Comparison Finsler Geometry (Springer Monographs in Mathematics)

✍ Scribed by Shin-ichi Ohta

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 324
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This monograph presents recent developments in comparison geometry and geometric analysis on Finsler manifolds. Generalizing the weighted Ricci curvature into the Finsler setting, the author systematically derives the fundamental geometric and analytic inequalities in the Finsler context. Relying only upon knowledge of differentiable manifolds, this treatment offers an accessible entry point to Finsler geometry for readers new to the area.

Divided into three parts, the book begins by establishing the fundamentals of Finsler geometry, including Jacobi fields and curvature tensors, variation formulas for arc length, and some classical comparison theorems. Part II goes on to introduce the weighted Ricci curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler manifolds. These tools allow the derivation of the Bochner–Weitzenböck formula and the corresponding Bochner inequality, gradient estimates, Bakry–Ledoux’s Gaussian isoperimetric inequality, and functional inequalities in the Finsler setting. Part III comprises advanced topics: a generalization of the classical Cheeger–Gromoll splitting theorem, the curvature-dimension condition, and the needle decomposition. Throughout, geometric descriptions illuminate the intuition behind the results, while exercises provide opportunities for active engagement.

Comparison Finsler Geometry offers an ideal gateway to the study of Finsler manifolds for graduate students and researchers. Knowledge of differentiable manifold theory is assumed, along with the fundamentals of functional analysis. Familiarity with Riemannian geometry is not required, though readers with a background in the area will find their insights are readily transferrable.

✦ Table of Contents

Preface
Contents
List of Definitions and Formulas
Part I
Part II
Part I Foundations of Finsler Geometry
1 Warm-Up: Norms and Inner Products
1.1 Norms and Inner Products
1.2 Three Characterizations of Inner Products
1.2.1 Sharp Uniform Convexity and Smoothness
1.2.2 Smoothness at the Origin
1.2.3 Center of Circumscribed Triangle
2 Finsler Manifolds
2.1 Minkowski Normed Spaces
2.2 Euler's Homogeneous Function Theorem
2.3 Finsler Manifolds
2.4 Asymmetric Distance and Geodesics
2.5 Reverse Finsler Structures
3 Properties of Geodesics
3.1 Fundamental and Cartan Tensors
3.2 Dual Norms and the Legendre Transformation
3.3 The Geodesic Equation
3.4 The Exponential Map
3.5 Completenesses and the Hopf–Rinow Theorem
4 Covariant Derivatives
4.1 The Geodesic Equation Revisited
4.2 Covariant Derivatives
4.3 Covariant Derivatives Along Curves
4.4 The Chern Connection
5 Curvature
5.1 Jacobi Fields and the Curvature Tensor
5.2 Properties of the Curvature Tensor
5.3 Flag and Ricci Curvatures and Their Characterizations
5.4 Further Properties of the Curvature Tensor
6 Examples of Finsler Manifolds
6.1 Minkowski Normed Spaces
6.2 Finsler Manifolds of Constant Curvature
6.3 Berwald Spaces
6.3.1 Isometry of Tangent Spaces and Its Applications
6.3.2 T-Curvature
6.3.3 Characterizations of Berwald Spaces
6.4 Randers Spaces
6.5 Hilbert and Funk Geometries
6.6 Teichmüller Space
7 Variation Formulas for Arclength
7.1 First Variation Formula
7.2 Second Variation Formula
7.3 Cut Points and Conjugate Points
8 Some Comparison Theorems
8.1 The Bonnet–Myers Theorem
8.2 The Cartan–Hadamard Theorem
8.3 Uniform Convexity and Smoothness
8.3.1 Background: k-Convexity and k-Concavity
8.3.2 Uniform Convexity and Smoothness Constants
8.3.3 T-Curvature Revisited
8.3.4 k-Concavity of (M,F)
8.3.5 k-Convexity of (M,F)
8.4 Busemann NPC for Berwald Spaces
Part II Geometry and Analysis of Weighted Ricci Curvature
9 Weighted Ricci Curvature
9.1 Measures on Finsler Manifolds
9.2 Riemannian Weighted Ricci Curvature
9.3 Finsler Weighted Ricci Curvature
9.4 Volume and Diameter Comparison Theorems
10 Examples of Measured Finsler Manifolds
10.1 Minkowski Normed Spaces
10.2 Berwald Spaces
10.3 Randers Spaces
10.3.1 Properties of the S-Curvature
10.3.2 Randers Spaces of Vanishing S-Curvature
10.4 Hilbert and Funk Geometries
11 The Nonlinear Laplacian
11.1 Energy Functional and Sobolev Spaces
11.2 Laplacian and Harmonic Functions
11.3 Laplacian Comparison Theorem
11.4 Linearized Laplacians
12 The Bochner–Weitzenböck Formula
12.1 Hessian
12.2 Pointwise Formula
12.3 Integrated Formula
12.4 Improved Bochner Inequality
13 Nonlinear Heat Flow
13.1 Global Solutions
13.2 Existence
13.3 Large Time Behavior
13.4 Regularity
13.5 Linearized Heat Semigroups and Their Adjoints
14 Gradient Estimates
14.1 L2-Gradient Estimate
14.2 L1-Gradient Estimate
14.3 Characterizations of Lower Ricci Curvature Bounds
14.4 The Li–Yau Estimates
15 Bakry–Ledoux Isoperimetric Inequality
15.1 Background
15.2 Poincaré–Lichnerowicz Inequality and Variance Decay
15.3 The Key Estimate
15.4 Proof of Theorem 15.1
16 Functional Inequalities
16.1 Logarithmic Sobolev Inequality
16.1.1 Entropy Decay
16.1.2 Logarithmic Sobolev Inequality
16.2 Beckner Inequality
16.3 Sobolev Inequality
16.3.1 Logarithmic Entropy-Energy and Nash Inequalities
16.3.2 Sharp Sobolev Inequality
16.3.3 Addendum to the Proof of Theorem 16.17
Part III Further Topics
17 Splitting Theorems
17.1 Busemann Functions
17.2 Diffeomorphic Splitting
17.3 The Berwald Case
18 Curvature-Dimension Condition
18.1 Optimal Transport Theory
18.2 Curvature-Dimension Condition
18.3 Brunn–Minkowski Inequality
18.4 Analytic Applications
18.4.1 Functional Inequalities
18.4.2 Concentration of Measures
18.5 Further Developments
18.5.1 Riemannian Curvature-Dimension Condition
18.5.2 Heat Flow as Gradient Flow
18.5.3 Measure Contraction Property
19 Needle Decompositions
19.1 Lipschitz Functions and Optimal Transports
19.1.1 Transport Rays
19.1.2 Cyclical Monotonicity
19.2 Construction of Needle Decompositions
19.2.1 Transport Sets
19.2.2 Disintegration
19.2.3 Conditioned Version
19.3 Properties of Needles
19.4 Isoperimetric Inequalities
19.5 Further Applications
References
Index