Nonlinear partial differential equations in differential geometry: [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah]
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| Format: | Conference Proceeding Book |
|---|---|
| Language: | English |
| Published: |
Providence, R.I.
American Mathematical Society
1996
|
| Series: | Park City Mathematics Institute: IAS/Park City Mathematics Series
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 339 S. |
| ISBN: | 0821804316 |
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MARC
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| 245 | 1 | 0 | |a Nonlinear partial differential equations in differential geometry |b [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] |c Robert Hardt ..., eds. |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 1996 | |
| 300 | |a XI, 339 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Park City Mathematics Institute: IAS/Park City Mathematics Series |v 2 | |
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| adam_text |
Contents
Preface xi
Introduction 1
Luis Caffarelli, A Priori Estimates and the Geometry of the Monge
Ampere Equation 5
Introduction 7
Part 1. Interior A Priori Estimates for Solutions of Fully Non linear Equa¬
tions 9
Statement of Main Theorems 10
Preliminary Tools 13
Harnack Inequality for Functions in S*(f) 19
W2'P Estimates 22
Holder Estimates 28
Part 2. Geometric Properties of the Monge Ampere Equation 33
A Localization Property 34
Part 3. A Priori Estimates of Solutions to Monge Ampere Equations 41
The Correct Invariant Measure 41
Part 4. Interior W2'p Estimates for Solutions of the Monge Ampere Equation 47
Preliminary Results 48
Approximation Lemmas 50
Tangent Paraboloids in Measure and Level Surface Estimates 53
References 63
Sun Yung Alice Chang, The Moser Trudinger Inequality and
Applications to Some Problems in Conformal Geometry 65
Introduction 67
Lecture 1. Some Background Material 69
Preliminaries 69
Rayleigh Quotients 71
Weyl's Asymptotic Formula 73
References 74
Lecture 2. Ray Singer Polyakov Formula on Compact Surfaces 75
Heat Kernel 75
Asymptotic Behavior of the Trace of the Heat Kernel 76
V
vi CONTENTS
Ray Singer Polyakov Log Determinant Formula on a Compact Surface
without Boundary 78
References 81
Lecture 3. Moser Onofri Inequality and Applications 83
Onofri's Inequality 83
Compactness of Isospectral Families 87
References 88
Lecture 4. Existence of Extremal Functions for Moser Inequality 89
Proof of Theorem 2, for n = 2 90
Proof of Lemma 1 92
References 94
Lecture 5. Beckner Adams Inequalities and Extremal Log determinants 95
Conformally covariant operators 95
Paneitz's Operator 99
References 100
Lecture 6. Isospectral Compactness on 3 manifolds and Relation to the Yam
abe Problem 101
Condition (*) 104
Proof of Theorem 2' 106
References 109
Lecture 7. Prescribing Curvature Function onSn 111
The Variation Functional FK 113
Bounds for S[w] 115
Some Sufficient Conditions on K or R 121
References 124
Richard Schoen, The Effect of Curvature on the Behavior of
Harmonic Functions and Mappings 127
Introduction 129
Lecture 1. Gradient Estimate and Comparison Theorems 133
Hessian Comparison Theorem 134
Laplacian Comparison Theorem 136
Proof of (*) 137
Lecture 2. Gradient Estimate Proof and Corollaries 141
Continuation of Proof 141
Harnack Inequality 144
A Liouville Corollary 145
Lecture 3. Harmonic Functions on Negatively Curved Manifolds 147
Dirichlet Problem 148
Related Results and Questions 149
Lecture 4. Harmonic Mapping into Singular Spaces 151
Motivation 151
Nonpositively Curved Metric Spaces 152
CONTENTS vii
A Two dimensional Result 154
Lecture 5. Energy Convexity of Maps to an NPC Metric Space 157
Geodesic Homotopy 157
Monotonicity 160
Lecture 6. The Order Function 163
Interior Gradient Bound 164
Existence of a Homotopy Minimizer 166
Lecture 7. Approximation and Smoothness Results for Harmonic Maps 167
Intrinsic Homogeneity 167
Homogeneous Minimizers 168
Intrinsic Differentiability 171
Lecture 8. Order 1 Points and Partial Regularity 173
Proof of Singular Set Estimate 174
Lecture 9. Rigidity Results via Harmonic Maps 179
Superrigidity 179
A Vanishing Theorem 181
References 182
Leon Simon, Singularities of Geometric Variational Problems 185
Lecture 1. Basic Introductory Material 187
Definition of Energy Minimizing Map 187
Definition of Regular and Singular Set 188
The Variational Equations 188
The Monotonicity Formula 190
The Regularity Theorem 192
Corollaries of the Regularity Theorem 192
A Further Remark on Upper Semicontinuity of the Density 196
Lecture 2. Tangent Maps 197
Definition of Tangent Map 197
Properties of Homogeneous Degree Zero Minimizers 198
Further Properties of Sing u 200
Lecture 3. The Top Dimensional Part of Sing u 205
Homogeneous Degree Zero if with dim S(ip) = n — 3 205
The Geometric Picture Near Points of Sing, u 208
Consequences of Uniqueness of Tangent Maps 210
Lecture 4. Recent Results Concerning Sing u 213
Statement of Main Known Results 213
Preliminary Remarks on the Method of Proof: "Blowing Up" 214
I? Estimates 216
Special Solutions of the Linearized Equation 219
Brief Sketch of the Proof of the Results of 4.1 220
References 222
viii CONTENTS
Leon Simon, Proof of the Basic Regularity Theorem for Harmonic
Maps 225
Lecture 1. Analytic Preliminaries 227
Holder Continuous Functions 228
Functions with L2 Gradient 230
Harmonic Functions 231
Harmonic Approximation Lemma 232
Lecture 2. A General Regularity Lemma 235
Statement of Main Regularity Lemma and Remarks 235
Proof of the Regularity Lemma 236
Lecture 3. The Reverse Poincare Inequality 239
A Lemma of Luckhaus, and Some Corollaries 239
Proof of the Reverse Poincare Inequality 243
Lecture 4. Completion of the Regularity Proof 247
A Further Property of Functions with 1? Gradient 247
Proof of Luckhaus' Lemma 248
Proof of CllQ and higher regularity 253
References 256
Michael Struwe, Geometric Evolution Problems 257
Introduction 259
Part 1. The Evolution of Harmonic Maps 261
Harmonic Maps 261
Bochner Identity 263
Homotopy and Dirichlet Problems 265
The Eells Sampson Result 267
Finite Time Blow up 269
Global Existence and Uniqueness of Partially Regular Weak Solutions for
m = 2 272
Applications 279
Existence of Global, Partially Regular Weak Solutions for m 3 283
The Monotonicity Formula 285
Convergence of Penalized Solutions 288
Nonuniqueness 290
Development of Singularities 292
Singularities of First and Second Kind 293
Part 2. The Evolution of Hypersurfaces by Mean Curvature 295
Mean Curvature Flow 295
Compact Surfaces 296
Entire Graphs 298
Generalized Motion by Mean Curvature 300
Uniqueness, Comparison Principles, Global Existence 302
Monotonicity Formula 304
Consequences of the Monotonicity Estimate 307
Singularities 309
CONTENTS ix
Part 3. Harmonic Maps of Minkowsky Space 311
The Cauchy Problem for Harmonic Maps 311
Local Existence 313
Global Existence 318
Finite time Blow up 320
Self similar Equivariant Solutions 320
Global Existence and Regularity for Equivariant Harmonic Maps for
m = 2 324
Equivariant Harmonic Maps with Convex Range 331
References 333 |
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| genre_facet | Konferenzschrift 1992 Park City Utah |
| id | DE-604.BV010960423 |
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| indexdate | 2024-07-20T05:14:54Z |
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| isbn | 0821804316 |
| language | English |
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| physical | XI, 339 S. |
| publishDate | 1996 |
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| spelling | Nonlinear partial differential equations in differential geometry [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] Robert Hardt ..., eds. Providence, R.I. American Mathematical Society 1996 XI, 339 S. txt rdacontent n rdamedia nc rdacarrier Park City Mathematics Institute: IAS/Park City Mathematics Series 2 Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 1992 Park City Utah gnd-content Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s Differentialgeometrie (DE-588)4012248-7 s DE-604 Hardt, Robert Sonstige oth Graduate Summer School Program on Nonlinear Partial Differential Equations in Differential Geometry 1992 Park City, Utah Sonstige (DE-588)5206764-6 oth Erscheint auch als Online-Ausgabe 978-1-4704-3901-9 Park City Mathematics Institute: IAS/Park City Mathematics Series 2 (DE-604)BV010402400 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007332065&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nonlinear partial differential equations in differential geometry [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] Park City Mathematics Institute: IAS/Park City Mathematics Series Differentialgeometrie (DE-588)4012248-7 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4128900-6 (DE-588)1071861417 |
| title | Nonlinear partial differential equations in differential geometry [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] |
| title_auth | Nonlinear partial differential equations in differential geometry [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] |
| title_exact_search | Nonlinear partial differential equations in differential geometry [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] |
| title_full | Nonlinear partial differential equations in differential geometry [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] Robert Hardt ..., eds. |
| title_fullStr | Nonlinear partial differential equations in differential geometry [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] Robert Hardt ..., eds. |
| title_full_unstemmed | Nonlinear partial differential equations in differential geometry [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] Robert Hardt ..., eds. |
| title_short | Nonlinear partial differential equations in differential geometry |
| title_sort | nonlinear partial differential equations in differential geometry lecture notes from the graduate summer school program on nonlinear partial differential equations in differential geometry held june 20 july 12 1992 in park city utah |
| title_sub | [... lecture notes from the Graduate Summer School program on Nonlinear Partial Differential Equations in Differential Geometry, held June 20 - July 12, 1992, in Park City, Utah] |
| topic | Differentialgeometrie (DE-588)4012248-7 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Differentialgeometrie Nichtlineare partielle Differentialgleichung Konferenzschrift 1992 Park City Utah |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007332065&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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