Geometry and Analysis on Finsler Spaces

Contents
Preface
1 Minkowski Spaces
	1.1 Minkowski Norm
	1.2 Legendre Transformations
	1.3 Cartan Torsion
2 Finsler Manifolds
	2.1 Finsler Metrics and Examples
	2.2 Cartan Tensor and Hilbert Form
		2.2.1 Fundamental Tensor and Cartan Tensor
		2.2.2 Hilbert Form
	2.3 Volume Measure
		2.3.1 Volume Form
		2.3.2 Busemann–Hausdorff Volume Form
		2.3.3 Holmes–Thompson Volume Form
	2.4 Divergence Lemma
3 Connections and Structure Equations
	3.1 Chern Connection
	3.2 Structure Equations
	3.3 Bianchi Identities and Ricci Identities
4 Curvature Invariant Quantities
	4.1 Riemannian Geometric Quantities
		4.1.1 Flag Curvature and Ricci Curvature
		4.1.2 Geodesic Fields
	4.2 Non-Riemannian Geometric Quantities
		4.2.1 Chern Curvature and Landsberg Curvature
		4.2.2 S-Curvature and the Related Curvatures
	4.3 Projectively Geometric Quantities
		4.3.1 Projective Equivalence
		4.3.2 Projectively Invariant Quantities
	4.4 Relations Among Geometric Invariant Quantities
	4.5 Some Rigidity Theorems
5 Theory of Geodesics
	5.1 Geodesics and Exponential Map
		5.1.1 Geodesics
		5.1.2 Exponential Map
		5.1.3 Geodesic Completeness
	5.2 Second Variation of Length
	5.3 Jacobi Fields and Conjugate Points
	5.4 First Conjugate Point and Cut Point
6 Comparison Theorems
	6.1 Gradient, Hessian and Finsler Laplacian
	6.2 Rauch Comparison Theorem
	6.3 Hessian and Laplacian Comparison Theorems
	6.4 Volume Comparison Theorem
7 Finsler Harmonic Functions
	7.1 Bochner–Weitzenböck Formula
		7.1.1 Pointwise Bochner–Weitzenböck Formula
		7.1.2 Integrated Bochner–Weitzenböck Formula and Inequality
	7.2 Mean Value Inequality
	7.3 Some Lp Liouville Theorems
		7.3.1 Lp(p > 1) Liouville Theorems
		7.3.2 Lp(0 < p ≤ 1) Liouville Theorems
	7.4 Gradient Estimates
		7.4.1 Compact Case
		7.4.2 Complete and Noncompact Case
8 The Eigenvalue Problem
	8.1 Eigenvalue and Eigenfunction
	8.2 Upper Bound Estimates for λ1
	8.3 Lower Bound Estimates for λ1
		8.3.1 General Lower Bound Estimates
		8.3.2 Sharp Lower Bound Estimates
	8.4 Characterizations of Extreme Manifolds
		8.4.1 Case 1. K = 0
		8.4.2 Case 2. K > 0
	8.5 Gradient Estimates for Eigenfunctions
9 Heat Flow on Finsler Manifolds
	9.1 Existence and Regularity
	9.2 Lp-Uniqueness
	9.3 Li-Yau’s Gradient Estimates
		9.3.1 Compact Case
		9.3.2 Complete and Noncompact Case
10 Appendix: Sobolev Spaces on Compact Finsler Manifolds
Bibliography