Some nonlinear problems in Riemannian geometry

✍ Scribed by Thierry Aubin

• Publisher: Springer
• Year: 1998
• Tongue: English
• Leaves: 411
• Series: Springer Monographs in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

During the last few years, the field of nonlinear problems has undergone great development. This book consisting of the updated Grundlehren volume 252 by the author and of a newly written part, 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 the most up-to-date methods for approaching it. The main objective of the book is to explain some methods and new techniques, and to apply them. It deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber.

✦ Table of Contents

Preface......Page 3
Contents......Page 9
1. Introduction to Differential Geometry ......Page 17
1.1. Tangent Space ......Page 18
1.3. Curvature ......Page 19
2. Riemannian Manifold ......Page 20
2.1. Metric Space ......Page 21
2.3. Sectional Curvature. Ricci Tensor. Scalar Curvature ......Page 22
2.4. Parallel Displacement. Geodesic ......Page 24
3. Exponential Mapping ......Page 25
4. The Hopf-Rinow Theorem ......Page 29
5.2. Second Variation of the Length Integral ......Page 31
5.3. Myers' Theorem ......Page 32
6. Jacobi Field ......Page 33
7. The Index Inequality ......Page 34
8. Estimates on the Components of the Metric Tensor ......Page 36
9. Integration over Riemannian Manifolds ......Page 39
10. Manifold with Boundary ......Page 41
11.1. Oriented Volume Element ......Page 42
11.2. Laplacian ......Page 43
11.3. Hodge Decomposition Theorem ......Page 45
11.4. Spectrum ......Page 47
1. First Definitions ......Page 48
2. Density Problems ......Page 49
3. Sobolev Imbedding Theorem ......Page 51
4. Sobolev's Proof ......Page 53
5. Proof by Gagliardo and Nirenberg ......Page 54
6. New Proof ......Page 55
7. Sobolev Imbedding Theorem for Riemannian Manifolds ......Page 60
9. Sobolev's Theorem for Compact Riemannian Manifolds with Boundary ......Page 66
10. The Kondrakov Theorem ......Page 69
11. Kondrakov's Theorem for Riemannian Manifolds ......Page 71
12. Examples ......Page 72
13. Improvement of the Best Constants ......Page 73
14. The Case of the Sphere ......Page 77
15. The Exceptional Case of the Sobolev Imbedding Theorem ......Page 79
16. Moser's Results ......Page 81
17. The Case of the Riemannian Manifolds ......Page 83
18. Problems of Traces ......Page 85
1. Differential Calculus ......Page 86
1.1. The Mean Value Theorem ......Page 87
1.3. Cauchy's Theorem ......Page 88
2.3. The Banach-Steinhaus Theorem ......Page 89
3.2. The Leray-Schauder Theorem ......Page 90
4. The Lebesgue Integral ......Page 91
4.1. Dominated Convergence Theorem ......Page 92
4.4. Rademacher's Theorem ......Page 93
5. The L,, Spaces ......Page 94
5.1. Regularization ......Page 96
5.2. Radon's Theorem ......Page 97
6. Elliptic Differential Operators ......Page 99
6.1. Weak Solution ......Page 100
6.2. Regularity Theorems ......Page 101
7.1. Holder's Inequality ......Page 104
7.3. Convolution Product ......Page 105
7.5. Korn-Lichtenstein Theorem ......Page 106
7.6. Interpolation Inequalities ......Page 109
8.2. Uniqueness Theorem ......Page 112
8.3. Maximum Principle for Nonlinear Elliptic Operator of Order Two ......Page 113
8.4. Generalized Maximum Principle ......Page 114
9. Best Constants ......Page 115
9.1. Applications to Sobolev's Spaces ......Page 116
1.1. First Nonzero Eigenvalue A of A ......Page 117
1.2. Existence Theorem for the Equation Aco = f ......Page 120
2.1. Parametrix ......Page 122
2.2. Green's Formula ......Page 123
2.3. Green's Function for Compact Manifolds ......Page 124
2.4. Green's Function for Compact Manifolds with Boundary ......Page 128
3.1. The First Eigenvalue ......Page 131
3.2. Locally Conformally Flat Manifolds ......Page 133
3.3. The Green Function of the Laplacian ......Page 135
3.4. Some Theorems ......Page 139
4.1. Elliptic Equations ......Page 141
4.2. Parabolic Equations ......Page 145
5. The Methods ......Page 150
6. The Best Constants ......Page 155
1. The Yamabe Problem ......Page 161
1.1. Yamabe's Method ......Page 162
1.3. Yamabe's Theorem ......Page 166
2. The Positive Case ......Page 168
3. The First Results ......Page 173
4.1. The Compact Locally Conformally Flat Manifolds ......Page 176
4.2. Schoen's Article ......Page 177
4.3. The Dimension 3, 4 and 5 ......Page 178
5. The Positive Mass ......Page 180
5.2. Schoen and Yau's Article ......Page 182
5.3. The Positive Energy ......Page 185
6.2. Hebey and Vaugon's Article ......Page 187
6.3. Topological Methods ......Page 188
7.1. Some Cases of Uniqueness ......Page 191
7.2. Particular Cases ......Page 192
7.4. Hebey-Vaugon's Approach ......Page 194
8.1. Topological Meaning of the Scalar Curvature ......Page 195
8.2. The Cherrier Problem ......Page 196
8.3. The Yamabe Problem on CR Manifolds ......Page 198
8.4. The Yamabe Problem on Non-compact Manifolds ......Page 199
8.5. The Yamabe Problem on Domains in Rn ......Page 201
8.6. The Equivariant Yamabe Problem ......Page 203
8.7. An Hard Open Problem ......Page 204
8.8. Berger's Problem ......Page 207
1.1. The General Problem ......Page 210
1.2. The Problem with Conformal Change of Metric ......Page 212
2. The Negative Case when M is Compact ......Page 213
3. The Zero Case when M is Compact ......Page 220
4. The Positive Case when M is Compact ......Page 225
5.1. The Problem ......Page 230
5.2. Study of the Sequence ......Page 232
5.3. The Points of Concentration ......Page 234
5.4. Consequences ......Page 237
5.5. Blow-up at a Point of Concentration ......Page 240
6.1. On Complete Non-compact Manifolds ......Page 243
6.2. On Compact Manifolds with Boundary ......Page 245
7. The Nirenberg Problem ......Page 246
8. First Results ......Page 247
8.1. Moser's Result ......Page 248
8.2. Kazdan and Warner Obstructions ......Page 249
8.3. A Nonlinear Fredholm Theorem ......Page 251
9. G-invariant Functions f ......Page 254
10.1. Functions f Close to a Constant ......Page 257
10.2. Dimension Two ......Page 259
10.3. Dimension n > 3 ......Page 261
11.1. Multiplicity ......Page 263
11.2. Density ......Page 264
11.3. The Problem on the Half Sphere ......Page 265
1. Kahler Manifolds ......Page 267
1.1. First Chern Class ......Page 268
1.2. Change of Kahler Metrics. Admissible Functions ......Page 269
2.1. Einstein-Kahler Metric ......Page 270
3.1. Reducing the Problem to Equations ......Page 271
3.2. The First Results ......Page 272
4.1. About Regularity ......Page 273
5. Theorem of Existence (the Negative Case) ......Page 274
6. Existence of Einstein-Kahler Metric ......Page 275
7. Theorem of Existence (the Null Case) ......Page 276
10. A Priori Estimate for App ......Page 279
11. A Priori Estimate for the Third Derivatives of Mixed Type ......Page 282
12. The Method of Lower and Upper Solutions ......Page 283
13. A Method for the Positive Case ......Page 285
14.1. The First Obstruction ......Page 287
14.3. A Further Obstruction ......Page 288
15.1. Definition of the Functionals I(cp) and J(W) ......Page 289
15.2. Some Inequalities ......Page 290
15.3. The CEstimate ......Page 292
15.4. Inequalities for the Dimension rn = I ......Page 293
15.5. Inequalities for the Exponential Function ......Page 294
16. Some Results ......Page 297
77. On Uniqueness ......Page 301
18. On Non-compact Kahler Manifolds ......Page 304
1.1. The Fundamental Hypothesis ......Page 305
1.2. Extra Hypothesis ......Page 306
1.3. Theorem of Existence ......Page 307
2.1. The First Estimates ......Page 308
2.2. C2-Estimate ......Page 309
2.3. C3-Estimate ......Page 312
3. The Radon Measure,li(sp) ......Page 317
4.1. Properties of f(ep) ......Page 322
5. Variational Problem ......Page 327
6.1. Bedford's and Taylor's Results ......Page 330
6.4. Some Properties of 7(v) ......Page 331
7. The Case of Radially Symmetric Fucntions ......Page 332
7.1. Variational Problem ......Page 333
8. A New Method ......Page 334
1.1. The Sectional Curvature ......Page 337
1.3. The Ricci Curvature ......Page 338
2.1. DeTurck's Result ......Page 339
2.3. DeTurck's Equations ......Page 340
2.4. Global Results ......Page 341
3.1. The Equation ......Page 342
3.2. Solution for a Short Time ......Page 343
3.3. Some Useful Results ......Page 346
3.4. Hamilton's Evolution Equations ......Page 349
4.1. Hamilton's Theorems ......Page 359
4.2. Pinched Theorems on the Concircular Curvature ......Page 360
5.1. On the Ricci Curvature ......Page 361
5.2. On the Concircular Curvature ......Page 362
1. Definitions and First Results ......Page 364
2.1. The Problem ......Page 367
2.2. Some Basic Results ......Page 368
2.3. Existence Results ......Page 370
3.1. Sobolev Spaces ......Page 372
3.2. The Results ......Page 373
4.1. General Results ......Page 375
4.2. Relaxed Energies ......Page 376
4.3. The Ginzburg-Landau Functional ......Page 377
Bibliography ......Page 381
Bibliography* ......Page 391
Subject Index ......Page 405
Notation ......Page 409