De Gruyter Expositions in Mathematics Volume 11 · Global Affine Differential Geometry of Hypersurfaces by Li, An-Min -- Read -- Imperial Library of Trantor

Index

Cover Title Contents Introduction 1 Preliminaries and basic structural aspects

1.1 Affine spaces

1.1.1 Basic notations for affine spaces 1.1.2 Volume and orientation 1.1.3 Dual determinant forms 1.1.4 Duality and cross-product construction 1.1.5 Affine mappings and transformation groups 1.1.6 Affine invariance and duality 1.1.7 Directional derivatives 1.1.8 Frames

1.2 Euclidean spaces

1.2.1 Basic notations for Euclidean spaces 1.2.2 Normed determinant forms 1.2.3 Different Euclidean structures 1.2.4 Isometries and orthogonal transformations

1.3 Differential geometric structures of affine and Euclidean space

1.3.1 Differential geometric equiaffine structures 1.3.2 Differential geometric Euclidean structures 1.3.3 ℝn+1 as a standard space

1.4 Klein’s Erlangen program 1.5 Motivation: A short sketch of the Euclidean hypersurface theory 1.6 Hypersurfaces in equiaffine space

1.6.1 Definition and notation 1.6.2 Bundles

1.7 Structural motivation for further investigations 1.8 Transversal fields and induced structures

1.8.1 Volume form 1.8.2 Weingarten structure equation 1.8.3 Gauß structure equation 1.8.4 Compatibility of the induced volume form and the connection 1.8.5 Relative normal fields

1.9 Conormal fields and induced structures

1.9.1 Volume form 1.9.2 Bilinear form

1.10 Relative normalizations 1.11 Nondegenerate hypersurfaces 1.12 Gauß structure equations for conormal fields

1.12.1 Compatibility of connections and volume forms in the conormal bundle 1.12.2 Compatibility of bilinear forms and connections; conjugate connections 1.12.3 Cubic form 1.12.4 Tchebychev form 1.12.5 Local notation 1.12.6 Relative Gauß maps

1.13 Affine invariance of the induced structures 1.14 A summary of relative hypersurface theory

1.14.1 Structure equations in terms of ∇ 1.14.2 Integrability conditions in terms of ∇

1.15 Special relative normalizations

1.15.1 Euclidean normalization as a relative normalization 1.15.2 Equiaffine normalization 1.15.3 Centroaffine normalization

1.16 Comparison of relative normalizations

1.16.1 Comparison of corresponding geometric properties 1.16.2 Gauge invariance

2 Local equiaffine hypersurface theory

2.1 Blaschke–Berwald metric and structure equations 2.2 The affine normal and the Fubini–Pick form

2.2.1 Affine normal 2.2.2 Fubini–Pick form 2.2.3 Affine curvatures 2.2.4 Geometric meaning of the affine normal

2.3 The equiaffine conormal

2.3.1 Properties of the equiaffine conormal 2.3.2 The affine support function

2.4 Hyperquadrics

2.4.1 Hyperquadrics 2.4.2 Hypersurfaces with vanishing Pick invariant

2.5 Integrability conditions and the local fundamental theorem

2.5.1 Relations between the coefficients 2.5.2 Integrability conditions 2.5.3 The fundamental theorem

2.6 Euclidean boundary points of locally convex immersed hypersurfaces 2.7 Graph immersions

2.7.1 Graph immersions with equiaffine normalization 2.7.2 Graph Immersions with Calabi metric

3 Affine hyperspheres

3.1 Definitions and basic results for affine hyperspheres

3.1.1 Definition of affine hyperspheres 3.1.2 Differential equations for affine hyperspheres 3.1.3 Calabi compositions

3.2 Affine hyperspheres with constant sectional curvature

3.2.1 Examples 3.2.2 Local classification of two-dimensional affine spheres with constant scalar curvature 3.2.3 Generalization to higher dimensions

3.3 Affine hypersurfaces with parallel Fubini–Pick form

3.3.1 Implications from a parallel Fubini–Pick form 3.3.2 Classification of affine hypersurfaces with parallel Fubini–Pick form

3.4 Affine completeness, Euclidean completeness, and Calabi completeness

3.4.1 Euclidean completeness and affine completeness 3.4.2 The equivalence between Calabi metrics and Euclidean metrics 3.4.3 Remarks

3.5 Affine complete elliptic affine hyperspheres 3.6 A differential inequality on a complete Riemannian manifold 3.7 Estimates of the Ricci curvatures of affine complete affine hyperspheres of parabolic or hyperbolic type 3.8 Qualitative classification of complete hyperbolic affine hyperspheres

3.8.1 Euclidean complete hyperbolic affine hyperspheres 3.8.2 Affine complete hyperbolic affine hyperspheres 3.8.3 Proof of the second part of the Calabi conjecture

3.9 Complete hyperbolic affine 2-spheres

3.9.1 A splitting of the Levi–Civita operator

4 Rigidity and uniqueness theorems

4.1 Integral formulas for affine hypersurfaces and their applications

4.1.1 Minkowski-type integral formulas for affine hypersurfaces 4.1.2 Characterization of ellipsoids 4.1.3 Some further characterizations of ellipsoids 4.1.4 Global solutions of a differential equation of Schrödinger type 4.1.5 Rigidity theorems for ovaloids 4.1.6 Hypersurfaces with boundary

4.2 The index method

4.2.1 Fields of line elements and nets 4.2.2 Vekua’s system of partial differential equations 4.2.3 Affine Weingarten surfaces 4.2.4 An affine analogue of the Cohn–Vossen theorem

5 Variational problems and affine maximal surfaces

5.1 Variational formulas for higher affine mean curvatures 5.2 Affine maximal hypersurfaces

5.2.1 Definitions and fundamental results 5.2.2 An affine analogue of the Weierstraß representation 5.2.3 Construction of affine maximal surfaces 5.2.4 Construction of improper (parabolic) affine spheres 5.2.5 Affine Bernstein problems

5.3 Differential inequalities for Δ (B) J, Δ (B) Φ , and Δ (C) Φ

5.3.1 Notations from E. Calabi 5.3.2 Computation of Δ (B) J 5.3.3 Computation of Δ (B) (‖B‖2 ) 5.3.4 Estimations of Δ (C) Φ and Δ (B) Φ

5.4 Proof of Calabi’s conjecture in dimension 2 5.5 Chern’s affine Bernstein conjecture

5.5.1 Some tools from p.d.e. theory 5.5.2 A partial result on Chern’s conjecture in arbitrary dimensions 5.5.3 Proof of Chern’s conjecture in dimension 2 5.5.4 Estimates for the determinant of the Hessian 5.5.5 First proof of Lemma 5.29

5.6 An analytic proof of Chern’s conjecture in dimension 2

5.6.1 Technical estimates 5.6.2 Estimates for ∑ fii 5.6.3 Estimates for the third order derivatives 5.6.4 Second proof of Lemma 5.29

5.7 An affine Bernstein problem with respect to the Calabi metric

6 Hypersurfaces with constant affine Gauß–Kronecker curvature

6.1 The affine Gauß–Kronecker curvature

6.1.1 Motivation 6.1.2 Main results

6.2 Splitting of the fourth order PDE Sn = const into two (second order) Monge–Ampère equations 6.3 Construction of Euclidean complete hypersurfaces with constant affine G-K curvature 6.4 Completeness with respect to the Blaschke metric

7 Geometric inequalities

7.1 The affine isoperimetric inequality

7.1.1 Steiner symmetrization 7.1.2 A characterization of ellipsoids 7.1.3 The affine isoperimetric inequality

7.2 Inequalities for higher affine mean curvatures

7.2.1 Mixed volumes 7.2.2 Integral inequalities for curvature functions 7.2.3 Total centroaffine area

A Basic concepts from differential geometry

A.1 Tensors and exterior algebra

A.1.1 Tensors A.1.2 Exterior algebra

A.2 Differentiable manifolds

A.2.1 Differentiable manifolds and submanifolds A.2.2 Tensor fields on manifolds A.2.3 Integration on manifolds

A.3 Affine connections and Riemannian geometry: Basic facts

A.3.1 Affine connections A.3.2 Riemann manifolds A.3.3 Manifolds of constant curvature, Einstein manifolds A.3.4 Examples A.3.5 Exponential mapping and completeness

A.4 Green’s formula

BLaplacian comparison theorem Bibliography Index

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