Affine Bernstein Problems and Monge-Ampere Equations

by Li, An-Min; Xu, Ruiwei; Simon, Udo; Jia, Fang

• ISBN13: 9789812814166
• ISBN10: 9812814167
• Format: Hardcover
• Copyright: 2010-07-31
• Publisher: World Scientific Pub Co Inc

Summary

In this monograph, the interplay between geometry and partial differential equations (PDEs) is of particular interest. It gives a selfcontained introduction to research in the last decade concerning global problems in the theory of submanifolds, leading to some types of Monge-Ampère equations. From the methodical point of view, it introduces the solution of certain Monge-Ampère equations via geometric modeling techniques. Here geometric modeling means the appropriate choice of a normalization and its induced geometry on a hypersurface defined by a local strongly convex global graph. For a better understanding of the modeling techniques, the authors give a selfcontained summary of relative hypersurface theory, they derive important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine constant mean curvature equation). Concerning modeling techniques, emphasis is on carefully structured proofs and exemplary comparisons between different modelings.

Table of Contents

Preface	p. v
Basic Tools	p. 1
Differentiable Manifolds	p. 1
Manifolds, connections and exterior calculus	p. 1
Curvature inequalities	p. 5
Geodesic balls and level sets	p. 6
Completeness and Maximum Principles	p. 7
Topology and curvature	p. 7
Maximum principles	p. 7
Comparison Theorems	p. 8
The Legendre Transformation	p. 9
Local Equiaffine Hypersurfaces	p. 11
Hypersurfaces in Unimodular Affine Space	p. 11
The ambient space	p. 11
Affine hypersurfaces	p. 13
Structure Equations and Berwald-Blaschke Metric	p. 14
Structure equations - preliminary version	p. 14
Covariant Gauß equations - preliminary	p. 16
The Affine Normalization	p. 16
The affine normal	p. 16
Affine shape operator and affine extrinsic curvature	p. 18
The affine conormal	p. 19
The conormal connection	p. 21
Affine Gauß mappings	p. 21
The Fubini-Pick Form	p. 22
Properties of the Fubini-Pick form	p. 23
The Pick invariant	p. 23
Structure equations - covariant notation	p. 23
The affine support function	p. 24
Integrability Conditions	p. 24
Integration via moving frames	p. 24
Covariant form of the integrability conditions	p. 26
Fundamental Theorem	p. 27
Graph Immersions with Unimodular Normalization	p. 27
Affine Spheres and Quadrics	p. 30
Affine hyperspheres	p. 30
Characterization of quadrics	p. 31
Local Relative Hypersurfaces	p. 33
Hypersurfaces with Arbitrary Normalization	p. 33
Structure equations	p. 33
Fundamental theorem for non-degenerate hypersurfaces	p. 35
Hypersurfaces with Relative Normalization	p. 35
Relative structure equations and basic invariants	p. 36
Relative integrability conditions	p. 38
Classical version of the integrability conditions	p. 38
Classical version of the fundamental theorem	p. 38
Examples of Relative Geometries	p. 39
The Euclidean normalization	p. 39
The equiaffine (Blaschke) normalization	p. 39
The centroaffine normalization	p. 40
Graph immersions with Calabi metric	p. 41
The family of conformal metrics G(¿)	p. 42
Comparison of different relative geometries	p. 43
Different versions of fundamental theorems	p. 43
Gauge Invariance and Relative Geometry	p. 43
The Theorem of Jörgens-Calabi-Pogorelov	p. 47
Affine Hyperspheres and their PDEs	p. 47
Improper affine hyperspheres	p. 47
Proper affine hyperspheres	p. 48
The Pick invariant on affine hyperspheres	p. 49
Completeness in Affine Geometry	p. 50
Affine completeness and Euclidean completeness	p. 50
The Cheng-Yau criterion for affine completeness	p. 51
Proof of the Estimate Lemma	p. 53
Topology and the equiaffine Gauß map	p. 56
Affine Complete Elliptic Affine Hyperspheres	p. 59
The Theorem of Jörgens-Calabi-Pogorelov	p. 59
An Extension of the Theorem of Jörgens-Calabi-Pogorelov	p. 61
Affine Kähler Ricci flat equation	p. 61
Tools from relative geometry	p. 63
Calculation of ¿¿ in terms of the Calabi metric	p. 63
Extension of the Theorem of Jörgens-Calabi-Pogorelov - Proof for n ≤ 4	p. 66
Comparison of two geometric proofs	p. 68
Technical tools for the proof in dimension n ≥ 5	p. 69
Proof of Theorem 4.5.1 - n ≥ 5	p. 79
A Cubic Form Differential Inequality with its Applications	p. 82
Calculation of ¿J in terms of the Calabi metric	p. 83
Proof of Theorem 4.6.2	p. 85
Affine Maximal Hypersurfaces	p. 89
The First Variation of the Equiaffine Volume Functional	p. 89
Affine Maximal Hypersurfaces	p. 92
Graph hypersurfaces	p. 92
The PDE for affine maximal hypersurfaces	p. 95
An Affine Analogue of the Weierstrass Representation	p. 96
The representation formula	p. 96
Examples	p. 99
Calabi's Computation of ¿J in Holomorphic Terms	p. 99
Computation of ¿ (J + B 2)	p. 104
Calabi's Conjecture	p. 105
Proof of Calabi's Conjecture for dimension n = 2	p. 106
Chern's Conjecture	p. 110
Technical estimates	p. 112
Estimates for the determinant of the Hessian	p. 114
Estimates for the third order derivatives	p. 121
Estimates for ¿ fii	p. 126
Proof of Theorem 5.6.2	p. 128
An Affine Bernstein Problem in Dimension 3	p. 131
Proof of Part I	p. 131
Proof of Part II: Affine blow-up analysis	p. 133
Another Method of Proof for some Fourth Order PDEs	p. 138
Euclidean Completeness and Calabi Completeness	p. 144
Hypersurfaces with Constant Affine Mean Curvature	p. 149
Classification	p. 149
Estimates for the determinant of the Hessian	p. 150
Proof of Theorem 6.1.1	p. 151
Proof of Theorem 6.1.2	p. 160
Hypersurfaces with Negative Constant Mean Curvature	p. 161
Proof of the existence of a solution	p. 165
Proof of the Euclidean completeness	p. 169
Bibliography	p. 173
Index	p. 179
Table of Contents provided by Ingram. All Rights Reserved.