Affine Bernstein Problems and Monge-Ampère Equations

Contents......Page 10
Preface......Page 6
1.1.1 Manifolds, connections and exterior calculus......Page 14
1.1.2 Riemannian manifolds......Page 17
1.1.3 Curvature inequalities......Page 18
1.1.4 Geodesic balls and level sets......Page 19
1.2.2 Maximum principles......Page 20
1.3 Comparison Theorems......Page 21
1.4 The Legendre Transformation......Page 22
2.1.1 The ambient space......Page 24
2.1.2 Affine hypersurfaces......Page 26
2.2.1 Structure equations - preliminary version......Page 27
2.3.1 The affine normal......Page 29
2.3.2 Affine shape operator and affine extrinsic curvature......Page 31
2.3.3 The affine conormal......Page 32
2.3.5 Affine Gau  mappings......Page 34
2.4 The Fubini-Pick Form......Page 35
2.4.3 Structure equations - covariant notation......Page 36
2.5.1 Integration via moving frames......Page 37
2.5.2 Covariant form of the integrability conditions......Page 39
2.7 Graph Immersions with Unimodular Normalization......Page 40
2.8.1 Affine hyperspheres......Page 43
2.8.2 Characterization of quadrics......Page 44
3.1.1 Structure equations......Page 46
3.2 Hypersurfaces with Relative Normalization......Page 48
3.2.1 Relative structure equations and basic invariants......Page 49
3.2.4 Classical version of the fundamental theorem......Page 51
3.3.2 The equiaffine (Blaschke) normalization......Page 52
3.3.3 The centroaffine normalization......Page 53
3.3.4 Graph immersions with Calabi metric......Page 54
3.3.5 The family of conformal metrics G( )......Page 55
3.4 Gauge Invariance and Relative Geometry......Page 56
4.1.1 Improper affine hyperspheres......Page 60
4.1.2 Proper affine hyperspheres......Page 61
4.1.3 The Pick invariant on affine hyperspheres......Page 62
4.2.1 Affine completeness and Euclidean completeness......Page 63
4.2.2 The Cheng-Yau criterion for affine completeness......Page 64
4.2.3 Proof of the Estimate Lemma......Page 66
4.2.4 Topology and the equia ne Gau  map......Page 69
4.4 The Theorem of Jorgens-Calabi-Pogorelov......Page 72
4.5.1 Affine Kahler Ricci flat equation......Page 74
4.5.3 Calculation of    in terms of the Calabi metric......Page 76
4.5.4 Extension of the Theorem of Jorgens-Calabi-Pogorelov proof for n   4......Page 79
4.5.5 Comparison of two geometric proofs......Page 81
4.5.6 Technical tools for the proof in dimension n   5......Page 82
I. A gradient estimate for......Page 83
II. Further estimates.......Page 85
4.5.7 Proof of Theorem 4.5.1 - n   5......Page 92
4.6 A Cubic Form Differential Inequality with its Applications......Page 95
4.6.1 Calculation of  J in terms of the Calabi metric......Page 96
4.6.2 Proof of Theorem 4.6.2......Page 98
5.1 The First Variation of the Equiaffine Volume Functional......Page 102
5.2.1 Graph hypersurfaces......Page 105
5.2.2 The PDE for affine maximal hypersurfaces......Page 108
5.3.1 The representation formula......Page 109
5.4 Calabi\'s Computation of  J in Holomorphic Terms......Page 112
5.4.1 Computation of   (J + kBk2)......Page 117
5.5 Calabi\'s Conjecture......Page 118
5.5.1 Proof of Calabi\'s Conjecture for dimension n = 2......Page 119
5.6 Chern\'s Conjecture......Page 123
5.6.1 Technical estimates......Page 125
5.6.2 Estimates for the determinant of the Hessian......Page 127
5.6.3 Estimates for the third order derivatives......Page 134
5.6.4 Estimates for   fii......Page 139
5.6.5 Proof of Theorem 5.6.2......Page 141
5.7.1 Proof of Part I......Page 144
5.7.2 Proof of Part II: Affine blow-up analysis......Page 146
5.8 Another Method of Proof for some Fourth Order PDEs......Page 151
5.9 Euclidean Completeness and Calabi Completeness......Page 157
6.1 Classification......Page 162
6.1.1 Estimates for the determinant of the Hessian......Page 163
6.1.2 Proof of Theorem 6.1.1......Page 164
6.1.3 Proof of Theorem 6.1.2......Page 173
6.2 Hypersurfaces with Negative Constant Mean Curvature......Page 174
6.2.1 Proof of the existence of a solution......Page 178
6.2.2 Proof of the Euclidean completeness......Page 182
Bibliography......Page 186
Index......Page 192