Affine Bernstein problems and monge-ampère equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore <<[u.a.]>>
World Scientific
2010
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 180 S. |
| ISBN: | 9789812814166 9812814167 |
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Record in the Search Index
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|---|---|
| adam_text | Titel: Affine Bernstein problems and monge-ampère equations
Autor: Li, An-Min
Jahr: 2010
Contents
Preface v
1. Basic Tools 1
1.1 Differentiable Manifolds........................ 1
1.1.1 Manifolds, connections and exterior calculus........ 1
1.1.2 Riemannian manifolds..................... 4
1.1.3 Curvature inequalities..................... 5
1.1.4 Geodesic balls and level sets................. 6
1.2 Completeness and Maximum Principles............... 7
1.2.1 Topology and curvature.................... 7
1.2.2 Maximum principles...................... 7
1.3 Comparison Theorems......................... 8
1.4 The Legendre Transformation..................... 9
2. Local EquiafHne Hypersurfaces 11
2.1 Hypersurfaces in Unimodular Affine Space.............. 11
2.1.1 The ambient space....................... 11
2.1.2 Affine hypersurfaces...................... 13
2.2 Structure Equations and Berwald-Blaschke Metric......... 14
2.2.1 Structure equations - preliminary version.......... 14
2.2.2 Covariant Gauß equations - preliminary .......... 16
2.3 The Affine Normalization....................... 16
2.3.1 The affine normal....................... 16
2.3.2 Affine shape operator and affine extrinsic curvature .... 18
2.3.3 The affine conormal...................... 19
2.3.4 The conormal connection................... 21
2.3.5 Affine Gauß mappings.................... 21
2.4 The Fubini-Pick Form......................... 22
2.4.1 Properties of the Fubini-Pick form.............. 23
2.4.2 The Pick invariant....................... 23
? Affine Bernstein Problems and Monge-Ampère Equations
2.4.3 Structure equations - covariant notation .......... 23
2.4.4 The affine support function.................. 24
2.5 Integrability Conditions........................ 24
2.5.1 Integration via moving frames................ 24
2.5.2 Covariant form of the integrability conditions ....... 26
2.6 Fundamental Theorem......................... 27
2.7 Graph Immersions with Unimodular Normalization......... 27
2.8 Affine Spheres and Quadrics...................... 30
2.8.1 Affine hyperspheres...................... 30
2.8.2 Characterization of quadrics................. 31
3. Local Relative Hypersurfaces 33
3.1 Hypersurfaces with Arbitrary Normalization............. 33
3.1.1 Structure equations...................... 33
3.1.2 Fundamental theorem for non-degenerate hypersurfaces . . 35
3.2 Hypersurfaces with Relative Normalization............. 35
3.2.1 Relative structure equations and basic invariants...... 36
3.2.2 Relative integrability conditions............... 38
3.2.3 Classical version of the integrability conditions....... 38
3.2.4 Classical version of the fundamental theorem........ 38
3.3 Examples of Relative Geometries................... 39
3.3.1 The Euclidean normalization................. 39
3.3.2 The equiaffine (Blaschke) normalization........... 39
3.3.3 The centroaffine normalization................ 40
3.3.4 Graph immersions with Calabi metric............ 41
3.3.5 The family of conformai metrics G(a)............ 42
3.3.6 Comparison of different relative geometries......... 43
3.3.7 Different versions of fundamental theorems......... 43
3.4 Gauge Invariance and Relative Geometry.............. 43
4. The Theorem of Jörgens-Calabi-Pogorelov 47
4.1 Affine Hyperspheres and their PDEs................. 47
4.1.1 Improper affine hyperspheres................. 47
4.1.2 Proper affine hyperspheres.................. 48
4.1.3 The Pick invariant on affine hyperspheres.......... 49
4.2 Completeness in Affine Geometry................... 50
4.2.1 Affine completeness and Euclidean completeness...... 50
4.2.2 The Cheng-Yau criterion for affine completeness...... 51
4.2.3 Proof of the Estimate Lemma................ 53
4.2.4 Topology and the equiaffine Gauß map........... 56
4.3 Affine Complete Elliptic Affine Hyperspheres............ 59
4.4 The Theorem of Jörgens-Calabi-Pogorelov.............. 59
Contents xi
4.5 An Extension of the Theorem of Jörgens-Calabi-Pogorelov..... 61
4.5.1 Affine Kahler Ricci flat equation............... 61
4.5.2 Tools from relative geometry................. 63
4.5.3 Calculation of ?F in terms of the Calabi metric...... 63
4.5.4 Extension of the Theorem of Jörgens-Calabi-Pogorelov -
proof for ? 4......................... 66
4.5.5 Comparison of two geometric proofs............. 68
4.5.6 Technical tools for the proof in dimension ? 5...... 69
4.5.7 Proof of Theorem 4.5.1 - ? 5................ 79
4.6 A Cubic Form Differential Inequality with its Applications..... 82
4.6.1 Calculation of ? J in terms of the Calabi metric...... 83
4.6.2 Proof of Theorem 4.6.2.................... 85
5. Affine Maximal Hypersurfaces 89
5.1 The First Variation of the Equiaffine Volume Functional...... 89
5.2 Affine Maximal Hypersurfaces .................... 92
5.2.1 Graph hypersurfaces ..................... 92
5.2.2 The PDE for affine maximal hypersurfaces......... 95
5.3 An Affine Analogue of the Weierstrass Representation....... 96
5.3.1 The representation formula.................. 96
5.3.2 Examples............................ 99
5.4 Calabi s Computation of ? J in Holomorphic Terms........ 99
5.4.1 Computation of ? ^J + ||ß||2l................ 104
5.5 Calabi s Conjecture........................... 105
5.5.1 Proof of Calabi s Conjecture for dimension ? = 2 ..... 106
5.6 Chern s Conjecture........................... 110
5.6.1 Technical estimates...................... 112
5.6.2 Estimates for the determinant of the Hessian........ 114
5.6.3 Estimates for the third order derivatives.......... 121
5.6.4 Estimates for ]? fa...................... 126
5.6.5 Proof of Theorem 5.6.2.................... 128
5.7 An Affine Bernstein Problem in Dimension 3............ 131
5.7.1 Proof of Part I......................... 131
5.7.2 Proof of Part II: Affine blow-up analysis.......... 133
5.8 Another Method of Proof for some Fourth Order PDEs ...... 138
5.9 Euclidean Completeness and Calabi Completeness......... 144
6. Hypersurfaces with Constant Affine Mean Curvature 149
6.1 Classification.............................. 149
6.1.1 Estimates for the determinant of the Hessian........ 150
6.1.2 Proof of Theorem 6.1.1.................... 151
6.1.3 Proof of Theorem 6.1.2.................... 160
xii Affine Bernstein Problems and Monge-Ampère Equations
6.2 Hypersurfaces with Negative Constant Mean Curvature...... 161
6.2.1 Proof of the existence of a solution ............. 165
6.2.2 Proof of the Euclidean completeness............. 169
Bibliography 173
Index 179
|
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| author_GND | (DE-588)113507054 |
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| ctrlnum | (OCoLC)699298475 (DE-599)BSZ305754831 |
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| institution | BVB |
| isbn | 9789812814166 9812814167 |
| language | English |
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| oclc_num | 699298475 |
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| owner_facet | DE-83 |
| physical | XII, 180 S. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Li, An-Min 1946- Verfasser (DE-588)113507054 aut Affine Bernstein problems and monge-ampère equations An-Min Li ... Singapore <<[u.a.]>> World Scientific 2010 XII, 180 S. txt rdacontent n rdamedia nc rdacarrier Monge-Ampère-Differentialgleichung (DE-588)4253327-2 gnd rswk-swf Globale Differentialgeometrie (DE-588)4021286-5 gnd rswk-swf Monge-Ampère-Differentialgleichung (DE-588)4253327-2 s Globale Differentialgeometrie (DE-588)4021286-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020485995&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Li, An-Min 1946- Affine Bernstein problems and monge-ampère equations Monge-Ampère-Differentialgleichung (DE-588)4253327-2 gnd Globale Differentialgeometrie (DE-588)4021286-5 gnd |
| subject_GND | (DE-588)4253327-2 (DE-588)4021286-5 |
| title | Affine Bernstein problems and monge-ampère equations |
| title_auth | Affine Bernstein problems and monge-ampère equations |
| title_exact_search | Affine Bernstein problems and monge-ampère equations |
| title_full | Affine Bernstein problems and monge-ampère equations An-Min Li ... |
| title_fullStr | Affine Bernstein problems and monge-ampère equations An-Min Li ... |
| title_full_unstemmed | Affine Bernstein problems and monge-ampère equations An-Min Li ... |
| title_short | Affine Bernstein problems and monge-ampère equations |
| title_sort | affine bernstein problems and monge ampere equations |
| topic | Monge-Ampère-Differentialgleichung (DE-588)4253327-2 gnd Globale Differentialgeometrie (DE-588)4021286-5 gnd |
| topic_facet | Monge-Ampère-Differentialgleichung Globale Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020485995&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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