Some nonlinear problems in Riemannian geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1998
|
| Schriftenreihe: | Springer monographs in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 395 S. |
| ISBN: | 3540607528 |
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| 100 | 1 | |a Aubin, Thierry |d 1942-2009 |e Verfasser |0 (DE-588)120040441 |4 aut | |
| 245 | 1 | 0 | |a Some nonlinear problems in Riemannian geometry |c Thierry Aubin |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1998 | |
| 300 | |a XVII, 395 S. | ||
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| 650 | 4 | |a Geometry, Riemannian | |
| 650 | 4 | |a Nonlinear theories | |
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Datensatz im Suchindex
| _version_ | 1805076508562161664 |
|---|---|
| adam_text |
Contents
Chapter 1
Riemannian Geometry
§1. Introduction to Differential Geometry 1
1.1. Tangent Space 2
1.2. Connection 3
1.3. Curvature 3
§2. Riemannian Manifold 4
2.1. Metric Space 5
2.2. Riemannian Connection 6
2.3. Sectional Curvature. Ricci Tensor. Scalar Curvature 6
2.4. Parallel Displacement. Geodesic 8
§3. Exponential Mapping 9
§4. The Hopf Rinow Theorem 13
§5. Second Variation of the Length Integral 15
5.1. Existence of Tubular Neighborhoods 15
5.2. Second Variation of the Length Integral 15
5.3. Myers' Theorem 16
§6. Jacobi Field 17
§7. The Index Inequality 18
§8. Estimates on the Components of the Metric Tensor 20
§9. Integration over Riemannian Manifolds 23
§10. Manifold with Boundary 25
10.1. Stokes' Formula 26
§11. Harmonic Forms 26
11.1. Oriented Volume Element 26
11.2. Laplacian 27
11.3. Hodge Decomposition Theorem 29
11.4. Spectrum 31
Chapter 2
Sobolev Spaces
§1. First Definitions 32
§2. Density Problems 33
§3. Sobolev Imbedding Theorem 35
§4. Sobolev's Proof 37
XII Contents
§5. Proof by Gagliardo and Nirenberg 38
§6. New Proof 39
§7. Sobolev Imbedding Theorem for Riemannian Manifolds 44
§8. Optimal Inequalities 50
§9. Sobolev's Theorem for Compact Riemannian Manifolds with
Boundary 50
§10. The Kondrakov Theorem 53
§11. Kondrakov's Theorem for Riemannian Manifolds 55
§12. Examples 56
§13. Improvement of the Best Constants 57
§14. The Case of the Sphere 61
§15. The Exceptional Case of the Sobolev Imbedding Theorem 63
§16. Moser's Results 65
§17. The Case of the Riemannian Manifolds 67
§18. Problems of Traces 69
Chapter 3
Background Material
§1. Differential Calculus 70
1.1. The Mean Value Theorem 71
1.2. Inverse Function Theorem 72
1.3. Cauchy's Theorem 72
§2. Four Basic Theorems of Functional Analysis 73
2.1. Hahn Banach Theorem 73
2.2. Open Mapping Theorem 73
2.3. The Banach Steinhaus Theorem 73
2.4. Ascoli's Theorem 74
§3. Weak Convergence. Compact Operators 74
3.1. Banach's Theorem 74
3.2. The Leray Schauder Theorem 74
3.3. The Fredholm Theorem 75
§4. The Lebesgue Integral 75
4.1. Dominated Convergence Theorem 76
4.2. Fatou's Theorem 77
4.3. The Second Lebesgue Theorem 77
4.4. Rademacher's Theorem 77
4.5. Fubini's Theorem 78
§5. The Lp Spaces 78
5.1. Regularization 80
5.2. Radon's Theorem 81
§6. Elliptic Differential Operators 83
6.1. Weak Solution 84
6.2. Regularity Theorems 85
6.3. The Schauder Interior Estimates 88
§7. Inequalities 88
7.1. Holder's Inequality 88
7.2. Clarkson's Inequalities 89
7.3. Convolution Product 89
Contents XIII
7.4. The Calderon Zygmund Inequality 90
7.5. Korn Lichtenstein Theorem 90
7.6. Interpolation Inequalities 93
§8. Maximum Principle 96
8.1. Hopf s Maximum Principle 96
8.2. Uniqueness Theorem 96
8.3. Maximum Principle for Nonlinear Elliptic Operator of
Order Two 97
8.4. Generalized Maximum Principle 98
§9. Best Constants 99
9.1. Applications to Sobolev's Spaces 100
Chapter 4
Complementary Material
§1. Linear Elliptic Equations 101
1.1. First Nonzero Eigenvalue A of A 101
1.2. Existence Theorem for the Equation Aip = f 104
§2. Green's Function of the Laplacian 106
2.1. Parametrix 106
2.2. Green's Formula 107
2.3. Green's Function for Compact Manifolds 108
2.4. Green's Function for Compact Manifolds with Boundary . 112
§3. Riemannian Geometry 115
3.1. The First Eigenvalue 115
3.2. Locally Conformally Flat Manifolds 117
3.3. The Green Function of the Laplacian 119
3.4. Some Theorems 123
§4. Partial Differential Equations 125
4.1. Elliptic Equations 125
4.2. Parabolic Equations 129
§5. The Methods 134
§6. The Best Constants 139
Chapter 5
The Yamabe Problem
§1. The Yamabe Problem 145
1.1. Yamabe's Method 146
1.2. Yamabe's Functional 150
1.3. Yamabe's Theorem 150
§2. The Positive Case 152
§3. The First Results 157
§4. The Remaining Cases 160
4.1. The Compact Locally Conformally Flat Manifolds 160
4.2. Schoen's Article 161
4.3. The Dimension 3, 4 and 5 162
§5. The Positive Mass 164
5.1. Positive Mass Theorem, the Low Dimensions 166
XIV Contents
5.2. Schoen and Yau's Article 166
5.3. The Positive Energy 169
§6. New Proofs for the Positive Case O 0) 171
6.1. Lee and Parker's Article 171
6.2. Hebey and Vaugon's Article 171
6.3. Topological Methods 172
6.4. Other Methods 175
§7. On the Number of Solutions 175
7.1. Some Cases of Uniqueness 175
7.2. Particular Cases 176
7.3. About Uniqueness 178
7.4. Hebey Vaugon's Approach 178
7.5. The Structure of the Set of Minimizers of J 179
§8. Other Problems 179
8.1. Topological Meaning of the Scalar Curvature 179
8.2. The Cherrier Problem 180
8.3. The Yamabe Problem on CR Manifolds 182
8.4. The Yamabe Problem on Non compact Manifolds 183
8.5. The Yamabe Problem on Domains in Rn 185
8.6. The Equivariant Yamabe Problem 187
8.7. An Hard Open Problem 188
8.8. Berger's Problem 191
Chapter 6
Prescribed Scalar Curvature
§1. The Problem 194
1.1. The General Problem 194
1.2. The Problem with Conformal Change of Metric 196
§2. The Negative Case when M is Compact 197
§3. The Zero Case when M is Compact 204
§4. The Positive Case when M is Compact 209
§5. The Method of Isometry Concentration 214
5.1. The Problem 214
5.2. Study of the Sequence { pqj} 216
5.3. The Points of Concentration 218
5.4. Consequences 221
5.5. Blow up at a Point of Concentration 224
§6. The Problem on Other Manifolds 227
6.1. On Complete Non compact Manifolds 227
6.2. On Compact Manifolds with Boundary 229
§7. The Nirenberg Problem 230
§8. First Results 231
8.1. Moser's Result 232
8.2. Kazdan and Warner Obstructions 233
8.3. A Nonlinear Fredholm Theorem 235
§9. G invariant Functions / 238
§10. The General Case 241
10.1. Functions / Close to a Constant 241
Contents XV
10.2. Dimension Two 243
10.3. Dimension n 3 245
10.4. Rotationally Symmetric Functions 247
§11. Related Problems 247
11.1. Multiplicity 247
11.2. Density 248
11.3. The Problem on the Half Sphere 249
Chapter 7
Einstein Kahler Metrics
§1. Kahler Manifolds 251
1.1. First Chern Class 252
1.2. Change of Kahler Metrics. Admissible Functions 253
§2. The Problems 254
2.1. Einstein Kahler Metric 254
2.2. Calabi's Conjecture 255
§3. The Method 255
3.1. Reducing the Problem to Equations 255
3.2. The First Results 256
§4. Complex Monge Ampere Equation 257
4.1. About Regularity 257
4.2. About Uniqueness 258
§5. Theorem of Existence (the Negative Case) 258
§6. Existence of Einstein Kahler Metric 259
§7. Theorem of Existence (the Null Case) 260
§8. Proof of Calabi's Conjecture 263
§9. The Positive Case 263
§10. A Priori Estimate for A^ 263
§11. A Priori Estimate for the Third Derivatives of Mixed Type 266
§12. The Method of Lower and Upper Solutions 267
§13. A Method for the Positive Case 269
§14. The Obstructions When Ci(M) 0 271
14.1. The First Obstruction 271
14.2. Futaki's Obstruction 272
14.3. A Further Obstruction 272
§15. The C° Estimate 273
15.1. Definition of the Functional I( p) and J( p) 273
15.2. Some Inequalities 274
15.3. The C° Estimate 276
15.4. Inequalities for the Dimension m=\ 277
15.5. Inequalities for the Exponential Function 278
§16. Some Results 281
§17. On Uniqueness 285
§18. On Non compact Kahler Manifolds 288
XVI Contents
Chapter 8
Monge Ampere Equations
§ 1. Monge Ampere Equations on Bounded Domains of Rn 289
1.1. The Fundamental Hypothesis 289
1.2. Extra Hypothesis 290
1.3. Theorem of Existence 291
§2. The Estimates 292
2.1. The First Estimates 292
2.2. C2 Estimate 293
2.3. C3 Estimate 296
§3. The Radon Measure J(( p) 301
§4. The Functional J(y ) 306
4.1. Properties of S(ip) 306
§5. Variational Problem 311
§6. The Complex Monge Ampere Equation 314
6.1. Bedford's and Taylor's Results 314
6.2. The Measure OT(p) 315
6.3. The Function 3( p) 315
6.4. Some Properties of 3( p) 315
§7. The Case of Radially Symmetric Fucntions 316
7.1. Variational Problem 317
7.2. An Open Problem 318
§8. A New Method 318
Chapter 9
The Ricci Curvature
§1. About the Different Types of Curvature 321
1.1. The Sectional Curvature 321
1.2. The Scalar Curvature 322
1.3. The Ricci Curvature 322
1.4. Two Dimension 323
§2. Prescribing the Ricci Curvature 323
2.1. DeTurck's Result 323
2.2. Some Computations 324
2.3. DeTurck's Equations 324
2.4. Global Results 325
§3. The Hamilton Evolution Equation 326
3.1. The Equation 326
3.2. Solution for a Short Time 327
3.3. Some Useful Results 330
3.4. Hamilton's Evolution Equations 333
§4. The Consequences of Hamilton's Work 343
4.1. Hamilton's Theorems 343
4.2. Pinched Theorems on the Concircular Curvature 344
§5. Recent Results 345
5.1. On the Ricci Curvature 345
5.2. On the Concircular Curvature 346
Contents XVII
Chapter 10
Harmonic Maps
§1. Definitions and First Results 348
§2. Existence Problems 351
2.1. The Problem 351
2.2. Some Basic Results 352
2.3. Existence Results 354
§3. Problems of Regularity 356
3.1. Sobolev Spaces 356
3.2. The Results 357
§4. The Case of dM f 0 359
4.1. General Results 359
4.2. Relaxed Energies 360
4.3. The Ginzburg Landau Functional 361
Bibliography 365
Bibliography* 375
Subject Index 389
Notation 393 |
| any_adam_object | 1 |
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| dewey-ones | 516 - Geometry |
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| indexdate | 2024-07-20T05:58:25Z |
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| isbn | 3540607528 |
| language | English |
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| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Aubin, Thierry 1942-2009 Verfasser (DE-588)120040441 aut Some nonlinear problems in Riemannian geometry Thierry Aubin Berlin [u.a.] Springer 1998 XVII, 395 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Differentiaalmeetkunde gtt Niet-lineaire problemen gtt Niet-lineaire theorieën gtt Riemann, Géométrie de ram Riemann-vlakken gtt Théories non linéaires ram Geometry, Riemannian Nonlinear theories Nichtlineare Theorie (DE-588)4251279-7 gnd rswk-swf Nichtlineare Differentialgeometrie (DE-588)4309230-5 gnd rswk-swf Nichtlineare Analysis (DE-588)4177490-5 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 s Nichtlineare Theorie (DE-588)4251279-7 s DE-604 Nichtlineare Differentialgeometrie (DE-588)4309230-5 s Nichtlineare Analysis (DE-588)4177490-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008066379&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Aubin, Thierry 1942-2009 Some nonlinear problems in Riemannian geometry Differentiaalmeetkunde gtt Niet-lineaire problemen gtt Niet-lineaire theorieën gtt Riemann, Géométrie de ram Riemann-vlakken gtt Théories non linéaires ram Geometry, Riemannian Nonlinear theories Nichtlineare Theorie (DE-588)4251279-7 gnd Nichtlineare Differentialgeometrie (DE-588)4309230-5 gnd Nichtlineare Analysis (DE-588)4177490-5 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)4251279-7 (DE-588)4309230-5 (DE-588)4177490-5 (DE-588)4128462-8 |
| title | Some nonlinear problems in Riemannian geometry |
| title_auth | Some nonlinear problems in Riemannian geometry |
| title_exact_search | Some nonlinear problems in Riemannian geometry |
| title_full | Some nonlinear problems in Riemannian geometry Thierry Aubin |
| title_fullStr | Some nonlinear problems in Riemannian geometry Thierry Aubin |
| title_full_unstemmed | Some nonlinear problems in Riemannian geometry Thierry Aubin |
| title_short | Some nonlinear problems in Riemannian geometry |
| title_sort | some nonlinear problems in riemannian geometry |
| topic | Differentiaalmeetkunde gtt Niet-lineaire problemen gtt Niet-lineaire theorieën gtt Riemann, Géométrie de ram Riemann-vlakken gtt Théories non linéaires ram Geometry, Riemannian Nonlinear theories Nichtlineare Theorie (DE-588)4251279-7 gnd Nichtlineare Differentialgeometrie (DE-588)4309230-5 gnd Nichtlineare Analysis (DE-588)4177490-5 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Differentiaalmeetkunde Niet-lineaire problemen Niet-lineaire theorieën Riemann, Géométrie de Riemann-vlakken Théories non linéaires Geometry, Riemannian Nonlinear theories Nichtlineare Theorie Nichtlineare Differentialgeometrie Nichtlineare Analysis Riemannsche Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008066379&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT aubinthierry somenonlinearproblemsinriemanniangeometry |