijkl vanish at P if tl, k f(P) = for all k satisfying 1 :s; k :s; nl2 - 1.

°

°

For other results when 1\72Wijkl l = 0 and l\7kWijk l(P)1 :f 0 for some k > 2 see [*168]. For the proof they use their test-functions (see 5.50) and the positive mass theorem. From Theorems 6.32 and 6.33 we get 6.34 Corollary. When n 2: 4, the set of the functions which are the scalar curvature of some conformal metric is C I dense in the set of the Coo functions which are positive somewhere.

§4. The Positive Case when M is Compact

213

Coo with sup I > 0, for any E > 0 there exists 1 satisfying and 1= sup I in a neighbourhood V of P a point where f(P) = sup f. At a point Q E V we can apply Theorem 6.32 if (M, g) is

I

Given

E

III - lile < E, sup 1 = sup I l

not locally conformally flat at P. Otherwise we can apply theorem 6.33. 6.35 Remark. All the results obtained are proved by using Theorem 6.28: Find sufficient conditions which imply that A, the inf of the functional, is smaller than A. When the function I is neither close to the constant function nor flat enough at a point where I attains its maximum, we suspect that A = A and that there is no minimizer. In this case we must use other methods (those used for the Nirenberg problem), for instance the method of isometry-concentration (Hebey's method) studied in the next paragraph or algebraic-topology methods (Bahri-Coron's method) studied in Chapter 5 (see 5.78). With this method, we can prove many results of the following type:

Theorem 6.35 (Aubin-Bahri [*11]). On a compact manifold (Mn,g), of dimension n > 4, let f be a C 2 function with only non-degenerate critical point Yo, Yl,'" Yk . We suppose fl/(Yi) > ofor 0::; i ::; t, 6.f(Yj) < Ofor l < j ::; k and I(yo) ~ ... ~ I(Yl) > I(Yl+l) ~ ... ~ I(Yk). Let Z be a pseudo-gradient for I which has the Morse-Smale property. For this pseudo-gradient we define X = Ui~l Ws(Yi), where Ws(Yi) is the stable manifold of Yi. Assume X is non-contractible, but contractible in KC for some positive real number c < I(Yl). There exists a constant Co independent of I such that if I(yo)/c ::; 1 + Co then I is the scalar curvature of some metric conformal to g. Here K C = {x E M/ I(x) ~ c} . The constant Co is of the order of 1. Let us remark that we do not assume that I is positive everywhere. On the sphere (8n , go), this theorem can be seen as a generalization of Chang and Yang's theorem (see 6.81). In [*7] Aubin-Bahri generalized this result. For compact manifolds of dimension four, (the sphere (84 , go) included), Ben Ayed-Chtioui-Hammami (E.N.LT. of Tunis) and Y. Chen (Rutgers University) obtained a nice result in [*34]. It is not a generalisation of the Bahri-Coron Theorem 6.87. The result is of a new type and the proof technically still more difficult. The hypothesis is different, although it is of the same kind.

214

6. Prescribed Scalar Curvature

§5. The Method of Isometry-Concentration 5.1. The Problem 6.36 Let (Mn, g) be any compact Coo riemannian manifold of dimension n ~ 3 and scalar curvature R ~ O. The manifold may have a smooth boundary aM. We consider a group of isometries G, which can be reduced to the identity. Given jaG-invariant Coo function on M, the problem is: Find a G-invariant metric g' conformal to 9 such that the scalar curvature R' = f. More precisely we want to find a G-invariant Coo solution of the equation (38)

O.

K2(n, 2) = 4w~2/n /n(n -

2), there exists A(C)

(44)

Using (43) and (44) in C times (42) yields

(45)

k2 1-.1.. .1.. 2( I IITJCP~i IIJv ( 1 - 2k _ 1C J.£qi Co qi (sup J) qi V q; ::; Canst. J

i.p~~ dV

I

N

»

§5. The Method of Isometry-Concentration

217

We suppose qi close to N, for instance 2 + 2/(n - 2) < qi < N. Then, it is possible to choose EO small enough so that the left hand side in (45) is positive for some ko > 1 + n/2. EO is independent of P and qi since 2

~

0< Ilqi ~ sup(l, R)lIrpqillHI ~ Const.. For k < ko (45) gives 1I1J0

j B(P,p)

and some p

4), a(x) and f(x) are radial junctions, equation (72) has a smooth solution if f(O) ~ O. The same conclusion holds when f(O) > 0, if

(n - 2)(n -

4)~f(0)

+ 8(n - l)a(O)f(O) < O.

6.60 On a smooth compact orientable Riemannian manifold (Mn, g) with boundary, the Cherrier Problem consists in finding g' conformal to 9 such that the scalar curvature of (Mn , g') and the mean curvature of 8M in (Mn , g') are given functions. The equation to solve is equation (2) (resp. (1) when n = 2) with non-linear Neumann boundary condition. We studied this problem in Chapter 5.

§7. The Nirenberg Problem 6.61 In 1969-70 Nirenberg posed the following problem: Given a (positive) smooth function f on (82 , go) ("close" to the constant function, if we want), is it the scalar curvature of a metric 9 conformal to 90 (gO is the standard metric whose sectional curvature is 1). Recall that if we write 9 in the form 9 = e'{) go, the problem is equivalent to solving the equation (75)

Since the radius (1/0:) of the sphere is chosen equal to 1, the scalar curvature

R= 20: 2 = 2.

Consider the operator r :

2. Given j a smooth function on (8n , 90), is it the scalar curvature function of a conformal metric 9 to 90. If we write 9 on the form 9 = rp4/(n-2)90 the problem is equivalent to exhibiting a positive solution of the equation

n- 1 4--/:l.rp + n(n - l)rp =

(76)

n-2

/

jrp(n+2) (n-2).

As before the differential of the operator

f' : rp

-+ rp-(n+2)/(n-2)

[4 n - 1 /:l.rp + n(n - l)rp]

n-2

is not invertible at rpl = 1, and the kernel of

n-l df 'PJ'lI) = 4 n _ 2 [/:l. 'lI - n'll] is the n+ 1 dimensional eigenspace corresponding to the first non zero eigenvalue of /:l..

§8. First Results 6.63 Let us try to solve the Nirenberg problem by a variational method. We consider the functional (77)

I(rp)

=

J

IVrpl2 dV + 4

J

rp dV

J

and the constraint je'P dV = 87r, where 47r is the volume of (82,90), Set /I = infI(rp) for rp E A = {rp E HI! J je'P dV = 87r}. First we have to prove that if rp E HI, e'P is integrable, and for the sequel, that the mapping HI 3 rp -+ e'P E L1 is compact (see Theorem 2.46). So if j is positive somewhere A is non empty. Then we must see if /I is finite. For this we need an inequality of the type (see 2.46 and Theorem 2.51): (78)

J

e'P dV ::; C(/-L) exp [/-L

J

IV rpl2 dV + V-I

J

rp dV]

which holds, on a compact manifold of dimension 2, for all rp E HI when /-L> /-L2 = 1/167r. Here V is the volume and C(/-L) a constant. On (82,90), (78) is valid with /-L 1/167r (C(/-L2) exists) and V 47r. Thus

=

87r::; supj

J

=

e'PdV::; Csupjexp[I(rp)/167r].

232

6. Prescribed Scalar Curvature

So v is finite. Unfortunately, the value of J.L2 does not enable us to prove that a minimizing sequence {epi} is bounded in HI. Indeed, I(epi) --+ v but we can have

6.64 In higher dimensions the variational method breaks down immediately. Consider the functional

J(ep) =

[4:

=~ JIV'epl2 dV +

n(n -

J [J

1) cp2 dV]

IcpN dV] -2/N

for ep E HI. By ~sing Aubin's test function (see 5.10) centered at P, a point where I is maximum, it is easy to show that

On the other hand if I == 1, we know the functions 'l1 for which l)w~n (see 5.58). For these functions if I ¢. Const., I lepN dV < sup I I epN dV. Thus if I is not constant, for any cp E HI, ep ¢. 0, J(cp) > inf J(ep). So the inf cannot be achieved. Nevertheless, J. Moser succeeded in solving the Nirenberg problem in the particular case when the function I is invariant under the antipodal map x --+ -x (82 is considered imbedded in R 3 ). J('l1) = n(n -

8.1. Moser's Result 6.65 Theorem (Moser [*245]). On (82,90) let I E Coo be a function invariant under the antipodal map x --+ -x. If sup I > 0, I is the scalar curvature of a metric conformal to 90. If ep satisfies (1), I leI{) dV = 81l'. So the condition sup I > 0 is both necessary and sufficient. As I is antipodally symmetric, we can pass to the quotient on P2 (1R). Now on P2 (1R) the problem of prescribed scalar curvature is entirely solved. The proof is written up in 6.25. The variational method works on P2(1R). The reason is that the volume of P2(1R) is half of that of the sphere. With V = 21l' in (78), it is possible to prove that a minimizing sequence is bounded in HI. Remark. For n ~ 3, we can consider the same problem as Moser. We will deal with this subject in a more general situation when I is invariant under a group of isometries (see §9), not only under the antipodal map.

233

§8. First Results

8.2. Kazdan and Warner Obstructions

6.66 Theorem (Kazdan and Warner [* 195]). Let F be the eigenspace corresponding to the first non zero eigenv Al = 2 of the laplacian of the unit sphere (S2,90). If

..) is convex. Similarly in the case n = 2. By the implicit function theorem it is proved that w p, up and Ap are continuous in p. In particular A : B - t ]Rn+1 is a continuous map. If Ap = 0 at some q E B with Iql < 1, equation (75) (resp. (76» has a solution, and / is the scalar curvature of some metric in [90]' Indeed Wq satisfies (91) with Aq = 0:

uf

Llwq + 2 =fqe Wq •

(93)

Thus W = (Wq -log IdetDt.pql) 0 t.pql is a solution of Equation (75), where t.pq = t.pQ,t with q = (1 - l/t)Q. Similarly when n ~ 3, u q satisfies (92) with Aq =0: (94)

n- 1

N-l

4-2 Lluq + n(n - l)uq = /qU q

n-

.

Thus v = (u qI det Dt.pq 1- 1/ N) 0 t.pq I is a solution of Equation (76). To finish the proof, suppose A does not vanish. Under the non-degeneracy condition 1 in 6.80, it is shown in [* 85] that G(P, t) does not vanish for t large enough and that deg(A, B, 0) = deg(G, B, 0). Thus the condition deg( G, B, 0) somewhere in B.

#- 0

implies the contradiction and A vanishes

243

§lO. The General Case 10.2. Dimension Two

6.82 Theorem (A. Chang and Yang [*81]). Let I > 0 be a Coo function on (82 ,90) with only nondegenerate critical points, where 1:::..1 does not vanish. Suppose I has p + 1 local maxima and q =f p saddle points where 1:::..1 > 0; then f is the scalar curvature of a metric in [90]. Recently Xu and Yang [*319] pointed out that we can remove the hypothesis

f >0.

Set n = {x E 8z/ I(x) > o} =f 0. Suppose I has only nondegenerate critical points where I:::..f(x) =f 0 when x E n. If, on n, I has p + 1 local maxima and q =f p saddle points where 1:::..1 > 0, then / is the scalar curvature of a metric in [90].

The critical points where / ~ 0 do not matter. This is not surprising, since concentration phenomena can happen only at points where / > 0 (see 6.42). Before these theorems, there were partial results in Chang and Yang [*81] and Chen and Ding [*88]. The proofs are quite different than that of Theorem 4.21 which was recently improved by removing the condition "close to constant". 6.83 Theorem (A. Chang, Gursky and Yang [*78]). Let / > 0 be a Coo function on (82 ,90), such that I:::../(Q) =f 0 whenever Q is a critical point of f. If deg(G, B, 0) =f 0, then / is the scalar curvature of a metric in [90]. This result generalizes Theorem 6.82: / may have degenerate critical points. Moreover the assumption is weaker. Indeed, when / has only nondegenerate critical points, the hypothesis q =f p (or p + 1 - q =f 1) is equivalent to the index counting condition: (95) Q critical,

~(Q»O

where k(Q) denotes the Morse index of / at Q, and it is shown in [*78] that (95) implies the hypothesis deg(G, B, 0) =f 0 in any dimension. For the proof of Theorem 6.83, consider the family of functions: (96)

/s = s/ + 2(1 -

s).

If So > 0 is small enough, we can apply Theorem 6.81. So there exists a Coo function Wso solution of (75) with / = Iso. Moreover it is shown in [* 85] that this solution is unique if So is small enough. Now we will solve for s E [so, 1] the following continuous family of equations (97) by using the method of topological degree. The critical points Q of /s are those of / and when s E [so, 1], II:::.. Is (Q)I = sll:::..f(Q)1 ~ sol I:::../(Q) \ ~ € for some € > O. Indeed suppose there is a sequence

244

6. Prescribed Scalar Curvature

Qi of critical points of I such that 6./(Qi) ---+ O. By passing to a subsequence, Qi ---+ Q which is a critical point of I where 6./(Q) = O. This is in contradiction with the hypothesis. Moreover I > 0 implies 0 < m 5 Is 5 M for some m and M independent of 8 E [80,1]. Thus we can apply Proposition 6.84 below to the solution of (97). These solutions satisfy Ilwl12,a < C for some constant C. Set

and consider the map: (98)

where Ps = log [J Ise w dV/87r]. We verify that Ws(w) 0 implies Ws w - Ps is a solution of (97). Conversely if Ws is a solution of (97), W Ws - Ws dV/47r satisfies Wiw) O. Now as w ---+ 6.- 1 (Jse W - ps ) +w is a Fredholm map n ---+ 2 ,a, continuous in 8 and 0 ¢ Ws (80) for 8 ~ 80 , deg(W s, n, 0) is well defined and independent of 8 for 8 2: 80. Equation (97) has a unique solution for 8 = 80; thus (97) has a solution for 8 = 1. For more details and the proof of the following proposition, see [*78] and [*85].

=

=

= J

c

=

6.84 Proposition (A. Chang, Gursky and Yang [*78]), Let I be a COO function on 8 2 and let ~ be the set of its critical points. Assume 6./(Q) " 0 when Q E ~ and 0 < m 5 15M for some m and M, then there exists a constant C which depends only on m, M and infQEe 16./(Q)I, such that any solution w of (75) satisfies Iwl 5 C.

J

First if f 5 M, by (4) J( O. Then there exists a constant k, which depends only on m, d, IIKllc2(S3)' a and the modulo of continuity ofV2 K on 8 3 such that

6.90 We can say that Bahii-Coron's result (6.87) and then Theorem 6.88 solve the problem of the existence of a positive solution of Equation (76) when n = 3 and f > O. Of course, we can hope to find some improvements as for dimension 2, in the case where f is not always positive. But in some sense, the hypothesis deg(G, B, 0) '1 0 or (95) is optimal, except if we find some more general topological assumption. Such hypothesis cannot be removed, since there are the Kazdan-Wamer obstructions. When n > 3, there is Theorem 6.81, and until recently, only partial results such as that of Bahri-Coron [*24]. In [*214] and [*215] Yan-Yan Li states existence results of positive solutions of Equation (76) when f is some positive function on (8n , 90)' When n =3 Li's result is similar to that of Bahri-Coron. But when n > 3, we have a new answer to the problem. As in 6.80, Li considers the leading part of fey) - f(q) in a neighbourhood of some critical point q of f. He supposes that for any q E 18 (the set of the critical points of j), there exists some real number f3 = f3(q) E ]n- 2, n[ such that the leading part R~(y) of fey) - f(q) expresses, in some geodesic normal coordinate system centered at q, in the form n

(100)

R{3(Y) =

n

~ ajIYjl{3,

where aj '10 and A(q) = ~ aj '1 O. j=1

j=1

be a positive a1 function which satisfies (100) at any q E 18. Then Equation (76) has a positive solution if ~ (_I)i(q) '1 (_l)n,

6.91 Theorem (Yan-Yan Li [*215]), On (8n ,90), n ~ 3, let

qE(!S with A(q) a the cardinality of GJ, and G I acts without fixed point. If (102)

(b/a)2/3 > 1 + b3 (

J1

dV/w3 sup 1)

I~

,

then 1 is the scalar curvature of at least two distinct me tries in [90] which are respectively G I-invariant and G 2 -invariant. Their energies are different.

6. Prescribed Scalar Curvature

248

Set gl = 0 on aWn. For the euclidean ball, H. Hamza established also some integrability conditions (see [* 155]).

Chapter 7

Einstein-Kahler Metrics

7.1 Introduction. In this chapter we shall use the continuity method and the method of upper and lower solutions to solve complex Monge-Ampere equations. But they can also be solved by the variational method. The difficulty is to obtain the a priori estimates; either method can be used indiscriminately. These equations arise in some geometric problems which will be explained. The results and proofs appeared in Aubin [11], [18] and [20], and Yau [277]. An exposition can also be found in Bourguignon [59] and [60]. We introduce some notation. Let g, w, W (respectively, g', w', W') denote the metric, the first fundamental form 7.2, and the Ricci form 7.4. For a compact manifold, V = J dV. In complex coordinates, d' and d" are defined by d' ep = 8>..epdz>" and d"ep = 8jlepdzjl. Also, let dCep = (d' - d")ep. Then ddcep = - 28>"jl ep dz>" /\ dzjl. First definitions. Let M 2m be a manifold of real even dimension 2m. We consider only local charts (n, ep), where n is considered to be homeomorphic by a map ep to an open set of em: ep(n). The complex coordinates are {z>"}, (>' = 1, 2, ... ,m). We write zX = z>". A complex manifold is a manifold which admits an atlas whose changes of coordinate charts are holomorphic. A complex manifold is analytic. A Hermitian metric 9 is a Riemannian metric whose components in a local chart satisfy for all II,/-L: gvlJ. = gOjl = 0, The first fundamental form of the Hermitian manifold is w = (i/27r)g>"jl dz A /\ dzjl, where 9 is a Hermitian metric.

§1. Kahler Manifolds 7.2 A Hermitian metric 9 is said to be Kahler: if the first fundamental form is closed: dw = O. A necessary and sufficient condition for 9 to be Kahler is that its components in a local chart satisfy, for all >., /-L, II,

252

7. Einstein-Kahler Metrics

On a Kahler manifold we consider the Riemannian connection (Lichnerowicz [184] and Kobayashi-Nomizu [167]). It is easy to verify that Christoffel's symbols of mixed type vanish. Only

r~IL = rxp' may be nonzero. Thus, if I E C 2 , then \1 )"p.1 = 8 Ap.I. On a Kahler manifold we will write t11 = -g).,p. 8)"p.1, which is half of the real Laplacian (warning!). Only the components of mixed type Ra!JAP. of the curvature tensor may be nonzero. It is easy to verify that the components of the Ricci tensor satisfy RAIL = R),p. = 0 and

where

Igl

is the determinant of the metric, g= gmt

gmm

In the real case we used the square of this determinant. ry = (i/2)m Igl dz 1 /\ dz I /\ ... /\ dz)" /\ dz>' /\ ... /\ dz m /\ dz m defines a global 2m-form. A complex manifold is orientable. 1.1. First Chern Class 7.3 'lI = (i/27r)R).,p. dz A /\ dzP. is called the Ricci form. According to (*), W is closed: d'll = O. Hence 'lI defines a cohomology class called the first Chern class: C 1(M). Recall that the cohomology class of 'lI is the set of the forms homologous to 'lI. Chern [91] defined the classes Cr(M) in an intrinsic way. For our purpose we only need to verify that Cl (M), so defined, does not depend on the metric. Indeed, let g' be another metric and 'lI' the corresponding Ricci form, let us prove that'll' - 'lI is homologous to zero. Since ry and ry' are positve 2m-forms, there exists I, a strictly positive function, such that ry' = Iry. Hence, according to (*), ,

t

).,-

'lI - 'lI = - 27r 8Ap.log I dz /\ dz IL , and the result follows from the following:

=

a)..p. dz)" /\ dzP. is homologous to zero if and only if there exists ajunction h such that a).,p. = 8).,p.h. For the necessity we suppose the manifold is compact.

Lemma. A 1-1 form I

Proof The sufficiency is established at once: 1= 8).,p.h dz )., /\ dz IL- = dd" h,

where d" h = 8p.h dzP..

Now let us consider I' a 1-1 form homologous to zero.

§1. Kahler Manifolds

253

Pick a function h such that !:l.h = _g>.ila>'il +Const (in fact the constant is zero), and define i' =a>'il dz>' /\ dz il with a>'il =8>.il h. t so ~ 9 >.p.(-a>.p. - a>.p. ) -- Cons, v v [9>'il(-a>'il - a>'il )] -- 9 >'il~ v v(-a>.p. - a>.p. ) -- 0

and 8"(i' -,) = g>'il\! >.(avp. - avp.) dz v = 0, since di' = d, = 0 implies 'ihavp. = \! va>.p. and \! >.avp. = \! va>.p.. Likewise, 8'(i'-,) = _g>'il\! il(a>'ii-a>'ii) dz ii = O. i' - , is homologous to zero and coclosed, so it vanishes (de Rham's theorem 1.72). On p-forms, the operators 8' and 8" are defined by 8' (_l)p-I * -I d' * and 8" = (_l)p-I * -I d" *, (see 1.69); they are, respectively, of type (-1,0) and (0,-1). •

=

1.2. Change of Kahler Metrics. Admissible Functions 7.4 Let us consider the change of Kahler metric: (1)

where I.{) E Coo is said to be admissible (so that g' is positive definite). Obviously g' is a Kahler metric, since (1) is satisfied. Let M(I.{) = 19'1lgl-l. Then dV' = M(I.{)dV. Since

1 + \!~I.{)

\!rl.{)

by expanding the determinant we find (la)

...

'\7~ I.{)

I

M(IIl) = 1 + ,rill + -1 \!VvI.{)

r

v-r

2

\!~I.{)

\!~I.{) where the last determinant has m rows and m columns. Remark. The first fundamental forms w' corresponding to the metrics g' defined by (1) belong to the same cohomology class (Lemma 7.3). Conversely, if two first fundamental forms belong to the same cohomology class, there exists a function I.{) such that the corresponding metrics satisfy (I).

A cohomology class, is said to be positive definite if there exists in , a Hermitian form (i/27r)C>.p. dz>' /\ dz il E , such that everywhere C>.p.ee > 0 for all vectors f o. A Kahler manifold M has at least one positive definite cohomology class defined by w. Thus the second Betti number, b2 (M), is nonzero.

e

254

7. Einstein-Kahler Metrics

If a Kahler manifold has only one positive definite cohomology class up to a proportionality constant, in particular if b2(M) = 1, the all Kahler metrics are proportional to one of the form (1).

7.5 Lemma. The Kahler manifolds (M, g') with M compact and g' defined by (1) have the same volume. Proof The determinants in (la) are divergences

\1>'

\1 >. cp

\1~cp

\1 J.tcp

\1~cp

\1I1CP

\1~cP

...

\1~cp

\1~cp

= \1~cP

\1~cp

. ..

\1~cp

\1~cp \1~cp \1~cP

\1~cP

Indeed the differentiation of the other columns gives zero, because on a Kahler manifold \1>'\1~cP = \1J.t\1~cp. So integrating (la) yields: V' = dV' = M(cp)dV = dV = V. •

Im

IM

IM

We can prove Lemma 7.5 by another method. Denote by wm (respectively, w'm) the m-fold tensor product of w (respectively, w'). w' = w - (i/47r) ddccp and

Since dw = 0, then by Stokes' formula,

V'

= 7rm!m

Jw,m

I w'm = I wm. Hence = 7rm!m

Jw m =V.

§2. The Problems 2.1. Einstein-Kahler Metric

7.6 Given a (compact) Kahler manifold M, does there exist an Einstein-Kahler metric on M? If 9 is an Einstein-Kahler metric, there is a real number k such that R>.p, = ky>.p,. The Ricci form q, = 2~R>'i-'dz>' /\ dzi-' is equal to k times the first fundamental form W, so kw E C t (M), the first Chern class and we have the following:

Proposition 7.6. A necessary condition for a compact Kahler manifold to carry an Einstein-Kahler metric is that the first Chern class is positive, negative or zero.

§3. The Method

255

We say that CI (M) is positive (resp. zero or negative) if there is a positive (1-1) form win CI(M) (resp. 0 E CI(M) or a negative (1-1) form, E CI(M)). It is easy to see that the three cases mutually exclude themselves. 2.2 Calabi's Conjecture 7.7 The Calabi conjecture ([73] and [74]), which is proved in 7.19, asserts that every form representing the first Chern class CI (M) is the Ricci form 'It' of some Kahler metric on a compact Kahler manifold (M, g). Let (i/21r)C)..p, dz).. 1\ dzP, belong to CI(M). According to Lemma 7.3, there exists an f E Coo such that C)..p, = R)..p, - 8)..p,f. Consider a change of metric of type (3), the components of the corresponding Ricci tensor in a local chart are:

So we shall have satisfies

R~p,

(2)

= C)..p" if there is an admissible function cp E Coo that

log M(cp) = f + k,

with k a constant.

By Lemma 7.5, we can compute k, k = log V -log J ef dV.

§3. The Method 3.1. Reducing the Problem to Equations 7.S If C 1(M) > 0, we consider as initial Kahler metric 9 some metric whose components g)..p, (in a complex chart) come from w = 2~g)..p,dz)..l\dzp, with wE C 1(M) as above. If C 1(M) < 0, we choose 9 such that, = -2~g)..p,dz)..1\ dzP, belongs to C 1(M). If C 1(M) is zero, we start with any Kahler metric. This case is a special case of the Calabi conjecture. We want to find a Kahler metric whose Ricci tensor vanishes, the zero-form belongs to C 1(M). Next we consider the new Kahler metric g' whose components are:

where cp is a Coo admissible function (see definition below). If AW E C 1(M) , since the Ricci form 'It = 2~ R)..p,dz).. 1\ dzP, E C 1(M), there exists, by Lemma 7.5, a COCJ function f such that (3)

256

7. Einstein-Kahler Metrics

If g' is an Einstein-Kahler metric, AW E C I(M) and we can choose w' homologous to w. So according to Lemma 7.5, g' is of the form (1) and R~p, = Ag~p, is equivalent to

since on a Kahler manifold, the components of the Ricci tensor are given by (5)

Definition. r.p admissible means that g' is positive definite. the C 2 admissible functions.

A will be the set of

If 9 is an Einstein-Kahler metric, it is proportional to a metric of the form (1), except in the null case when there are more than one positive (1-1) cohomology class. Then we proved that the problem is equivalent to solve the equation logM(r.p) = r.p+ f

(6)

log M(r.p)

=f

+k

10gM(r.p) = -r.p+ f

(7)

if CI(M)

< 0,

if CI(M)

=0,

if CI(M)

> 0,

where M(r.p) = Ig' 0 g-II = Ig'llgl-1 and f is some Coo function. The proof is not difficult. MUltiplying (4) by gAP, and integrating yield

6.[Ar.p+logM(r.p)+f] =0. Thus (8)

Ar.p + log M(r.p) + f

= Const.,

which is nothing else than equation (2) when A = 0, or equations (6) and (7), where the unknown function is r.p - Const., when A = -lor + 1. 3.2. The First Results 7.9 Equation (2) is the equation of the Calabi conjecture [*70]. T. Aubin [18], [20] and S.T. Yau [277] solved the two first equations (2) and (6), when A ~ 0. Theorem 7.9. IfCI(M) < 0, there exists an Einstein-Kahler metric unique up to an homothety. IfCI(M) = 0, there exists a unique Einstein-Kahler metric (up to an homothety) in each positive (1-1) cohomology class. For the proof, it is possible to use the variational method as in the original proof (Aubin [20] and [18]), but here the continuity method is easier.

257

§4. Complex Monge-Ampere Equation

7.10 The continuity method. Let E(t.p) = 0 be the equation to solve. We proceed in three steps: a) We find a continuous family of equations En with T E [0,1], such that EJ = E and Eo( t.p) =0 is a known equation which has one solution t.po. b) We prove that the set c.B = {T E [O,I]jEAt.p) = 0 has a solution} is open. For this, in general, we apply the inverse function theorem to the map t.p --7 ET(t.p) in well chosen Banach spaces. c) We prove that the set c.B is closed. For this we have to establish a priori estimates.

r:

§4. Complex Monge-Ampere Equation 7.11 More generally, we can consider an equation of the type M(t.p) = exp[F(t.p, x)],

(9)

where I x M 3 (t, x) --7 F(t, x) is a Coo function on I x M (or only C 3 ), with I an interval of R (9) is called a Monge-Ampere Equation of complex type. 4.1. About Regularity 7.12 Proposition. ifF is in Coo, then a C 2 solution 0/(9) is Coo admissible. F is only cr+o r 2:: 1, 0 < a < 1, the solution is c 2+r+o.

If

Proof At Q a point of M, where t.p, a C 2 solution of (9), has a minimum, 8).J.. t.p( Q) 2:: 0 for all directions A. So at Q, g' is positive definite. By continuity, no eigenvalue of g' can be zero since M( t.p) > O. Hence t.p is admissible. Consider the following mapping of the C2 admissible functions to Co: (10)

r

r:

cp

--7

F(cp, x) -logM(cp).

is continuously differentiable. Let

dr 0 has a Coo admissible solution. This problem is not yet solved. It is more difficult than the two preceding cases. First, since the linear map dr

0 may have many solutions (see Aubin [20] pp. 85 and 86). For instance, on the complex projective space, log M(cp) = -.AI cp has many solutions; these solutions come from the infinitesimal hoi om orphic transformations which are not isometries. Worse, we know that some Equations (8) have no solution, since if we blow up one or two points of projective space, the manifold obtained cannot carry an Einstein-Kahler metric according to a theorem of Lichnerowicz [185] p. 156 (see Yau [275]). We will see below that there has been great progress in the positive case (§ 13 and the ones following).

§10. A Priori Estimate for

~,tlr.p\l p.tlr.p.

Recall g,AP. are the components of the inverse matrix of «gAP. + 8Ap.r.p». Differentiating (8) yields:

.\\l vr.p + \l vf (20)

=\1 v log M(r.p) =g'OI{;\l v\l 0I{;r.p

-.\tlr.p - tlf = g'OI{;\1V\l v\l 0I(;r.p - g'OIP.g'-Y(;\1v\1 -yp.r.p\l v\l 0I(;r.p·

But from 1.13, (21)

tl'tlr.p - g'OI(;\1V\l v\l 0I(;r.p = ROI(;Ap. \l0l(; r.pg,AP. - RAp. \1~r.pg'vp.

=E

and there exists a constant C such that E satisfies (22)

Write tl'r.p = _g'AP.(g~p. - gAP.) = g,AP.gAP. - m and observe that (23)

g'OI{; g,AP.\lV\l 0IP.r.p\l v\l A(;r.p ~ (m - tlr.p)-l g,AP.\l Atlr.p\l p.tlr.p.

To verify this inequality, we have only to expand

[em - tlr.p)\1 v\l A(;r.p + \1 Atlr.pg~(;] x [(rn - tlr.p)\l 'Y \1 0IP.r.p + \l p.tlr.pg~'Y]g'OI(; g'AP.gV'Y ~ O.

(19)-(23) lead to

(24)

tl' A ~ k(rn - g,)..p.g)..p.) - (rn - tlr.p)-I(E - .\tlr.p - tlf).

At a point P where A has a maximum, tl' A ~ O. We find, using (22), (25)

(k - C)g,vp.gvp. ~ (rn - tlr.p)-l(.\tlr.p + tlf) + mk.

Since the arithmetic mean is greater than or equal to the geometric mean,

265

§10. A Priori Estimate for I:::.t.p

Thus at P inequality (25) yields (26)

(k - C)g,v[LgV[L

:S mk - A+ (A + I:::.! /m)e-(>"'P+f)fm.

However, g'V[Lgvji, 2: m[M(cp)]-\/m, so that (27)

m(k - C) - A -

fl.! /m :S (mk -

A)e(>"'P+ f)/m.

Pick k such that m(k - C) 2: 1 + SUp(A, 0) + sup(I:::.j)/m, expressions (26) and (27) lead to: There exists a constant Ko such that at P (28)

Ko depends on Ao, F\, and the curvature through C. In an orthonormal chart at P for which 8v[Lcp = 0 if 1/ :f J.L

Taking the sum and using (28) yields at P

Hence everywhere

where K is a constant depending on Ko and Fo. The inequality of Proposition 7.21 now follows since k 2: A/m. If fJl is bounded in CO that is Icpl :S ko, using (29) for all cp E fJl we have I:::.cp uniformly bounded: II:::.cp I :S k\. Therefore, in an orthonormal chart adapted to cp (8v fi, = 0 if /I :f J.L),

and (1 + 8JJ.[Lcp)-1 :S (m + k l )m-l[M(cp)]-1 < (m + kdm-1ekol>"I+!. Thus the metrics g~, cp E fJl, are equivalent to g; for all directions J.L e-kol>"l-sup!(m + k )l-m g I

< g'JJ.JJ._ < (m + k)g I f.LJJ.'

JJ.JJ. -

•

266

7. Einstein-Kahler Metrics

§11. A Priori Estimate for the Third Derivatives of Mixed Type 7.22 Once we have uniform bounds for Icpl and I~cpl, to obtain estimates for the third derivatives of mixed type, consider (30)

The choice of this norm instead of a simpler equivalent norm (in the metric g, for instance) imposes itself on those who make the computation. We now give the result; the reader can find the details of the calculation in Aubin (11) pp. 410 and 411.

Lemma. _~/lcpI2 = g'AfL g,a(J g'ab g'cd[(\7fLabcCP - \7fL'Y bCP \7a6ccpgl'Y6) (conjugate expression)

+ (\7 AabcCP - \7 AbpCP\7 avccpg'PV - \7 AVCcp\7 pbacpg'PV) x (conjugate expression)] _ g'cd(2gla6 gl'YP g'ab

+ g,a(J gla6 g''Yb)\7 abccp\7 Padcp[\7 'Y6(>'CP + f) - R'Y6] + gla pg,abg'cd[\7 /'3adcp\7 abc(>'cp + f) + \7 OIbcCP\7 /'3ad(>'CP + f)] + gIAfL gla /'3glab gICd[\7 (3adr - -In(RVcM- \7 aJ1.- v In - In r + R~bfLa \7 Apcr + R~fLa \7 AbvCP) + conjugate expression + g,a(J g,cd[\7 (Jadcp(g'AfL\7 AR~fLa - glab\7 aRcb)

+ conjugate expression]. Hence there exists a constant k2 which depends on >'0, curvature such that

IlaJllco, 1if6'lic3, and the

(31)

Proposition. There exists a constant k3, depending only on >'0, 11~llc(). and the curvature, such that \7 AfLvCP \7 AfLv cP :S k3, for all cP E ~.

11f6'llc3,

Proof Equations (20) and (21) give (32)

~' ~cP = g'OIfL 9 1'1/'3 \71,1 \7 'YfLCP\7 1,1\7 a(JCP - >'~CP - ~f + E.

As all metrics g' are equivalent (Proposition 7.21), there exists a constant B such that g'a fL gl'Y/'3\7 V\7'YfLCP\7 v \7 apCP ~ BI1P12. Let h

> 0 be a real number. According to (31),

>0

267

§12. The Method of Lower and Upper Solutions

Picking h (33)

= 2k2 B- 1, we get

~/(1?fJ12 - h~ep) ~ -(k2/2)1?fJ1 2 + k2/2 + 2k2B-I(A~ep + ~f - E).

At a point P where nonnegative. Thus

17Jl12 -

1?fJ(P)1 2 So by Proposition 7.21,

h~ep has a maximum, the first member of (33) is

~ 1 + 4B-l(A~ep(P) + ~f(P) - E(P)).

1?fJ(P)j2

~ Const. Hence everywhere,

1?fJ1 2 ~ Const.

7.23 Proposition. .on a compact Kahler manifold, let £f be a set ofe5 admissible functions, f!4 bounded in CO. Let (ep, A) E £f x [-Ao, AO] with Ao a constant, and let f = log M(ep) - Aep.

If the set rc of corresponding functions f

in

e 2M for all 0: E]O, 1[.

is bounded in e

3,

then f!4 is bounded

Proof According to Proposition 7.22, the third derivatives of mixed type of the functions ep E f!4 are uniformly bounded. Hence there exists a constant k such that for all ep E £f IV' ~epl ~ k since the gradient of ~ep involves only third derivatives of mixed type. By the properties of Green's function (Theorem 4.13), for any 0: E]O, 1[, £f is bounded in e 2+0"{t is positive definite. By the implicit function theorem, the map (t, rp) ---+ [t, f(t, rp)] is a diffeomorphism of a neighbourhood of (T, rpr) in lR x onto an open set of lR x C3+0: . So, if we can solve the equation ret, rp) = f at t =T, we can solve it when t is in a neighbourhood of T.

e

7.28 Now, let us complete part a) of the continuity method. There is a difficulty: we cannot consider equation (37) at t = even if f is chosen so that J ef dV = J dV, because Eo will have an infinity of solutions rpo (the solution is unique up to a constant) and, according to (39), the map is not invertible with respect to rp at (0, rpo). This is the reason why we consider E t for t E [c,l] with c > 0, but we have to prove the existence of rpe: for some small c. For this we consider the map lR x e 3 (t,rp) ---+ trp+ 10gM(rp) + f3 rpdV E C 3+0:, where f3 > is a given real number. is continuously differentiable and its partial differential with respect to rp

°,

r

r:

°

J

r

IS

[Dcpr(t,rp)]N·)=t'l1-~~'l1+f3! 'l1dV.

r is invertible even at t = 0. Since equation (2) has a unique solution up to a constant, the equation log M (rp) + f3 J rp dV = f has a unique solution 0, carries an Einstein-Kahler metric.

282

7. Einstein-Kahler Metrics

7.51 Proposition (Real [*275]). aG(lPm(C) = 1 where G is the compact subgroup of Aut Pm(C) generated by the permutations O'j,k of the homogeneous coordinates together with the transformations Ij,9, j = 1, 2, ... m and e E [0,211"]

O'k,j :

[zo, ... Zj, ••. , Zk, ... Zm]

-+

[zo, ... ,Zk, ••• , Zj, ... , Zm].

Proof. The Kahler potential is K =(m + l)log(1 + 2::1 Xi), where Xi = Izd 2, in Uo defined by ZO ::fO, the usual metric is g).ji. o).oji.K, (0). %z).). Since idd"(K + 0 and without holomorphic vector field has an Einstein-Kahler metric.

J

In [*71] and [*72] Calabi studied the functional R2 dV when g belongs to a given cohomology class. Note that J RdV = Const. since J RdV = 1I"m J 'lI /\ wm - l , where 'lI is the Ricci form and w the first fundamental form (see 7.1). Let [w] be a fixed class of Kahler metrics. The Euler-Lagrange equation of 8(g) = R2 dV when g E [w] is 'Va 'V{3R = 0 (or equivalently 'Va 'V fJR = 0). That is to say, the real vector field on M

J

x =gajl ('V)..R a~jl + 'V jlR a~>.. ) generates a holomorphic flow (possibility trivial, if R is constant). After this, Calabi proved that, if 9 is a critical point of 8(g), then the second variation of 8(g) with respect to any infinitesimal defonnation with 89afJ = aa{3u is effectively positive definite (it is zero if and only if oga{3 is induced by a holomorphic flow). The problem of minimizing J R2 dV for all Kahler metrics in a given class is very hard. Solving it when Cl(M) > 0 and [w] Cl(M) would prove the conjecture. Indeed, if R = Const. and RXji = 9Xji + aXji!, we have! = Const. and 9 is an Einstein-Kahler metric. To illustrate his study on 8(g), Calabi [*71] minimized 8(g) on Pm(C) blown up at one point. This Calabi conjecture is proved for m = 2 (Theorem 7.53). In [* 302] TIan discusses the problem when m > 2.

=

7.55 Fermat hypersurfaces Xm,p. Xm,p

={(ZQ, ... Zm+l) E Pm+l(C)/z& + ... + Z~+l =o}

where p is an integer satisfying 0 < p :::; m+ 1. Cl (Xm,p) > 0, the restriction of K =(m+2 - p) log(lzo)12+ ... + IZm+d 2) to Xm,p is the potential of a Kahlerian metric whose first fundamental form belongs to C l (Xm,p). Tian [*300] and Siu [*291] prove that Xm,m+l and Xm,m have an EinsteinKahler metric. Tian proves that oG(Xm,p) > m/(m + 1) if p m or m + 1. Here G is generated by (J'j,k and Ij,8 with () E [0,211"] (see 7.51). Siu applies his method. cp being an admissible function, Siu [*291] considers restricting cp to algebraic curves in M. When m = 1 we saw (§15.4) that we can obtain the CO-estimate by using the Green function. If the curves, considered by Siu, are invariant under a large group of automorphisms of M, the CO-estimate obtained is sharp enough to infer the existence of an Einstein-Kahler metric (compare with 7.41 and 7.42, (3 is larger when the volume V'is smaller or when k is larger).

=

7.56 Theorem (Nadel [*248], Real [*274]). The Fermat hypersurfaces Xm,p with 1 + m/2 :::; p :::; m + 1 have an Einstein-Kahler metric.

§17. On Uniqueness

285

Real proves that aa(Xm,p) ~ 1 when p ~ 1 + m/2, by using Proposition 7.52; he then applies Theorem 7.50. For the proof of Theorem 7.56, Nadel uses the following: 7.57 Theorem (Nadel [*248]). Let (M, g) be a compact Kahler manifold with C 1 (M) > 0 and let G be a compact group of automorphisms of M. If M does not admit a G-invariant multiplier ideal sheaf, M admits an Einstein-Kahler metric. The proof proceed by contradiction. If M does not admit an EinsteinKahler metric the CO-estimate fails to hold. We saw that inequality (60) with a > m/(m + 1) implies the CO-estimate for the functions .jl + 8>'jlcp (cp is supposed to be admissible for 90)' First we introduce the functionals J(ep, cp)

=V-I

J(cp,0)

J

(cp - cp) [wo(cpr - wo(cp)m] ,

=- L(cp, cp) + V-I

J

and

(cp - cp)wo(cp)m,

with ep and cp admissible functions for 90, V the volume and L(ep, cp) = V-lib

[J

CPtwo(cpt)m] dt

where CPt = 8cpt!8t, (t, x) ---+ CPt(x) being a smooth function satisfying CPa = cP and CPb = cpo We verify that L(cp, cp) does not depend on the choice of the family CPt, as M(cp, cp) defined by (67)

M(cp, cp)

= V-I

jb

[J

(m - Rt)CPtWo(cpt)m] dt,

where R t is the scalar curvature of the metric 90(CPt). When cp = 0, we recognize Aubin's functionals J(ep) and J(cp) (see 7.34) in J(ep,O) and J(cp,O) respectively. Bando and Mabuchi prove many properties of these functionals such as (41) and

7.61 The family of generalized Aubin' s equations on (M,90) is defined by (69)

Wo Wo

=J = V (we suppose the where f is the function satisfying (3) and Jef manifold is positively oriented). (37) is the original family of equations. For t = 0, equation (69) has a unique solution CPo. CPo satisfies L(O, CPo) = 0, and the Ricci form of woe CPo) is WOo

287

§17. On Uniqueness

Lemma 7.61 (Bando-Mabuchi [* 31)). Let {'Pt} be a Coo family of solutions of (69) on [a, b] (0 ~ a < b ~ 1), then

d

dt [l(O,'Pt) - J(O,'Pt)] 2: O.

(70)

Proof A computation leads to

According to Theorem 4.20, the right hand side is

~

O.

Theorem 7.61 (Bando-Mabuchi [*31]). Let {'Pt} be a COO family of solutions of(69) on [a,b] (O:S; a < b:S; 1), then (71)

dp,(t)

d1 = -(1 -

d

t) dt [1(0, 'Pt) - J(O, 'Pt)]

:s; 0

where p,(t) = M(O, 'Pt). Proof Multiplying (38) by the inverse of the metric gO('Pt), we have R t = m + (l - t)!:lwo(cpd'Pt. (71) follows from (68), since

7.62 Theorem (Bando-Mabuchi [*31]). Any solution 'PT of (69), 0 < T < 1, uniquely extends to a smooth family {'Pt} of solutions of (69), 0 :s; t < T + € for some c > O. In particular (69) admits at most one solution at t = T. Moreover if p,(t) is bounded from below T + c = 1.

Proof According to Aubin (see 7.27), the solution uniquely extends locally. We prove, by contradiction, that it extends until t = 0 (see [*31]). Moreover if we suppose that there are two smooth families {'Pt} and {cpd of solutions of (69) satisfying 'PT =CPT> the set ~ of the t, for which 'Pt =CPt, is open. But it is also closed since the families are smooth. Thus ~ = [0, T + c[. For the last part of the theorem, the hypothesis p,(t) ~ K implies that 1(0, 'Pt) - J(O, 'Pt) is bounded from above. The rest of the proof is similar to that of the first part. 7.63 Sketch of the proof of Theorem 7.60. Suppose (M,go) admits an EinsteinKahler metric g. Then any w in 0, the orbit of wunder Aut(M), is EinsteinKahler. Now any w E 0 is of the form w wo({iJ) for some Coo function {iJ, since wo and wbelong to C) (M).

=

288

7. Einstein-Kahler Metrics

If the first positive eigenvalue >'1 of the Laplacian ~ on (M,go(-/i;») is equal to 1, there is a necessary condition to extend -/i; = -/i;1 to a smooth family -/i;t of solutions of (69). Indeed v = (it -/i;t)t=1 must satisfy (~ - l)v = -/i;. Thus -/i; 1, Equation (3) has a solution and Theorem 8.5 is proved. If 0'0 S; 1, let us suppose for a moment the following, which we will prove shortly: the set of the functions CPa for 0' E [0, O'o[ is bounded in C 3(n). Then there exist CPaQ E C2hen) for some ~ E ]0, 1[ and a sequence (J'i -+ 0'0 such that CPa; -+ CPaQ in C2+%(n). Since CPaj satisfies (5), letting i -+ 00, we see that CPaQ satisfies (5) with 0' = 0'0' But now we can apply the inverse function theorem at CPao and find a neighborhood 30f 0'0 such that Equation (5) has a solution when 0' E 3. This contradicts the definition of 0'0 • Now we have to establish the estimates, the hardest part of the proof.

§2. The Estimates 2.1. The First Estimates 8.6 Co and C l estimates. Henceforth, when no confusion is possible. we drop the subscript 0'. Then cP = CPa E Clb (as do the constants introduced in the proof). Proof. Calabi [75] p. 113 establishes the following inequality in the special case F(x, q» = 0:

9

iiV .. R > 2(n + 1) R2 I) n(n - 1) .

He introduced A ijk = r ijk = 1- Oijkq>. A ijk is symmetric with respect to its subscripts and we can verify the following equalities: (15)

where F is written in place of F(x, q» for simplicity. A computation similar to that of Calabi (see Pogorelov [235] p. 39) leads to

C 1 and C 2 are two constants and the indices are raised using gii, for instance = gj(r itk are Christoffel's symbols of the Riemannian connection. Moreover:

r{k

AiikViikF = Aijk Oi(Ojk F - qk Ot F )

om F ) rJi om F].

- Aijkrt){O(k F - r(k - Aijkrf,Joj(F -

Thus

297

§2. The Estimates

According to (15) for some constant C 3 we get:

•

and inequality (14) follows.

8.10 Interior C 3-estimate. In this paragraph we assume that the derivatives of cp up to order two are estimated. A term which involves only derivatives of cp of order at most two is called a bounded term. First of all note that Oi[gijM(cp)] = O.Indeed,

Interchanging t and i in the last term, we obtain the result. Multiply (12) by h2M(cp) and integrate over n. Since oi[gijM(cp)] = 0, integrating by parts twice leads to:

rh2 f gimgj( OijkCP om(kcpM(cp) dx :s; Const.

In

k; 1

Set R = aga.Pgikgj( 0a.ijCP opuqJ. Since the metrics ga are equivalent, the preceding inequality implies (16)

{ h2 R dx :s; Const.

Sn

It is possible to show that R dx :s; Const, but that yields nothing more here. Let us prove by induction that for all integer p: (17)

Assume (17) holds for a given p. Multiplying (14) by h2P +2RP-IJM(cp) and integrating by parts over Q lead to: (1 - p) {h2P+2gijRP-2 oiR OjRJM(cp) dx

- {gii oih2p+2Rp-l OjRJM(cp) dx

~ _2_

r h2p+2RP+ IJM(cp) dx + C Ir h2P + RP-l/2JM(cp) dx.

n - 1 In

n

2

298

8. Monge-Ampere Equations

Integrating the second integral by parts again gives, by (15),

Thus,since(17)holdsfor p = 1 (inequality 16),(17) holds for all p. Accordingly, for any compact set K c Q and any integer p, IllPaIIHf(K) ~ Const for all 0" E [0. 1]. By the regularity theorem (3.56), for all r > 0:

In particular the third derivatives of lPa are uniformly bounded on K. 8.11 C 3 -estimate on the boundary. a) Recall (8.7), where we defined L = X k Ok with X E g. Differentiating (9) twice with respect to L gives: LZF(x, IP) = _gitgjkL(o(klP)L(oijlP) + gijL 2(Oijq». Next we compute gij oijL zIP. Since

then gij(OijL21P) = gijL2(Oijq» + 4gij(i\Xt)Xk OjtklP + gij oiixtXk) O(kq> + gij oij(X t OtXk) Okq> + 2 0i(X k OkXi).

Thus (19)

gij oijL 21P = (X k Ojtkq>gi j

+ 2 O(Xi)(XA. Oi.(Jiq>g(J( + 20iXt)

+ bounded terms. Consequently, there exists a constant

:J.

such that

Since L 2", and h vanish on the boundary L 2", + rxh ~ 0 on Q and on the boundary (20)

on, by the maximum principle

299

§2. The Estimates

b) If we get an inequality in the opposite direction, the third derivatives will be estimated on the boundary. Indeed, then we have on the boundary: (21) Consider PEOn and use the coordinates in (8.7). Differentiating (18) with respect to L yields L3({) = XiXiX k Ojik({) + bounded terms. On on, L3({) = L 3 (t/I + y) = L 3 y, so (O~22({)P is estimated. Likewise the third derivatives with respect to coordinates Xi with i ~ 2 are estimated. By (21), (Olij({)P is estimated for i andj ~ 2. To get an estimate for (Ollj({)P,j > 1, we differentiate (9) with respect to X j' This yields: gl1 0l1jC{J

= bounded terms.

Because the metrics gij are equivalent, there exists a constant Yf > 0 such that gl1 ~ Yf > 0 and the estimate of (0 lljC{J)P follows. Finally, differentiating (9) with respect to Xl yields gIl O~ 11 (() = bounded terms. Hence all the third derivatives are estimated on the boundary. c) It remains to find an upper bound for OvL 21/1 on the boundary. For the present such a bound is established only in the case n = 2. From now on, n = 2; consequently the dimension of on is equal to one. Consider a vector field X tangent to en and of norm one on an. Since the second derivatives of ({) are estimated, there exists M such that IL 21/11 ~ M on on. Recall that 1/1 = C{J - y. Let p be an integer that we will choose later and set

Let Ken be a compact set such that IIXII > 1/2 on g:ZP oa.pef on n - K to obtain g"P O"/i ef

=

-p(1 + p(p

n-

K. Compute

+ M + L 21/1)-P-lg"P o7pL21/1 + 1)(1 + M + L 21/1)-p-2g:t./J 0"L 21/1 opL21/1.

Using (19) gives (21)

gr1.P Or1.pef

= p(1 + M + L 21/1)-p-2[(p +

l)g2P oa.UI/I opL21/1 L 21/1)(X k OJ'kC{Jgi j + 20,X i)

- (1 + M + X (X'" O).Pi({)gPI

En -

+ 2 OiX')] + bounded terms.

At Q K suppose that X is in the direction of X2' According to (10), there exists a constant ko such that

8. Monge-Ampere Equations

300

Thus there is a constant ki such that (XIt OjlltCl'gi j + 2 o,Xi)(X" O).(JiCl'g" + 2 OjX') ~ k i (1 + I022ICl'I 2 + la 222 C1'I 2 ). Moreover,

where k2 is a constant which depends on K and on the preceding estimates, much as did ko and k i . Pick p such that (p + 1)k2 ~ (1 + 2M)k l . Since the third derivatives are bounded on K (by 8.9), (21) shows that (f satisfies the following inequality on

n:

g«' o«,(f ~ Const. Hence there exists a constant f such that (22) since the metrics g(1 are equivalent. Because (f and according to the maximum principle.

while on

~

are constant on

on,

an: ov(f + f ovh ~ O. This gives -P(1

+ M + L 21/1)-P-I ovLzI/I

~

-f ovh.

Hence we obtain the desired inequality (23)

c3 -estimate on n. By inequality (14) there exist two positive constants a and a such that:

..

2 n-

2

2

g'JVjjR ~ --1 [(R - a) - a ].

on

n.

Two cases can occur: R attains its maximum on or in In the first case we saw that R is bounded, while if R attains its maximum at PEn, according to the preceding inequality:

R

~

supR n

~

a+a

§3. The Radon Measure Jt(cp)

301

8.12 Using the method of lower and upper solutions (as in 7.25), and using Aubin [23] p. 374 for the estimates, we can prove the following. Theorem. Let a c IR" be strictly convex. If there exist a strictly convex upper solution Yo E C2(n) satisfying (4) and a strictly convex lower solution "Ii E CCO(n) satisfying (6) with "Ii ~ Yo, the Dirichlet problem (3) has a strictly convex solution qJ belonging to CCO(ri) when n = 2, and qJ satisfies y 1 ~ qJ ~ Yo. Coronary. If there exists Yo as above and if lim, __ co[ It 1-" exp f(x, t)] = 0, then the Dirichlet problem (3) has a strictly convex solution qJ E CCO(n) when n = 2.

Proof. By Proposition (8.2), for C( large enough "11 = Y + cch is a strictly convex lower solution satisfying (6). Thus we can choose C( so that "Ii ~ 10. Moreover "Ii E CCO(n). The preceding theorem now implies the stated result.

§3. The Radon Measure JIt(q» 8.13 Definition. For a c 2 convex function on a, we set .,H(qJ)

= M(qJ) dx 1

"

dx 2

"

••• "

dx"

and we define the Radon measure: Co(O)

3'" L",.it(qJ), --+

where Co(O) denote the set of continuous functions with compact support. This definition extends to convex functions in general, according to Alexandrov. Here we will follow the analytic approach of Rauch and Taylor [242]. First let us remark that for a C2 convex function

We are going to show by induction that d(8 h qJ) " d(8 i2 qJ) " •.. " d(8 i,.,qJ) defines a current when qJ is a con vex function and 1 ~ i 1 < i2 < ... < im ~ n. Let us begin with m = 1. Let ro be a continuous (n - I)-form with compact 1 f,(x) dx 1 " ••• " dx' " ... " dx". For such support Ken, and ro = rowe set llrollo = SUPl sIs" sUPxEKIf,(x)l. When qJ is a convex function we will define

L;=

8. Monge-Ampere Equations

302

Choose a COO positive function f with compact support Ken which is equal to one on K. Now consider a sequence of convex COO functions f{J j which converges to f{J uniformly on K. We have

But since

f{Jj

is convex,

(24)

and

Thus there exists a constant C 1 such that (25) The constants C2 (a EN) introduced depend on K and f only. In particular, they are independent of f{Jj and w. Let Wk be a sequence of Coo (n - 1)-forms which converges uniformly to w. We choose Wk such that supp Wk c K. We define fn d(8 j f{J) 1\ Wk as a distribution and letj - oc,

Then by (25)

Sn

so that d(8 i ({J) 1\ Wk is a Cauchy sequence which converges to a limit independent of the sequence Wk' We call this limit d(8 j f{J) 1\ w. It satisfies:

Sn

Sn

Now suppose we have defined d(8 j1 f{J) 1\ d(8 jz f{J) 1\ ••• 1\ d(8 im f{J) 1\ OJ for all families of integers 1 ~ i1 < i2 < ... < im ~ n,(m < n)andallcontinuous

§3. The Radon Measure ..It(t.p)

303

(n - m)-forms wwith support contained in K. Suppose we have proved that

there exists a constant C 2 such that:

As before, !!wl!o is equal to the sup of the components of w. Let W be a continuous (n - m - i)-form with support contained in K, and let Wk be a sequence of c~ (n - m - i)-forms which converge uniformly to W: Ilwk - wl!o -+ 0 when k -+x. We pick Wk such that supp Wk C K. Consider (m + i) integers i ~ it < i2 < ... < im+l ~ n. We want to define [(w) = d(OilqJ) 1\ ••. 1\ d(Oim+ IqJ) 1\ W. First we define d(Oi1qJ) 1\ ... 1\ d(Oim+ lqJ) 1\ Wk = [(Wk) by integrating by parts. For instance, if! E.@(Q), then by definition:

In

(m

So

+ i) L!(X) d(OilqJ)

=

II i q= t

1\ ... 1\

qJ d(Oi1qJ)

1\ " ' 1 \

d(Cim+1qJ)

1\

dxm+2

d(Oiq_1qJ)

1\

d(i\f)

1\

dx m+ 3

1\

1\ ... 1\

d(Oiq_1qJ)

dx"

1\ ..• 1\

0

x d(o·Im+ 1't" rn)

1\

dx m+ 2

1\ ... 1\

dx".

This equality holds if qJ E Coo. When qJ is only convex the integrals of the second term make sense by our assumption, with the continuous (n - m)forms

Then letting j

-+ 00,

On the other hand,

IL! d(OilqJ)

1\ ••. 1\

d(Oim+ lqJ)

1\

dx m + 2 1\

... 1\

dx"

I

~ sup!!! {ID(X)I dx, whereD{x) = det«o,ikqJj»is the (m + l)-determinantwith 1 ~ I,k ~ m + 1. But because qJj is convex, 2ID(x)1 ~ D 1{x) + Dix) where DI(x) = det«o,kqJj» and D2 = det«oikilqJ).

304

8. Monge-Ampere Equations

Indeed this inequality is the same as (24) on Am + 1(1R"). The extension to Am + 1 of the quadratic form defined by «0 ih cp) is a positive quadratic form whose components are the (m + I)-determinants extracted from «OihCP) with 1 5, i, h 5, n. Thus there exists a constant C 3 such that

F or instance, according to (26):

tYD1(X) dx =

t

y d(Ol CPj} /\ d(02 cp) /\ ...

= tCPjd(OlCP) /\ d(02CP)

1\

d(om+ 1 cp) /\ dxm+ 2

1 \ ' " 1\

/\ d(Om+lY) /\ dXm+2 /\ ...

1\

1\ ... 1\

dx"

d(omCPj)

dx"

5,

C3 [suplcp j lr+ 1•

Consequently, there exists a constant C 4 such that

Lettingj

-+ 00,

by (27):

Therefore [(£Ok) is a Cauchy sequence which converges to f(w) and this limit is independent of the sequence Wk' By induction we have thus defined the Radon measure vH(cp) when cP is a convex function. Moreover, for any compact set K c 0 there exists a constant C 5 such that (28)

for alll/l E C(O) with supp

l/I

c K.

8.14 Proposition. Let {cpp} be a sequence of convex functions on n which converges uniformly to cP on n. Then jl(cpp) -+ vI/(cp) vaguely (i.e. for all l/I E Co(O), l/IvI/(cpp) -+ l/Ij/(cp».

In

In

305

§3. The Radon Measure Jt(cp)

Proof. As in the preceding paragraph, the proof proceeds by induction. We use the same notation. Let W be a continuous (n - I)-form and let {w,J be a sequence of Coo (n - 1)-forms with support contained in K, and which converges uniformly to w. Obviously

By definition

But according to (25) this convergence is uniform in p:

Thus we can interchange the limits in p and in k:

I Ld(8 qJ ) 1\ W j

+ I L d(8 qJ) j

Ld(8 j qJ p)

1\

1\

WI!5; I Ld(8 qJ ) 1\ (w - Wt)1

w" - Ld(8 j qJ p )

j

1\

w"

I + I Ld(8 qJ j

p ) 1\

(w -

Wk)

I·

These three terms are smaller than 6 > 0 if we choose k such that Ilwk - wllo < 6/C1> and then p large enough. This proves the result for m = 1. We suppose now that it is true for some m < n, so we assume: for all continuous (n - m)-form OJ with support included in K,

We then prove the result for m

+ 1 in a similar way to the m =

1 case.

•

8. Monge-Ampere Equations

306

§4. The Functional J (qJ ) 8.15 Definition. For a we set:

c2 convex function which is zero on the boundary an J(ep) = -n

(29)

In

epM(ep) dx.

This definition makes sense if the convex function is not C 2 • Indeed, for a < 0 set = {x E n/ep(x) ~ a}. According to the preceding paragraph

na

-1

na

epA(ep) = rea)

makes sense. Since A( ep) is a non-negative Radon measure and ep ~ 0, rea) is an increasing function. We define

J(ep)

=

-nlim a--+O

r epj{(ep).

J~a

Of course J( ep) may be infinite. The set of the convex functions for which J( ep) is finite will play surely an important role. 4.1. Properties of J(ep) 8.16 Let us suppose that ep E C 2(Q) is a strictly convex function. As before, set gij = Oijep and let gij be the components of the inverse matrix of«gij»' We will prove that (30)

Ollep M(cp) =

oinep

L

OI2CP

... 01nep onnep

1

= n!L

°iiep Oijep

oijep Ojjep

°ikep

°ikep

where the sum is extended to all n x n determinants obtained when the n subscripts i,j, ... , k run from 1 to n. If two subscripts are equal the determinant is zero, otherwise it is equal to M(ep). Thus we see that M(ep) is a divergence:

307

§4. The Functional J(ep)

When we differentiate a column other than the first, we get a determinant which is zero. Thus integrating (29) by parts leads to (30). For the preceding proof we supposed that C{J E C 3 , but the result is true if C{J is only C 2 . In this case we approximate C{J in C2 by strictly convex Coo functions (p. We have

By passing to the limit we get the result. We will now show that the functional J(C{J) is convex on the set of the strictly convex functions C{J E C2 (Q). Let I/! E C2(Q) be a function which vanishes on the boundary. C{J + tl/! is strictly convex for It I small enough and the second derivative of J(C{J + tl/!) with respect to t at t = 0 is equal to

+ 1)

n(n

t

gij

Oil/! Ojl/!M(C{J) dx

~ O.

Indeed, we have n terms of the form

1

(n-l)!

IiI/!

0ii I/! oij I/!

oijC{J

OilC{J

OjjlP

dx

n

o ik I/!

=

tl/!

o;[gijM(C{J) Ojl/!] dx

ouC{J

and n(n - 1)/2 terms of the form

1 (n-l)!

IinC{J

0ii I/! oij I/!

Oij I/! ojj I/!

Oil C{J

0ik I/!

o jk I/!

0k/C{J

OikC{J

OJ/C{J

dx,

...

°kkC{J

which are equal to

(n

~ I)! I

t

OilP

Oi I/! OJ I/!

oijl/! OilC{J Ojjl/! OJ/C{J

OiklP

Ok I/!

ojk I/! °k/C{J

ouC{J

a I/!

OikC{J

Ojl qJ

dx.

This maybe rewritten as

(n

~ I)! I

LOil/!

OiC{J OjC{J

ij

Oi/C{J

ojj I/!

OJ/C{J

°kC{J ajk I/!

°k/lP

dx,

OulP

8. Monge-Ampere Equations

308

which is equal to In'" aj[gijM(ep) aj '"J dx. Altogether we have n(n terms. Integrating by parts gives the result.

+ 1)/2

S.17 Remark.lfn is a ball of radius rand ep a C 2 radially symmetric function vanishing on the boundary: ep(x) = g(llxll) with g(r) = O. The integrand in (30) is equal to Ig'l ft + 1p1-ft with p = Ilxll. Thus in this case

Letf be a function belonging to H~+ 1([0, rJ). f is Holder continuous. Indeed: (31)lf(b)-f(a)l=

b 1 S 1 fb fa1'(s)ds al1'(s)lft+lds 1 (ft+ 11

1

Thus, fer) = 0 makes sense, and if ep(x) equal to wn- 1 Io 1 l' In+ 1 dp.

=

1)

Ib-alft/(n+l)

f(IIxll) we can define J(ep) to be

S.lS Proposition. Let {epp} pe~, be a sequence ofconvexfunctions on n, vanishing on an, which converges uniformly to ep on Q. Then J(ep) slim inf J(epp). p-x

Thus, J( ep) is lower semi-continuous. Proof. We use the notation of 8.14. For a < 0, according to (28):

thus

In(al (ep -

epp)j(epp)

-+

Taking the lim inf when p -+ - n

r

0 when p -+

x;

In(al

00.

But we can write

leads to

ep~lt( ep)

by Proposition 8.14. Then letting a

-+

slim inf J( epp), p-oo

0 we get the desired result.

•

309

§4. The Functional J(cp)

8.19 In this paragraph we will recall some properties of convex bodies which will be useful in 8.20. As previously, 0 is a strictly convex bounded set of jRn.1f Q E a~, w(Q) will denote the direction of the vector PH where P is the origin of the coordinates of IRn (which we assume is inside Q) and H(Q) the projection of P on the tangent plane of ao at Q. We identify ro with a point on the sphere §n-l(l). We let t(Q) be the length of PH. For the proof of the next theorem, we are going to define a symmetrization procedure which is not the usual one of 2.11. It will be a consequence of the general inequality of Minkowski (see Buseman [71] p. 48) which applies to the convex bodies 0 and B and asserts that (32) where Jl. denote the Euclidean measure and dro the element of measure on the unit sphere. Since equality holds for the ball, a consequence of (32) is:

Proposition 8.19. Among the convex bodies with Sen t dw given, the ball has the greatest volume. 8.20 Let d and D be the inradius and circumradius of o.

Theorem. Let lp E C 2(!l) be a convexfunction which vanishes on an. There exists a radially symmetric function (jJ E C1(Br) (d :S r :S D), vanishing for Ilx /I = T, with the following properties: ~)

(33)

ft) Y)

ip has the same extrema as lp; J(ip):s J(lp); Jl.(Oa):S Jl.(fi a ) for all a < 0,

where fia = {x E Btl ip(x) :S a}. Proof. Let m be the minimum of cpo On [m,O] we define the function p by = (l/ron - I ) Sena t dw for m :S a < o. p is strictly increasing and Cion ]m, 0[. Indeed, p(O) = (ljWn -l) Son t dro and pea)

(34)

p'ea) = _1_

wn -

1

i

ana

IVdro I for a E ]m, 0[. lp

p'ea) is a continuous strictly positive function which goes to infinity as m because IVlp(x)l-+ 0 as d(x, Q".) -+ O. Moreover, when a -+ 0,

a -+

i

1 p'(a ) -+--dw -. ron - 1 onlVlp I

Thus a -+ pea) is invertible. Let g be its inverse function: a = g(p).

310

8. Monge-Ampere Equations

an

On I satisfies d ~ t ~ D; hence d ~ r = p(O) ~ D. If }l(nm) ::f: 0, p(m) > O. In this case we set g(p) = m for 0 ~ p ~ p(m). Thus g E C 1([0, r]), g'(O) = 0, and g' ~ O. Consider now the radially symmetric function ~(x) = g(IIxll). Obviously ~) is true and y) holds by Proposition 8.19 since

f

(35)

tdw = p(a)w"-l

COa

It remains to prove

= f~ tdw. COa

P). By (30)

where Rj(Q) are the principal radii of curvature of

because d(j = (36)

ni; t R

J(~) =

j

dw and

fOda m

ana at Q E ana. Thus

f~ IV~I"dw =

0),,-1

fOlg,[p(a)]I"da

COa

m

Applying Holder's inequality yields

0),,-1

=

f ana

dw ~

(f

aOa

IVqJl

,,)1/("+ 1)(f dw

COa

dw

)"/("+

-IVI qJ

1)

.

Consequently by (34):

Integrating with respect to a over [m, 0] gives P).

8.21 Theorem. All convex functions cp, which are zero on (37)

where D is the circumradius of n.

an, satisfy;

311

§S. Variational Problem

Proof. If cp E C2(1l), Theorem 8.20 shows that it is enough to prove the result for radially symmetric function cp. Setcp(x) = f(llxll). Wehavef(O) = inf cpandf(t) = owith t S D. Using(31) gives

And we get the result from Theorem 8.20. If cp is a convex function but not necessarily C2 • we consider a sequence of convex Coo functions !/I i which converges uniformly to cp on nIl' where a < 0 is close to zero. We can suppose that on onll • !/Ii > 2a. By Proposition 8.14,

Su,. !/Ii.,ll(!/I J .... Sn,. cp.,ll( cp).

Applying (37) to the function inf(O, !/Ii - 2a) leads to linf(!/Ii - 2a)l n + 1 letting i ....

00

~ DIIW;!{ -n La!/li.,ll(I/Ii)}

•

and then a .... 0 yields (37).

§5. Variational Problem 8.22 Let!(t) E Ck(] - 00,0]) (k ~ 0) be a strictly positive function when t#-O and greater than some e > 0 for t < to, to some real number. Set F(t) = S':III j(u) du and consider the functional f defined on the set of continuous functions on the unit ball B by: f(I/I)

= fsF(I/I(X»

dx.

We are interested in the following problem: Minimize $(1/1) over the set.s;l of convex functions which are zero on oB and which satisfy f(I/I) = i, for i > 0 some given real number. Theorem 8.22. The inf of $(1/1) for all 1/1 E.N, which we call m, is attained by a radially symmetric convex function 1/10 E Ck+2(B) which vanishes on oB and which satisfies $(1/10) = m, f(I/Io) = i, and for some v > 0 (38)

Proof. IX) First of all .s;I is not empty. More precisely if !/I ~ 0 (1/1 ~ 0) is a continuous function on B there exists a unique real number J1.0 > 0 for which f(J1.ol/l) = i. Indeed for J1. > 0 (39)

0/Af(J1.I/I) = - Ll/lj(J1.I/I) dx > 0

8. Monge-Ampere Equations

312

because the integrand is strictly negative somewhere. It is easy to verify that the hypotheses imply f(ltl/l) -4 00 as It -4 XJ and f(J1.I/I) -4 0 as It -4 O. Hence Ito exists and is unique.

f1) One can show that the inf of ~(I/I) for 1/1 Ed and that for 1/1 Ed ('\ C 2 (i3) are equal. The reader will find the details in Aubin [23J p. 370.

y) Now let 1/1 Ed ('\ C 2(1j) and ~ be the corresponding radially symmetric functions introduced in 8.20. Then [(~) ~ f(I/I). Indeed consider ~, the radially symmetric function such that J1.(Qa) = J1.(o'a) for all a < 0, where o'a = {x E BI~(x) :s; a}. ByTheorem8.l9,Il(na):s; It(na).Thus~:s; ~onBandthereforef(~) ~ f(~) since f is decreasing in 1/1. See (39). Moreover obviously, [(~) = [(1/1). Hence there exists Ito :s; 1 such that [(J1.0~) = rel/l) = l. See ex). But according to Theorem 8.20, ~(~):s; ~(I/J). Thus ~(lto~) = 1t"o+1~(~):s; ~(I/I). Therefore m is equal to the inf of ~(I/I) for all radially symmetric functions 1/1 Ed ('\ C 2(B). b) It remains for us to solve a variational problem in one dimension. That is

the aim of the following.

Jb

8.23 Theorem. The inf of ~(g) = (On-1 Ig'(r)ln+ 1 dr for all nonpositive functions 9 E H~+ 1([0, 1J) which vanish at r = 1 and which satisfy f(g) = (On-1 gF(g)r n - 1 dr = l is attained by a com:exfunction goEC k + 2 ([0, 1]) which is a solution of Equation (38) with v > 0, go satisfying go(O) = 0 and go(1) = o.

Proof. Since this is similar to several proofs done previously, we only sketch it. We already saw that this problem makes sense: 9 is Holder continuous on [0, 1] (see 8.17), and there exist functions 9 satisfying [(g) = l (see 8.22, ex». Let {gJ be a minimizing sequence. These functions are equicontinuous by (31). Applying Ascoli's theorem 3.15 there exists a subsequence of the {gil which converges uniformly to a continuous function go. Thus [(go) = l, go :s; 0, and go(1) = O. Moreover a subsequence converges weakly to go in H~+ 1([0, 1]). Thus by 3.17 the inf of /(g) is attained by go. Writing the Euler equation yields

(40)

for some real v and all function 1/1 E H~+ 1([0, 1J) vanishing at r = 1. Picking 1/1 = go we see that v > 0 (v = 0 is impossible because this would imply ~(go) = oand consequently go == 0).

§5. Variational Problem

313

I'i

We now prove that go(r) is equal to g(r) = [v I~ !(go(t»t n- l dtJl/n duo g(1) = 0, g E el([O, 1J), and (iJ,n(r», = v](go(r»rn - l . Thus for all integrable functions "I on [0, 1J :

This implies Ig~I"-lg~ sidering the expression

= g,n,

so g~

= g'

(' g~(r) = [ v o j(go(c»cn -

J

°°

g'

since

l

dt

~

0. Hence go

]l/n

= g.

Con-

,

°

°

we see that g~(I) > since go $ and therefore goer) < for r < 1. Thus g~(r) # for r > and go is 2 on ]0, 1] where g'O > 0. Moreover as r -+ 0, g~(r) '" [(v/n)l(go(O))] linr. Thus go E 2([0, 1]) and it is convex. If] E ek ,

°

go

e

E C 2 +k.

e

•

8.24 Corollary. Let f(x, t) be a eoo function on B x ] - oc, 0]. There exists a real number Vo > 0 such that the equation M(cp) = v exp f(x, cp),

(41)

has a strictly convex solution

cp/oB = 0

iP E COO(E) when

Proof. For some f. > 0 define JCt) = [; S~ lcu) du and the functional r(I/!) =

n = 2 and

+ SUP.~Eii exp f(x,

°< v =:;; Yo. c). Consider F(c)

SB F(I/!(x» dx as in 8.22.

=

By Theorem 8.22, there exist Vo > 0 and a convex function I/!o E COO(B) satisfying I/!%B

= O.

°

Obviously I/! is a strictly convex lower solution of (41) for v =:;; vo: M(I/!o) ~ Vo exp f(x, I/!o(x»

and I/!o is strictly convex since M(I/!o) > O. Then we choose p > 0 small enough so that Yo = P(llxl1 2 - 1) is an upper solution of(41) greater than t/lo, where v=:;; Vo is given. Using Theorem 8.12 we obtain the stated result.

314

8. Monge-Ampere Equations

§6. The Complex Monge-Ampere Equation 8.25 The problem. We cannot end this paragraph without discussing the complex Monge-Ampere Equation. Definition. A function cp with value in [ - X), + 00[, (q> ¢ - (0) is plurisubharmonic if it is lower semi-continuous and if the restriction of cp to any complex line is either a subharmonic function or else equal to - 00. In case q> is C2 , q> is plurisubharmonic if the Hermitian form a).jicp dz). dzji is nonnegative. Henceforth n will be a strictly pseudoconvex bounded open set in em defined by a strictly plurisubharmonic function hE CCO(Q): h/an = 0 and let u E Ck(oQ)(k 20). We consider the Dirichlet Problem

det«J).pcp))

(42)

= f(x, cp),

cp/an = u

where f(x, cp) is a non-negative function such that f1/m(X, cp) E C(Q x IR), r 2 O. Aji cp denotes the second derivative of q> with respect to z)' and zP (1 ~ A, J.l ~ m). This problem was studied by Bedford and Taylor [29] and [30], who use a very special method. They consider the upper envelope of the set of plurisu bharmonic functions which are lower solutions of (42). They first prove that this upper envelope is a solution of (42) in a generalized sense and then try to prove that this solution is regular.

a

Remark 8.25. As boundary condition we can use cp(x) -+ + 00 when x -+ an. Cheng and Yau [90] solve a problem of this kind: They set q> = U - log( - h) and write the problem using the Kahler metric g defined by g).ji = - a).ji loge - h). They now solve a Monge-Ampere equation on a complete Kahler manifold, using the continuity method. The estimates are obtained by using the methods for compact Kahler manifolds (Chapter 7) thanks to their generalized maximum principle (Theorem 3.76). 6.1. Bedford and Taylor's Results 8.26 Bedford and Taylor [29] consider the case where the function f does not depend on cp and in Bedford and Taylor [30] the following is proved: Theorem. Iff t/m(x, cp) is convex and nondecreasing in cp, then there exists a unique plurisubharmonic function cp E C(Q) which solves the Dirichlet problem (42) in a generalized sense. If k ~ 2 and f t/m(x, cp) is Lipschitz on Q x IR, then the solution cp is Lipschitz on Q. In case n is the unit ball B, if in addition r = k = 2, then the function q> has

second partial derivatives almost everywhere which are locally bounded.

§6. The Complex Monge-Ampere Equation

315

6.2. The Measure 9Jl( cp)

8.27 Recall that 0 is a strictly pseudoconvex bounded open set in em. For a continuous pI uri sub harmonic function cp on 0, it is possible to define a measure 9Jl(cp) which is equal to

in case cp is C 2 • Recall Am is m-times the exterior product and d"cp = aJ.cp dzJ.. The method used in an earlier article by Chern, Levine, and Nirenberg [93] is similar to that of 8.13. The main point is: for all compact K c: 0 there is a constant CCK) such that S1\ Am dd" cp :$; CCK) sUPai cp 1m for all plurisubharmonic function cp E C2(O).

6.3. The Functional 3(cp) 8.28 For a plurisubharmonic function cp E CCQ) n C 2(n) which is zero on the boundary an we set:

(43)

If the plurisubharmonic function belongs only to C(n) we can extend this definition by the same procedure as in 8.15. For a < 0 set Oa = {x E O/cp(x) ~ a}. Since 9Jl(cp) is a non-negative Radon measure cp9Jl(cp) = rea) makes sense and is an increasing function of a. We define

-Sa.

3(cp)

= -mlim

a .... O

r cp9Jl(cp).

In,.

The set of the continuous plurisubharmonic functions on for which ~(cp) is finite will play an important role.

n vanishing on an

6.4. Some Properties of 3( cp)

8.29 Suppose cp E C 2(n) is a strictly plurisubharmonic function vanishing on

an Set gAli = aAll cp and let gAil be the components of the inverse matrix of

«g AJj»' Integrating (43) by parts leads to:

316

8. Monge-Ampere Equations

Thus ~(lP) is the integral on n of the square of the gradient of lP in the Kahler metric g;"il. To carry out the integration by parts we need, in fact, lP E C 3 ; however we obtain the result for lP E C 2 by a density argument. Proposition. ~(lP) is convex on the set of the strictly plurisubharmonicfunctions lP E C 2(n) vanishing on an.

+ tt/! is strictly plurisubharmonic for It I small enough. Thus we have to verify that the second derivative with respect to t at t = 0 of ~(lP + tt/!) is non-negative. Integrating by parts enough times yields the following expression for this second derivative: Proof. Let t/! E C 2(n) be a function which is zero on the boundary. lP

[:t22

~(lP + tt/!)],=o = m(m - l)i m

t

gAP

a;.,t/! 0ilt/! det«oAillP» dz 1

1\

dz 1

1\

dz 2

1\ .•. 1\

dz m.

•

This is obviously non-negative.

~(lP) is lower semi-continuous: if {lPp}pE"" is a sequence of plurisubharmonic functions continuous on n and vanishing on an which converges uniformly to lP on then

Theorem.

n

~(lP)

slim inf 3( lP p). p-oo

Proof. It is similar to that in 8.18. lP, being the uniform limit of the lPP' is continuous on vanishes on an, and also is plurisubharmonic according to the definition. Thus ~(lP) makes sense. •

n,

§7. The Case of Radially Symmetric Functions 8.30 If n is a ball of radius r in

n

em and lP a C 2 radially symmetric function

vanishing on the boundary, we can write lP(z) = plurisubharmonic on g( I z II) with g( r) = o. In this paragraph we suppose that lP has these properties. In this case, by (44)

"'( )

"lP =

i 4 '2 (,)m-l m2m-l it t

W2m-l

0

1

9

9 2r

2 r

d = .21 W2m- log'm+ 1..111 r r

and we can apply Proposition 2.48 with q = m

+ 1.

dr

317

§7. The Case of Radially Symmetric Functions

Similarly we can associate to 9 the function t/I defined on the ball Br of IRm + 1 by t/I(x) = g( I x II). Then :J( ep) = tW2m- 1w';; 111 Vt/l II: ~ i and we can apply all the results of 2.46-2.50 by noting that r 2m - 1 :5 t m - 1 r m • In particular, from Theorem 2.47 we get

Theorem 8.30. If ep satisfies :J(ep) :5 1, then

where the constant C depends only on m and where

From Corollary 2.49 we get

Corollary 8.30. Set

em = 2m m(m + 1) - 2m- lW2~ _ l' Then all ep satisfy

where C depends only on n. Proposition 8.30. Let .sd be a set of functions ep for which :J( ep) :5 Const. Then the set {e 0 and Sij the euclidean tensor. 8.34 We can prove that equation (46) has a unique solution belonging to CB'([Rn) for which the tensor gij = oij({J - (p/e)cpSij defines a Riemannian metric. 3

See 8.35 and 8.36 for new results.

319

§8. A New Method

The sketch of P. L. Lions' proof is the following. First he solves in C:(lRn) an approximated equation of equation (46):

with A. > O. For that he proves the existence of a solution in Hf(Rn) of an associated stochastic control problem; and for the regularity he uses the results of Evans (112b). These results yield uniform bounds with respect to A. for the second derivatives. So he obtains a solution qJ, E C:(Rn) of equation (46), the regularity being given by Evans' results. Instead of using the diffusion processes, it is possible by using the continuity method to solve directly equation (46). In any case we must construct a sub solution qJo E C:(lRn) of (46) such that (gO)ij = OijqJO - (pje)qJolfijdefines a Riemannian metric. Then the maximum principle implies that any solution of (46) satisfies qJo :::; qJ < O. For t ~ 0 a parameter we consider the equation:

and for

0: E

]0, 1[ the operator f:

Ci· 7 (Rn)

::J

e 3 qJ -+ log det( (OijqJ - ~ qJth"ij)) E C8(Rn)

where e is the subset of functions qJ for which «gi) is a positive definite bilinear form. f is continuously differentiable and its differential df at qJ is invertible. Indeed the equation If)

(48) has a unique solution belonging to ci' "(R"). To establish this result we consider for instance the solution I/Ik E C 2•2 (B k) of (48) on Bk for Dirichlet's data equal to zero on the boundary. It is easy to prove that the set of functions {I/I k} kE ~ is uniformly bounded, then we use the Schauder Interior Estimates 3.61. At x E IRn we have for k > Ilxll + 2:

with K = Bx( 1) and the constant C independant on x. It follows that a subsequence of I/Ik converges to a solution of (48) which belongs to ci' Z(IR"). The generalized maximum principle 3.76 implies the uniqueness assertion. By Theorem 3.56 the solution belongs to C:(R").

320

8. Monge-Ampere Equations

The inverse function theorem 3.10 establishes that the set ~ of the t E [0, 1], for which (47) has a solution, is open in [0, 1]. To prove that ~ is closed we need uniform estimates of the solutions of (47) in C 3(lRft). To get them we do similar computations to those done at the beginning of this chapter, but here we use the generalized maximum principle 3.76. 8.35 Pogorelov, Cheng and Yau get Coo approximated solutions of (45) by by analysis. geometrical considerations. Here we get the functions C{J1c = It is the distinction between both proofs, because we proceed as Pogorelov [235] to have uniform estimates in C 3(K) with the compact K c: n. That is why Lions' result is not an improvement for equation (45).

"'tin

But we can apply the method to the Dirichlet problem for the complex Monge-Ampere equation 8.25. By a similar approach, we get Coo approximated solutions of the complex equation. Then we need estimates. The CO and C 1 estimates are not difficult to obtain. But ifthere exists Aubin's estimate 7.22 for the third derivatives of mixed type (and so for the gradient of the laplacian) when we assume the estimate for the laplacian, there is no complex equivalent of Pogorelov's C 2 -estimate [235 pp. 73-75]. This is still an open problem. 8.36 Note added in proofs. In June 1982, Cafarelli L., Nirenberg L. and Spruck 1. proved [*66] that in the general case they have obtained the estimate of the third derivatives of the functions ' /\ dz il belongs to C j (M) the first Chern class. Moreover, in each positive cohomology class there is a solution g, which is unique up to a homothetic change of metric. 9.13 Myers' theorem 9.6 gives obstructions for a compact manifold to carry a metric with positive Ricci curvature. On the other hand, there is no obstruction for a manifold to carry a metric with negative Ricci curvature (see Lohkamp' s result in 9.44). In Kazdan [* 194] we find other cases of non existence, such as: Theorem 9.13 (DeTurck-Koiso [* 112]). On a compact manifold (M, g), if the Ricci curvature is positive, the tensor CRij is not the Ricci tensor of any metric for c large enough. We may take c > 1 if Rij is the Ricci tensor of an Einstein metric, or if the sectional curvature of Rij considered as a metric on M, IS

:S

1/(n - 1).

When 0 :S c < 1, we can conjecture that there is no metric with Ricci (g) = cR ij , and Cao-DeTurck [*75] proved that there is no conformally fiat metric with this property. DeTurck-Koiso [* 112] also established some results of uniqueness for Ricci curvature.

326

9. The Ricci Curvature

§3. The Hamilton Evolution Equation 3.1. The Equation 9.14 One of the most famous problems in geometry is:

The Poincare conjecture. A compact simply-connected Riemannian manifold (M, g) of dimension n = 3 is diffeomorphic to 8 3 . To attack this problem we can think of trying to deform the initial metric to an Einstein metric. If we succeeded we would get a metric of constant curvature since the Weyl tensor vanishes identically when n = 3. And we know (9.2) that a compact simply connected Riemannian manifold with constant curvature is isometric to the sphere. In his theorem (9.37) Hamilton supposes that the Ricci curvature of the initial metric is positive. Of course a hypothesis of this type is necessary since 8 2 x C has non-negative Ricci curvature. (C is the circle). Actually we don't know how to express the hypothesis "simply connected" of the Poincare conjecture, by means of Riemannian invariants. 9.15 To carry out this idea, R. Hamilton [*151] introduced the following evolution equation:

a at

-g-tJ- = (2rjn)g·· - 2RtJ tJ

(11)

where gij and Rij are the components of the metric gt and the Ricci tensor of gt in a local chart.(To simplify we drop the subscript t when there is no ambiguity). The solution gt of this equation will be a smooth family of metrics on the compact manifold M, and r is the average of the scalar curvature R: r = RdVj dV. Because gt = ~v'i9Tgij =(r-R)v'i9T the volume of (M, gt) is constant. In order to make the computations easier, R. Hamilton [*151] considered the evolution equation

I

v'i9T

I

8Z?

(12)

Proposition 9.15. Suppose gt is a solution of (12). We define the function met)

so that (M, ['it) has volume 1 with 9t equation (11) with l instead of t.

= m(t)gt. Set l =Io

t

m(s) ds, then 9 satisfies

Proof First of all, in a homo thetic change of metric, the Ricci curvature remains unchanged: Rij = R ij . So f = RdV = [m(t)]n/2-1 RdV. But by hypothesis

I

1=

J

I

dV = [m(t)f /2

J

dV,

§3. The Hamilton Evolution Equation

327

hence

~m/(t) 2 met)

=

using (12). Thus f (11 ).

-~[m(t)r/2J9ija9ij dV= 2

at

= ~m/(t)/m2(t).

[m(t)]n/2JRdV

Now we verify that

a_I a _

m'(tL

al9ij = met) at 9ij = m2(t)9ij

!it

satisfies equation

- 2Rij .

9.16 Let {xi} be a normal coordinate system at P E M (see 1.25). We will write equation (12) at P in this local chart. According to the expression of the components of the curvature tensor (1):

If the coordinate system would not normal at P, there would be, in the expression of R ij , additional terms involving only 9ij, 9 kl and quadratic in the first derivatives ai 9jk. So from (13) we get the lin~arization DE(9) of the right hand side of (12): E(9) = -2 Ricci(9). We have DE(9) = -2DE(9) where E(9) was defined in (2). Equation (12) is not strictly parabolic, as (2) is not strictly elliptic (See 9.8).

3.2. Solution for a Short Time 9.17 Theorem (Hamilton [* 151], DeTurck [* 111]). On any compact Riemaniann manifold (M, 90), the evolution equation (12) has a unique solution for a short time with initial metric 90 at t = O. For the proof Hamilton used the Nash-Moser inverse function theorem [* 150], some special technique is required because equation (12) is not strictly parabolic. When this proof appeared, DeTurck [* 109] had already solved the

local existence of metrics with prescribed Ricci curvature (that we saw above §2), and then he gave a proof of Theorem 9.17 which uses Theorem 4.51 for parabolic equations. DeTurck's idea is to show that (12) is equivalent to a strictly parabolic equation (15) when n = 3 or (16) for n > 3. Let c be any constant such that Lij ~j + C9ij is positive definite at any point of (M, 90). So L -1 exists. Recall Definition 9.9: For a symmetric tensor h={h ij }, (Gh)ij = hij - ! l l hkl9ij

=

=

=

and for a I-form v {Vi}, (O*V)ij 'ViVj + 'VjVi. The second Bianchi identity implies (see 1.20): (14)

9. The Ricci Curvature

328

9.18 When n

=3, DeTurck [* 111] considers the following parabolic equation: !gij

= -2 [Rij -

a at Lij = - t:l.Lij

(15)

g(x,O)

(8* [L- 18GL])

ij]

- 2c( Lij - Cgij) - [Q(L - cg)

L j

=go(x), L(x, 0) =Ricci(go)(x) + cgo(x)

where the unknown is the pair [gij(X, t), Lij(x, t)]. Q(S) is some quadratic expression in S using the metric. This system is strictly parabolic. Indeed by (8) we have that the symbol of the right hand side of the first equation with respect to g is the symbol of minus the laplacian. Hence from Theorem 4.51 (15) has a unique solution for a short time. We have to show that this solution solves (12). For this DeTurck considers the quantities

A computation gives the evolution equations for u and P. It is a parabolic system which admits the solution u == 0 and P == O. As the initial conditions are P(x,O) =0 and 8G L(x, 0) =0, we have indeed P == 0 since the solution is unique. Since any solution of (12) is a solution of (16), the resulting solution of (12) is unique. When n ~ 4, the Weyl tensor does not vanish identically, and equation (15) involves the curvature tensor. We must introduce a new unknown Tijkl. The parabolic equation to consider is of the form. !gij =

(16)

:t !

Lij

-2[~j -

= -t:l.Lij -

(8*[L- 18GL])ij] 2c(Lij - cgij) + 2gPT gqsTipqjLrs - 2g pq LpiLjq

Tijkl = -t:l.Tijkl + quadratic expression in 1ijkl using the metric

g(x,O) = go(x), L(x,O) = Ricci(go)(x) + cgo(x), T(x, 0) = Riem(go)

where Riem(go) is the curvature tensor of go. Thanks to (8) it is obvious that this system is strictly parabolic. Hence (16) has a unique solution for a short time. We prove that this solution satisfies (12) by the same way as above for the dimension 3. The evolution equation for u, P and S =T - Riem(g) is strictly parabolic and admits the solution u == 0, P == 0 and S == O. So (12) has a solution. This solution is unique since any solution of (12) is a solution of (16).

§3. The Hamilton Evolution Equation

329

9.19 DeTurck found a simpler proof of the existence, for a short time, of solutions for the evolution equation (12). Since his proof is unpublished, we reproduce it now. As before, DeTurck replace (12) with a strictly parabolic equation. Let Tij be any symmetric tensor field on M which has the property that Tij is invertible (as a map from Tp(M) to Tp(M») at every point P of M. One could, for instance, take T equal to go. Then the equation (17)

has a unique solution for small time by the parabolique existence Theorem 4.51 For the notations 8*, B, see Definition 9.9. The proof that the right side of (17) is elliptic appears in [* 109], see also (8) in 9.9. The introduction of T breaks the diffeomorphism-invariance of (12) and renders (17) parabolic. To show how to get solutions of (12) from those of (17), we need the following two results. Proposition 9.19. Let v(y, t)(y E M, t E JR+) be a time-varying vector field on M. Then for small t, there exists a unique family of diffeomorphisms CPt : M ---+ M such that 8 0 and 8 > 0 are small and Id =9 if F = T(M) and where (Id)ijkl = t(gikgjl - gilgjk) if F = A2 (M). These are the bundle F for which we will use Theorem 9.20. We assert that, for some 8 > 0, Tf > 0 on [0, 8] and for every c > O. Then letting c - 0 yields Tt ~ 0 on [0, 8], hence on [0, T]. If not there is a first time 0(0 < 0 ~ 8) and a unit vector v E Fx o for some (xo E M) such that T6(v) = o. Thus (p(T6' go)(v), v) ~ O. As N is a polynomial, IIp(T', g) - p(T, g)1I ~ C1 11T' - Til for some constant Cl which depends only on max(IIT'II, IITID. Then

(19)

(No(v), v) ~ -C2c8

We extend v in a neighbourhood of Xo to a vector field denoted V, in such a way that v is independent of t and such that 'Vv(xo) = O. Set I(t, x) = (Tfv, v). Then I ~ 0 on [0,0] x M and at (0, xo), I = 0, %f ~ 0, df = 0 and f11 ~ o.

§3. The Hamilton Evolution Equation

331

This implies among other things (VTe(ii), ii)xo (18) gives

= 0 and (b.Te(ii), ii)x o ::; O. But

aj k a at = -(b.T(ii), ii) + (u VkT(ii), ii) + (N(ii), v) + c(8 + t) at (v, v) + c(ii, ii). At (0, xo) we get

(N(ii),ii) ::; -c[l - 2C8]

(20)

where C ~ C 2 is chosen so that gt (v, v) ::; C. If we choose 8 < 1/3C, (20) is in contradiction with (19). 9.21 Theorem (Hamilton [* 151]). Interpolation inequality for tensors. Let (Mn' g) a compact Riemannian manifold and let p, q, r be real numbers ~ 1. If l/r = lip + l/q, any tensor field Ton M satisfies [ / IVTI 2r dV] l/r

(21)

::; [Vn + 2(r - 1)] [/ IV 2TIP dV] lip [/ IT\q dV] l/q. If p = r

~

I and q = 00, then T satisfies I

[ / IVTI 2p dV] P ::;

(22)

I

[Vn + 2(P - 1)] s~ ITI [/ IV 2TIP dV] p.

Proof. Set T = (To,), a multi-index. Integrating by parts yields: / \\7T\2r dV = / ViTa V iT a l\7TI 2(r-l) dV (23)

= - / T aV iV i T a l\7T\2(r-l) dV - 2(r - 1)

J

T a\7i\7jTt3\7 iTQ \7 j Tt3\\7T\2(r-2) dV.

Now (24)

ITaviViTal2 ::; nlTI21V 2TI2

(25)

TQViVjTt3ViTQVjTt3::; ITIIV 2TIIVTI 2.

In fact, expanding ITt3Vi VjTa -).. ViTa VjTt312 ~ 0 yields a polynomial in ).. of order 2. The nonpositivity of the discriminant is of this polynomial gives (25). To verify (24), we write \TQ\7 iVjTQ - TQ\7kVkTagijlnl2 ~ 0

332

9. The Ricci Curvature

and

IT,BV'iV'jTa - Ta V' i V'jT,B12 ~ O. The first inequality is ITaV'kV' kTa 12 ~ nlTaV'i V' j Ta 12 and the second

ITaV'iV'jTaI2 ~

ITI21V'iV'jTaI 2.

Putting (24) and (25) in (23) implies (26)

JIV'TI2r

dV

~

[2(r - 1) + v'n]

As 1/p + 1/ q + (r - 1)/r

JIV'TI2r

JITIIV'2TIIV'TI 2(r-l)

= 1 the Holder inequality then implies

dV ::; [2(r - 1) + v'n] X

dV.

[/

[/ 1V'2TI P dV] lip

ITlql dVr/ q[/ IV'TI 2r dVr-I/r

which is (21). Similarly (26) implies (22) when q = 00. 9.22 Corollary (Hamilton [*151]). Let (Mn , g) be a compact Riemannian manifold and let mEN. There exists a constant C(n, m) independent of 9 such that any tensor field T satisfies (27)

/

lV'kTI2m/k dV

for all integers k with 1 ~ k

~ C(n, m) s~ ITI 2(m/k-l) / lV'm Tl 2dV ~

m - 1.

Proof Set f(O) = SUPM ITI and f(k) = [J lV'kTI2m/k dV]k/2m. Applying (21) to the tensor field (V'ilh .. ik_lTa) with p = k~' q = ;:\ and r = m/k yields (28) where we can choose C depending only on nand m. But (28) implies, as we will see below,

f(k) ~ cm(m-k) [f(O)] I-k/m [f(m)] kim

(29)

which is (27). Let us now prove (29). Set g(k) (28) we have 2g(k) - g(k + 1) - g(k - 1) ~ 2a.

L j [2g(j) - g(j + k

1) - g(j -

= log f(k)

1)]

and a

= log C.

By

~ k(k + l)a.

j=1

As (k + l)g(k) - kg(k + 1) - g(O) =

E;=l j[2g(j) -

g(j + 1) - g(j - 1)] we

find (30)

(1 + l/k)g(k) - g(k + 1) ~ (k + l)a + g(O)/k.

§3. The Hamilton Evolution Equation

333

And since (m- k)g(k+ l)-(m- k-1)g(k)-g(m) = ~;:k~l(m- j)[2g0)g0 + 1) - g0 - 1)] ~ (m - k)(m - k - l)a, we can sum this inequality with (m - k) times (30) to get

m m-k kg(k) ~ g(m)+m(m - k)a+ -k-g(O) which implies (29). 3.4. Hamilton's Evolution Equations 9.23 Theorem (Hamilton [* 151] p.274). If get) satisfies the evolution equation (12) on [0, T[, then the curvature tensor Rijkl(t), the Ricci tensor Rij(t) and the scalar curvature R(t) satisfy the following equations: (31)

a

= -fl.Rijkl + 2 (Bijkl

at Rijkl

- Bijlk - Biljk + B ikjl )

- gPq(RpjklRqi + RipklRqj + RijplRqk + RijkpRql)

(32) (33)

:t Rik = -fl.Rk

+ 2g Pa gqb RpiqkRab - 2gpq RpiRqk

~R= -fl.R + 2gijllRkR'1 at J

where Bijkl

=gPagqb RpiqjRakbl. Recall fl.Rijkl =- '\lv'\l vRijkl.

a

'1

a

. 1 a

.

'1

•

Proof As at Rik = gJ at Rijkl- RijklgJPg q atgpq, contractmg (31) by gJ gIves (32). Similarly contracting (32) yields (33). In normal coordinates for get) at P:

! rji

As

%t rji

= gmk (akRjl - ajRkl - aIRjk)'

is a tensor field in the local chart

:trji =gmk('\lkRjl- "VjRkl

(34)

- '\lIRjk ).

According to (1) we have :t Rij m 1 = Vi ( :t rji )

-

'\l j

(

:t

r~ ).

Thus we get (35)

a

at Rjkl = '\likRjl

- Rr

- '\lilRjk - '\ljkRi/

Rjml -

+ '\ljlRk

Ri Rijmk

since '\ljiRkl - '\lijRkl - 2RkmRjml = RijmkRmI - RkmRjm l . Differentiating the second Bianchi identity we obtain fl.Rijkl = Vm'\liRjmkl + '\lmVjRmikl.

334

9. The Ricci Curvature

Permuting the covariant derivatives and using the contracted second Bianchi identity we obtain (31) from (35).

c

~ 0 then R(t) ~ C for 0 ~ t < 9.24 Theorem (Hamilton [*151]). If R(O) ~ In dimension n = 3, if Rij(O) ~ 0 then Rij(t) ~ 0 for 0 ~ t < T. Moreover, when n = 3, if Rij(O) - aR(O)gij(O) ~ 0 with R(O) > 0 then for 0 ~ t < T, Rij(t) - aR(t)gij(t) ~ O. T.

Proof. The maximum principle for the heat equation implies the first result since according to (33)

a

-at R+D..R >- O.

When n = 3 the curvarture tensor expresses itself in terms of the Ricci curvature and R since the Weyl tensor W ijkl vanishes. Thus (32) becomes

~R at .. +D..R .. --

(36)

~J

~J

N ~J..

with N ij = -6gkl~IRjk + 3RRij - (R 2 - 2RkIRkl)gij. We remark now that if RijUi =0 with lui ::f 0, Nijuiu j = (2Rij R ij - R2)lu1 2 ~ O. Indeed if the eigenvalues of «Rij » are >., J.-L and zero 2Rij R ij - R2 = (>,_J.-L)2, zero is an eigenvalue since we suppose RijU i = O. Theorem 9.20 then implies the second assertion. For the third we apply the same theorem to the tensor field Tij = Rij / R - agij. Indeed we verify that

~T. + IV .. at + D..T·tJ = ~gkIVkRVIT' R ~J

~J

~J

. wIth N- ij = 2aRij + N ij - 2R -2 RklR kl R ij . As before RijUi = 0 implies NijUiU j ~ O.

9.25 The curvature tensor defines a linear operator on the space A2(M) of two differential forms (Wij): Riem(g)(w) = RijklWkl dx i 1\ dx j . Theorem 9.25 (Margerin [*233]). IfRiem(go) is positive, Riem(gt) remains positive for all 0 ~ t < T. The smallest eigenvalue >'t ofRiem(gt) satisfies >'t ~ >'0·

Proof. One more time we apply Theorem 9.20. This time F = A 2(M), T t = Riem(gd, u =0 and Nt is given by (31). We verify that Riem(g)(w) = 0 implies (Nt(w), w) ~ O. We prove the second part of the theorem by using Theorem 9.20 with T t defined by

§3. The Hamilton Evolution Equation

335

9.26 The solution gt of (12) exists on [0, T[ for some Proposition 9.17. Theorem 9.26 (Hamilton [* 151]). Proof. Set f(x, t)

= ne/(n -

If Flo

2et). ~

~

e> 0, then T

T

::;

> 0 according to nl2e.

=2f2 In thus

a

at (R - f) ~ -b.(R - f)

2

+ ;,(R + f)(R - f)

since IR· .. 12 > 'Rij > R2/n. t:) - E.g n tJ - 0 implies R t JAs R - f ~ 0 at t = 0, R - f remains ~ 0 on [0, T[. But f(x, t) t --4 nl2e, so T ::; nl2e.

--4

00

when

9.27 Let us return to the normalized equation (11). In 9.15 we have written g(l) for the solution of (11). The key point is to prove that the solution get) exists for all t ~ 0 and converges to a smooth metric when t --4 00. E. Hebey pointed out to me that, altough the following theorem is not explicitly stated in Hamilton, all the ingredients needed for its proof were proved in Hamilton. This theorem is basic in the works of Hamilton [*151], [*179], Huisken [* 179], Margerin [*233], [*234] and Nishikawa [*259]. Let Z be the concircular curvature tensor (37)

R

Zijkl = Rijkl - n(n _ 1) (gikgjl - gilgjk).

(M,g) has a constant sectional curvature if and only if Z(g) =

o.

Theorem 9.27. Let (Mn , go) be a compact Riemannian manifold of dimension n ~ 3 and scalar curvature Ro > O. If there exist positive constants n, j3, " independent oft such that on M and for all t E [0, T[ (38)

Rij(t) - nRtgij(t) ~ 0

(39)

IZ(9t)1 ::; j3R!-'Y

then the solution g(t) of equation (11) exists for alit> 0, and g(t) converges to a metric with constant positive sectional curvature when t --4 00. The proof is given in 9.36 and uses many results that we give now, following Hebey (private communication). 9.28 Proposition (Analogous to theorem 11.1 of Hamilton [* 151]). Under the hypothesis of Theorem 9.27, for any ry E ]0, I/n[, there exists a constant e(ry) independent of t such that (40)

336

9. The Ricci Curvature

Proof Ro > 0 implies there exists c > 0 such that R t

~ c according to Theorem 9.24. Thus IZ(gt)1 ~ GIRt with G) = (3c'Y. Set A (t) = I'VRI 2R- I - T/R2. Using (33) a computation gives:

~A=-LlA+~'V at R where

0
aRg-~J' ..

"''''''J -

M

=0

and

:::; C < 00 for any f E [0, f[ and f = +00.

Proof. Under dilations inequality (38) and IZII R are unchanged. Hence (48) comes from (38), Lemma 9.31 and 9.32. As ~j ~ aRgij > 0, we have if :::; Un-I (jn In where if and (j denote the volume and the diameter of (M, g). Now by definition if = 1 and according to Myers' theorem J2 :::; (n - 1)-11'2 lainfM R, thus infM R :::; constant. (48) then implies sUPM R :::; C < 00 and consequently f :::; C. Moreover df = m(f) dt and m(t)f(f) = Tt (see the proof of Proposition 9.15) yield JOT f(f) d(f) = JOT Tt dt which is equal to +00, according to Lemma 9.32.Thus f = +00. 9.34 Lemma. Under the hypothesis of Theorem 9.27, there exists a constant C > 0 such that infM R(f) ~ C for all f.

Proof. According to Lemma 9.33, IZ(f)1 < cR(f) for some c < 2n(~-I) when is large enough (t> TI)' The sectional curvature f< of (M, g) then satisfies

f

for f> T1 • By (48) there exists T2 such that for f

> T2

1[ 2 1) inf R(t) -_] :::; R(t) -_ < ( 2 1) inf R(t). -_

(50)

-4

n(n -

M

n n-

M

Now let us consider the Riemannian universal cover (M, g) of (M, g). According to the Klingenberg Theorem, the injectivity radius 8 of (M, g) satisfies 8 ~ A

(2

7r n(n-I)

infM R-)-1/2 . So there exists a constant C such that

Vol(M) ~ C(infM R) -n/2 since the sectional curvature is bounded, and we get infM R ~ [C/II]2/n where II is the number of elements in the fundamental group of (M, g). Indeed Vol(M, g) = II Vol(M, g) = II.

341

§3. The Hamilton Evolution Equation

Moreover, by Myers' Theorem v is finite since Rij > 0 (48), and of course v does not depend on the metric. 9.35 Lemma. Under the assumptions of Theorem 9.27, there exist two positive real numbers C and 8 such that for alit:

Proof From (33) and (44), we get

%t R ~ -b.R+ Z~2 and

(51)

for some constant A > O. Set A = IZl zR-z. As A is homogeneous (unchanged under dilations), A = A = IZl zR-z. _ 8A 4 i We compute B - at + b.A - Ii \1 iR\1 A. B

= R-Z~IZIZ -

at

2IZI ZR- 3aR - 21\1ZI ZR- Z

at

+ R- 2b.IZI 2 + 2R- 4 1Z1 2(1\1 RI2 - Rb.R), B (52)

Since (53)

2) _ ~ + 16Al/Z] n(n - 1) n

< AR [4(n -

+ R-zl\1RI2(2A - A) - 2R- 21\1ZI 2.

9 =m(t)g and dt = met) dt, we get - aA - 4 - -- .B = -_ + b.A - -=- \1iR\1~A at R

~ AR(16Al/Z _

4 ) + R- 2IV R12(2A - A). n(n - 1)

By (48), there exists s > 0 such that for t ~ s

Al/2et) = IZ(t)I/R(t) ~ inf( JA/2, 1/8n(n -1)). Set 0 = ~~~~b' (53) yields B ~ -2oA and by the maximum principle 26t e A(l) ~ e26s sup As for t ~ s. Hence for alIt, IZtle26t ~ C some constant since R is bounded by (49). The proof of the second part of Lemma 9.35 is similar. By virtue of (43) we have

342

9. The Ricci Curvature

~ (I\7RI2) < _R-2~(I\7Rj2) +C R- 21\7.R. 12 at R3 R 3 ~ Jk - 2R- 4 1\7 R12( -~R + 21Rii12)

~(R-31\7RI2) < _~(I\7RI2) at R3

(54)

+ i\7i(I\7RI2)\7'R R R3 t

21\7R. 12_2 1V'RI 4 +CR3 t Jk R5 Set f = R- 3 1\7RI 2+kR- 2IZI 2 for some constant k

2 _4R- 4 Inv RI 2IR··1 1J' > O. (52) and (54) yield

a

4 \7 i f\7 R ~ [ C31\7 Rjkl 2 - 2kl\7ZI 2] R- 2 atf + ~f - R i i

_ ~1\7RI2 _ 4k lZI2 ( nR2

R

- R- 21\7RI 2 ( A Pick k large enough , k

>

_

40

R

)

21~r).

n(n-I)(3n-2)c 4(n-2)2 3·

As R- 31\7RI 2 is homogeneous! = f and for have by (45):

a- --

1

n(n - 1)

4 - . -- -

2

t ~

8 (8

- -

defined above) we -

---,;;f + ~f - -=- \71f\7·R < Rf < -28f· at R 1 n(n - 1) Thus ! e 28t ~

Ct some constant. Hence

and sup M R(t) - infM R(t) :S C2d(t)e 28t . But we saw in 9.33 that the diameter d(t) of (M, g(t») is uniformly bounded, thus the result follows.

9.36 Proof of Theorem 9.27. We have

according to Lemma 9.35. The metrics g(t) are all uniformly equivalent and converge to some metric goo as t --+ 00 in CO. Using (21), (22) and (27), we see that all the derivatives of get) are uniformly bounded and get) converge to goo in the Coo topology when t --+ 00. (48) together with lemma 9.34 then implies Z(goo) = O. Thus goo has constant positive sectional curvature.

343

§4. The Consequences of Hamilton's Work

§4. The Consequences of Hamilton's Work 4.1. Hamilton's Theorems 9.37 Theorem (Hamilton [* 151 D. A compact Riemannian manifold of dimension

3, which has strictly positive Ricci curvature, carries a metric of constant positive sectional curvature. It is thus diffeomorphic to a quotient of 8 3• Proof. As Ricci (go) > 0, we have

Ro >

0 and there exists a such that Ricci (go) ~ aRogo· By Theorem 9.24, Rij(t) ~ aR(t)gij(t) for 0 ::; t < T. In order to apply the main Theorem 9.27, which implies the announced result, we have only to show that (39) IZ(gt)1 ::; {3R!--r for some positive constants {3, 'Y, this for all t E [0, T[. In dimension 3, inequality (39) is equivalent to IEij(t)1 ::; {3R!--r for some positive constants {3,'Y, since IZI 2 = IWij kL!2 + n~2lEijl2 with the Weyl tensor Wijk1 == O. Set A = R;aIR ij (t)j2 - Iii-a /3 =R;aIEij (t)1 2 with 1 < a < 2. A computation, using (32), (33) and the expression of the Weyl tensor, leads to (see Hamilton [* 152] p.285):

(55)

:t

A + boA::; 2(a - 1)R- 1V i RV i A + 2R- 1- a [(2 - a)IRijl2lEijl2 - 2Q]

where Q = l~jl4 + R[R(R2 - 5IRijI2)/2 +2Rij R k iRi]. According to Lemma 9.38 below Q ~ a21Rij 121Eij 12. Pick a such that 2 - a ::; 2a 2 , we get %t A + boA::; 2(a - 1)R- 1V i RV i A. By the maximum principle At ::; Ao for all t E [0, T[. This is the inequality we need.

Pick normal coordinates at x E M such that Rij(x) is diagonal. Let >. ~ f.L ~ v ~ 0 be the eigenvalues of Rij(x). We have R(x)

= >. + J-L + v,

!R ij (x)!2

= >.2 + f.L2 + v 2

and Q(x) = (>.2 + f.L2 + v 2)2 + (>' + f.L + v)

x

[(>' + f.L + v)(>'f.L + >.v + Vf.L -

2>.2 - 2f.L2 - 2v2) + 2>.3 + 2f.L3 + 2v 3]

Q(x) = (>. - f.L)2 [>.2 + (>. + f.L)(J-L - v)] + v 2(>. - v)(f.L - v).

Since both sides of the inequality that we wish to prove, are homogeneous of degree 4 in >., f.L, v, we can suppose >.2 + J-L2 + lI 2 = 1. This implies R2 = (>. + J-L + v)2 ~ 1, and since Rij ~ aRgij , v ~ a. Now Q(x) ~ >.2(>. - J-L)2 + v 2(f.L - v)2 ~ a 2 [(>' - f.L)2 + (J-L - v)2] and IEij 12 = [(>' - f.L)2 + (>' - v)2 + (J-L - v)2] ::; (>. - f.L)2 + (J-L - v)2. Thus the inequality is proved.

t

9. The Ricci Curvature

344

9.39 Theorem. A compact manifold ofdimension 3, for which the Ricci curvature is non-negative and strictly positive at some point, is diffeomorphic to a quotient of 8 3 . The proof comes at once from Theorem 9.37 together with the following result (Aubin [21]): If the Ricci curvature of a compact Riemannian manifold is non-negative and positive somewhere, then the manifold carries a metric with positive Ricci curvature. 9.40 Theorem (Hamilton [* 152]). A compact Riemannian manifold of dimension 4, whose curvature tensor is strictly positive, carries a metric of constant positive sectional curvature. It is therefore diffeomorphic to 8 4 or JID4(lR). Curvature tensor strictly positive means that the bilinear form on the two-forms, defined by (cp, 'l1) ---+ Riem( '1', 'l1) = Rijkl ben) > 0 are two constants depending only on n.

So, as for the scalar curvature, when n 2:: 3, the negative sign for the Ricci curvature has no topological meaning. Previously Gao and Yau [* l36] proved that any compact Riemannian manifold of dimension 3 has a metric with negative Ricci curvature. But the proof of Lohkamp is quite different and begins with the existence on ]R3, then on ]Rn, of a metric 9n which satisfies Ricci (9n) < 0 on a ball B, and 9n = [ the euclidean metric outside B. Surgical techniques are used. Then, using some deformation techniques, Lohkamp exhibits from 9n a metric 9 which satisfies (57). Lohkamp ([*226], [*227]) studied the space of all metrics with negative Ricci curvature. He also proved the following results. 9.45 Theorem (Lohkamp [*226]). A Riemannian manifold Mn ofdimension n 2:: 3 carries a complete metric 9 with negative Ricci curvature and finite volume. 9.46 Theorem (Lohkamp [*226]). A subgroup G of the group ofdiffeomorphisms of a compact manifold Mn(n 2:: 3) is the isometry group of (Mn, 9) for some metric 9 with negative Ricci curvature, if and only if G is finite.

Bochner's result asserts that the isometry group of a compact manifold with negative Ricci curvature is finite, Lohkamp proved the converse. 5.2. On the Concircular Curvature 9.47 The concircular curvature tensor Z is defined in 9.27. We saw (see Theorem 9.42) that under an hypothesis of the type (56): IZI 2 < C(n)R with R > 0, we can prove the existence of a metric with positive constant sectional curvature on a compact manifold. Instead of to have (56) satisfied at each point of M n , we can ask the following question: Can we get similar results with only integral assumptions on IZI? The components Zijkl of the tensor Z express itself in terms of Wijkl (W the Weyl tensor) and of Eij = Rij - R9ij!n. If one of the two orthogonal components of Z vanishes, Theorem 9.48 gives a first answer to this question.

§5. Recent Results

347

On a compact Riemannian manifold (Mn, go) n 2: 3, a Yamabe metric is a metric 9 such that dV(g) 1 and such that R(g) dV(g) ~ R(g)dV(g) for all metric 9 E [g] (the conformal class of g) with dV(g) = 1. We know that there always exists at least one Yamabe metric in each conformal class and that the scalar curvature R(g) is constant. If 9 is Einstein, 9 is unique in [g].

J

=

J

J

J

9.48 Theorem (Hebey-Vaugon [*170]). Let (Mn , go) be a compact Riemannian manifold with n 2: 3 and conformal invariant p,( [go]) > 0 (see 5.8). We suppose either [go] has an Einstein metric or go is locally conformally fiat. Then there exists a positive constant C(n), with depends only on n, so that if for some Yamabe metric 9 E [go], IIZ(g)II~,n/2 < C(n)R2(g), then (Mn, g) is isometric to a quotient of 8 n endowed with the standard metric. Here IIZ(g)ll g ,n/2 = [J IZ(g)ln/2dV(g)]2/n. If [go] has an Einstein metric, we can pick C(n) = [en - 2)/20(n - 1)]2 when 3 ~ n ~ 9 and C(n) = (2/5nf when n 2: 10. If go is locally conformally flat C(3) = 25/63 , C(4) = 6/64 and C(n) = 4/n(n - l)(n - 2) when n 2: 5 suffices. The last constant is optimal. Indeed on (8n - 1 x C, g), 9 the product metric with volume 1, IZ(g)1 2 = 4R2/n(n - 1)(n - 2), and 9 is a Yamabe metric when the radius of C is small enough.

Corollary 9.48 (Hebey-Vaugon [* 170]). P4(R) and 8 4, with their standard metrics, are the only locally conformally fiat manifold of dimension 4, which have positive scalar curvature and positive Euler-Poincare caracteristic. In particular if (M4, g) is not diffeomorphic to P4(R) or 8 4, M4 does not carry an Einstein metric if 9 is locally conformally fiat with R(g) > O.

Chapter 10

Harmonic Maps

§1. Definitions and First Results 10.1 Let (M, g) and (M, g) be two Coo riemannian manifolds, M of dimension nand M of dimension m. M will be compact with boundary or without and {xi}(1 :c:::; i :c:::; n) will denote local coordinates of x in a neighbourhood of a point P E M and yO: (1 :c:::; 0:' :c:::; m) local coordinates of y in a neighbourhood of j(P) E M. We consider j E C 2(M, M) the set of the maps of class C 2 of Minto M. Definition 10.1. The first fundamental form of j is h = 1* g. Its components are hij = ado:ajf{39a{3 where adO: = ~r. The energy density of f at x is e(f)x = Hhijgijt and the energy of the map f is defined by E(f) = JM e(f)dV. As 9 is positive definite, the eigenvalues of h are non negative and E(f) if and only if f is a constant map.

=0

10.2 Definition. The tension field T(f) of the map f is a mapping of Minto T(M) defined as follows. T(f)x E Tf(x)(M) and its components are:

T'Y(f)x = -b:.r(x) + gij(x)f'~{3(j(x»)odO:(x)8jf{3(x).

(1)

Proposition 10.2 (Eells-Sampson [* 124]). The Euler equation for E is T(f) = O. For any v E C (M, T(M») satisfying vex) E Tf(x)(M) and vex) = 0 for x E aM in case aM f. 0:

E'(f)'v= - IM9a{3(j(X»)TO:(f)xv(3(X)dV.

(2)

Proof

E'Cf)' v =

~ 1M O'Ygo:(3(jCX»)v'YCX)gi j (x)odO: Ojj(3 dV + 1M gO:(3 (jCX) )gi j (x) Oi V O:(x)Ojj{3(x) dV.

Integrating by parts the second integral in the right hand side, we get

§1. Definitions and First Results

E'(f)· v =

349

iM 9ex /3(J(X»)V ex (X)b. f /3(X)dV + ~ iM (r~/39Aex + r~ex9A(3) f(x) V'Y(X)gij (x)8d ex (x)8j f/3(x) dV

-i

v'Y (X)gij (x)8j f/3(x)8 ex 9'Y/3 (J(x) )8d lX (x) dV.

Since 8 ex 9'Y{3 = r:{39A 'Y + r~ex9A/3' the symmetry beetwen a and by gij gives the result (2).

f3 induced

10.3 Definition. A harmonic map f E C 2(M, M) is a critical point of E (Definition 10.1). That is to say, f satisfies r(f) = O. We can introduce the harmonic maps in another way. Suppose f is an immersion; f is injective on D a neighbourhood of P. Let Y be a vector field on D; Y = f* Y can be extended to a neighbourhood of f(P). For X belonging to Tx(D), we set X = f*X. We verify that VxY is well defined and that 'V' x Y - f*('7 x Y) = ax(X, Y) is bilinear in X and Y. Indeed (3)

ax(X, Y) = [8;jf'Y(x) - r~jakf'Y(X) + r:{3 (J(x) ) Bdex(x)Bj f/3(x)] Xiyj aB . y'Y

We call ax the second fundamental form of f at x. It is a bilinear form on Tx(M) with values in Tf(x)(M). The tension field r(f) is the trace of ax for g. f is totally geodesic if ax = 0 for all x E M and f is harmonic if r(f) = o.

10.4 Proposition (Ishiara [* 183]). (i) f is totally geodesic if and only if for any C 2 convex function r.p defined on an open set () c M, r.p 0 f is convex on f -1 «(). (ii) j is harmonic if and only if for any r.p as above, r.pof is subharmonic. Proof We suppose that the coordinates {xi} are normal at x and that the coordinates {ylX} are normal at f(x). (4)

aij(r.p 0 f)x = (Bex{3r.p)f(x)Bd ex (x)Bj f{3(x) + (Bexr.p)f(x)BijflX(x).

If Bijjex(x) = 0, (Hess r.p)f(x) ~ 0 implies Hess (r.p 0 f)x ?: O. Conversly if Bijjex(x) f. 0, we can exhibit a convex function r.p such that r.p 0 f is not convex. From (4), we get if f is harmonic

= gi j (x)(Bex /3r.p)f(x)Bd ex (x)8j f{3(x) f)x ::; o. Conversly if r(f)x f. 0, we

-t1(r.p 0 f)x

r.p convex implies t1( r.p 0 convex function r.p such that t1( r.p 0 f)x > O.

10.5 Proposition. A C 2 harmonic map f is Coo.

can exhibit a

10. Harmonic Maps

350

f satisfies r(f) = 0 which is in local coordinates an elliptic equation. By the standard theorems of regularity f E Coo. Examples 10.5. If (M, g) is (IRm, E), we can choose the coordinates {yo O. Invent. Math. 89 (1987) 225-246. [*301] Tian G.- A Harnack type inequality for certain complex Monge-Ampere equations. J. Differ. Geom. 29 (1989) 481-488. [*302] Tian G.- On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990) 101-172. [*303] Tian G. and Yau S-T.- Kahler-Einstein metrics on complex surfaces with C J > O. Comment. Math. Phys. 112 (1987) 175-203. [*304] Tian G. and Yau S-T.- Complete Kahler manifolds with zero Ricci curvature II. Invent. Math. 106 (1991) 27-60. [*305] Tian G. and Yau S-T.- Existence of Kahler-Einstein metrics on complete Kahler manifolds and their applications to algebraic geometry. Proc. Con! San Diego. Adv. SeT. Math. Phys. 1 (1987) 574-629. [*306] Tian G. and Yau S-T.- Complete Kahler manifolds with zero Ricci curvature II. Invent. Math. 106 (1991) 27-60. [*307] Uhlenbeck K - Harmonic maps; a direct method in the calculus of variations. Bull. Am. Math. Soc. 76 (1970) 1082-1087. [*308] Uhlenbeck K- Morse theory by perturbation methods with applications to harmonic maps. Trans. Am. Math. Soc. 267 (1981) 569-583. [*309] Vaugon M.- Equations differentielles nonlineaires sur les varietes riemanniennes compactes I et II. Bull. Sci. Math. 106 (1982) 351-367; 107 (1983) 371-391. [*310] Vaugon M.- Transformation conforme de la courbure scalaire sur la sphere. Ann. Inst. Henri Poincare 3 (1986) 55-65. [*311] Vaugon M.- Transformation conforme de la courbure scalaire sur une variete riemannienne compacte. J. Funct. Anal. 71 (1987) 182-194. [*312] Vazquez J.L. et Veron L.- Solutions positives d'equations elliptiques semilineaires sur des varietes Riemanniennes compactes. C.R. Acad. Sci. Paris 312 (1991) 811-815. [*313] Veron L.- Semilinear elliptic equations with uniform blow-up on the boundary. J. Anal. Math. 59 (1992) 231-250. [* 314] Veron L.- Singularities of solutions of second order quasilinear equations. (to appear) [* 315] Webster S.- Pseudoherrnitian structures on a real hypersurface. J. Differ. Geom. 13 (1978) 25-41. [*316] White B.- Homothopy classes in Sobolev spaces and energy minimizing maps. Bull. Am. Math. Soc. 13 (1985) 166-168. [* 317] White B.- Nonunique tangent maps at isolated singularities of harmonic maps. Bull. Am. Math. Soc. 26 (1992) 125-129. [*318] Witten E.- A new proof of the positive energy theorem. Commun. Math. Phys. 80 (1981) 381-402. [* 319] Xu X. and Yang P.- Remarks on prescribing Gauss curvature. Trans. Am. Math. Soc. 336 (1993) 831-840. [*320] Ye R.- Global existence and convergence of Yamabe flow. J. Differ. Geom. 39 (1994) 35-50. [*321] Yeung S.- Compactification of Kahler manifolds with negative Ricci curvature. Invent. Math. 106 (1991) 13-25.

Subject Index

absolutely continuous 77 adjoint operator 27, 75, 126 admissible function 253 Alexandroff-Bakelman-Pucci theorem 139 almost everywhere 76 a priori estimate 257, 263, 266, 270, 273, 289 arc length 5 Ascoli's theorem 74 asymptotically flat 164 atlas definition 1 c1asse C k 1 equivalent 1 Banach space 70 -Steinhaus theorem 73 's theorem 74 Berger's problem 191 best constants 43, 58, 60, 65, 68 definition 99 in Sobolev spaces 41, 100, 139, 140, 153, 236 Betti numbers 29, 30, 255 Bianchi's identities 6 bifurcation theory 136 blow-up 142, 224 boundary 25 bracket 2 Calabi's conjecture 255, 263 Calderon-Zygmund inequality 90 Cauchy's theorem 72 chart 1 Cheeger's theorem 124 Chern class 252 Cherrier's problem 180 Christoffel symbols 3 Clarkson's inequalities 89

closed (see differentiable) co-differential 27 cohomology class definition 252 positive definite 255 compact mapping 74 complex manifold 251 concircular curvature tensor 335, 346 conformal normal coordinates 158, 159 conjugate point 17, 18 connection 3 Riemannian 6 continuity method 257, 258, 269, 270, 290 convex ball 11, 19 function 291, 306 inequality for 309 set 289 convolution product 89 coordinates 1 geodesic 9 normal 7 covariant derivative 4 covering manifold 24 critical point 40 CR structure 182 curvature definition 3 Ricci 7, 252, 346 scalar 7, 145, 179, 194, 196 sectional 7, 321 tensor 4, 6, 252 cut locus 14, 18 degenerate 40 degree (Leray-Schauder) 135 density problem 33 --+ 35 diffeomorphism 72 differentiable 71 form 2 closed 28

390

Subject Index

co closed 28 co-exact 28 exact 28 harmonic 28 homologous 28 manifold 1 mapping 1 differential operator 83, 126 elliptic 83, 84, 125 leading part of 83 linear 83, 125 strongly elliptic 126 uniformly elliptic 83 Dirichlet problem 289, 290, 314 distance 5 divergence form 85 dual space 70 eigenfuncti on 10 1, 102 eigenvalue definition 75, 102 of the Laplacian 31, 101, 102 eigenvector 75 Einstein-Kahler metric 254, 256, 259 Einstein metric 7 elliptic (see differential operator and linear) elliptic equation (definition) 125 energy at P 168 of a functional 247 of a map 348 equicontinuous 74 equivariant Yamabe problem 187 Euler equation 42, 102, 103, 148, 192, 193 Euler-Poincare characteristic X 29, 191 exact (see differentiable) 28 exceptional case 63, 68 exponential mapping 9 exterior differential p-form 2 differentiation 3 Fatou's theorem 77 finiteness theorem (Cheeger) 124 first Chern class 252, 255 first eigenvalue 115, 116, 190 first fundamental form 251 fixed point method 74, 134 form (see differentiable) formal adjoint 84 Fredholm alternative 126

mapping 132 nonlinear theorem 236, 237 theorem 75 Fubini's theorem 78 functional 105, 146, 150, 191,306,315 Gauss-Bonnet theorem 191 geodesic 8, 11 coordinates 9 global scalar product 28 Green's formula 107 Green's function 120, 122, 123, 162, 163, 167, 168 definition 107 existence properties 108, 112 Gromov's compactness theorem 125 Gromov's isoperimetric inequality 119 Hahn-Banach theorem 73 harmonic (see differentiable and spherical) harmonic map 349 weakly harmonic map 350 Harnack inequality 127,275 Hausdorff dimension 357 heat operator 129 solution 130 Hermitian metric 251 Hilbert space 70 Hodge decomposition theorem 29 Holder's inequality 88 homeomorphism 72 homologous 28 Hopf-Rinow theorem 13 imbedding 1 immersion 1 implicite funtion theorem 72 improvement of the best constants 57, 68 index form 18 inequality 19 inequality Clarkson's 89 Holder's 88 index 19 interpolation 93 isoperimetric 40 Minkowski 309 optimal 50 plurisubharmonic function 316, 317

391

Subject Index inequality (Alexandrov-Bakelman-Pucci) 128 injectivity radius 15, 124, 125 inner product 70 integrable 76 integration over Riemannian manifolds 23,24,29 interpolation inequalities 93 inverse function theorem 72 isometry - concentration 134, 214 isoperimetric function 119 isoperimetric inequality 40, 119 Jacobi field 17 Jacobian matrix 71 Kahler Einstein metric (see Einstein-Kahler) 254, 256, 259 manifold 251, 252 metric 251, 253 Kondrakov theorem 53, 55 Kronecker's symbol 4 Kom-Lichtenstein theorem 90 Laplacian 27, 106 leading part 83 Lebesgue integral 75, 77 measure 77 theorems 76, 77 length arc 5 minimizing 11 second variation 15 Leray-Schauder degree 135 Leray-Schauder theorem 74 Lichnerowicz's theorem 115 linear elliptic equation 101, 113 existence of solution for 104, 105, 113 mapping 70 tangent mapping 2 locally conformally flat manifold 117, 160 lower solution (see subsolution) 97 method of 198,200,202,267 Lp spaces 78 manifold complete 13, 14, 16, 45 complex 251

definition 1 differentiable Riemannian 4 with boundary 25 mapping differentiable rank 1 mass 165 maximum principle 96, 97, 138 for parabolic equations 130 generalized 98, 330 in narrow domains 138 second part 139 when u ;::; 0 139 mean value theorem 71 measurable 76 measure 76 methods 134 --t 138 metric Einstein 7 hermitian 251 Kahler 251 Riemannian 4 space 5 tensor, estimates on 20 minimax methods 134 minimizing curve 11 Monge-Ampere equation 257,269, 314 Moser's theorem 65, 209, 232 moutain pass lemma 134 moving planes method 136, 137, 138 Myer's theorem 16 Nash imbedding theorem 123 Neumann problem 88 Nirenberg's problem 230 norm 70 normal coordinates 7 normed space 70 Obata's theorem 176 open mapping theorem 73 optimal inequalities 50 orientable 23 orientation 23 Palais-Smale condition 134 parabolic equation 132 linear (solution) 131 local existence for nonlinear 133 solution of the Eells-Sampson equation 132

392

Subject Index

parallel displacement 8 vector field 8 parametrix for the Laplacian 106 plurisubharmonic function 314 inequality 316, 317 Pohozaev identity 229 point of concentration 134, 141,218, 219 definition 215 positive mass 166 precompact 74 principal symbol 125 projective space complex 260, 263 real 23, 209 Rademacker's theorem 77 Radon measure 75, 301, 315 Radon's theorem 81 rank 71 reflexive 70, 81 regularity interior 85, 86 up to the boundary 87 regularity theorem 127 regularization 80 reflexive 70 relaxed energy 360 Ricci curvature 7, 322, 323, 346 form 252 tensor 7 Riemannian connection 6 manifold 4 metric 4 Sard theorem 123 scalar curvature 7, 145, 179 Schauder fixed point theorem 74 interior estimates 88 Schouten's tensor 117 second variation of the length integral 15 sectional curvature 7, 321 semi-norm 73 sense of distribution 84 sliding method 138 Sobolev imbedding theorem 35, 44, 45, 50 proof 37 ~ 39, 46, 47, 49, 51

lemma 37 space 32 spectrum 31 spherical harmonic 233, 235, 236 steepest descent 134 subsolution (see lower solution) 97 supersolution (see upper solution and lower solution) 97 tangent space 2 vector 2 Taylor's formula 72 tension field 348, 349 tensor field 2 test functions Aubin's 155, 159 Escobar's 181 Hebey and Vaugon' s 171 Schoen's 161 topological methods 172, 213 torsion 3 totally geodesic 349 trace 69 tubular neighborhood 15 uniformly convex 81 locally finite cover 48 upper solution (see supersolution)

97

variational problem 42, 101, 105, 146, 150,154,204,209,311,317 vector field 2 volume element oriented 26 Riemannian 30 weak convergence 74 derivative 81,84 solution 84, 85 weakly sequentially compact Weyl's tensor 117 Whitney's theorem 4 Yamabe's equation 146 problem 145 theorem 147, 150 Yamabe problem 145 functional J(0

-

means strictly positive means strictly negative means negative or zero means positive or zero means compact manifold without boundary unless we say otherwise (resp. aij > 0) for a bilinea~ form means aij~i~j ~ 0 for any vector ~ (resp. aijee > 0 for any vector ~ =10).

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