Comparison Finsler Geometry

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.

本論文介紹了Finsler流形上比較幾何學和幾何分析的最新發展。作者將加權Ricci曲率推廣到Finsler設置中，系統地推導了Finsler背景下的基本幾何和分析不等式。本書只依賴於對可微流形的知識，為初次接觸Finsler幾何的讀者提供了一個易於理解的入門點。

本書分為三個部分，首先建立了Finsler幾何的基礎，包括Jacobi場和曲率張量、弧長變分公式以及一些經典的比較定理。第二部分介紹了加權Ricci曲率、非線性Laplacian和Finsler流形上的非線性熱流。這些工具使得可以推導出Bochner-Weitzenböck公式和相應的Bochner不等式、梯度估計、Bakry-Ledoux的高斯等周不等式以及Finsler設置中的函數不等式。第三部分包括高級主題：對經典的Cheeger-Gromoll分裂定理的推廣、曲率-維度條件和針分解。在整個過程中，幾何描述使結果背後的直覺變得清晰，而練習提供了積極參與的機會。

《比較Finsler幾何學》為研究生和研究人員提供了研究Finsler流形的理想入門。假設讀者具備可微流形理論的知識以及基礎的函數分析知識。雖然不需要熟悉黎曼幾何，但在該領域有背景的讀者會發現他們的見解很容易轉移到Finsler幾何中。