Nonlinear geometric analysis on Finsler manifolds

TL;DR

This survey reviews recent advances in comparison geometry and geometric analysis on Finsler manifolds with weighted Ricci curvature bounds, focusing on nonlinear techniques and their applications.

Contribution

It provides a concise overview of the development of weighted Ricci curvature and introduces nonlinear Gamma-calculus methods on Finsler manifolds.

Findings

Discussion of gradient estimates and functional inequalities

Explanation of isoperimetric inequalities in Finsler geometry

Summary of the nonlinear Gamma-calculus approach

Abstract

This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear analogue of the $\Gamma$-calculus based on the Bochner inequality. In the latter we discuss some gradient estimates, functional inequalities, and isoperimetric inequalities.