A semigroup approach to Finsler geometry: Bakry-Ledoux's isoperimetric inequality

TL;DR

This paper extends the semigroup approach to Finsler geometry, establishing gradient estimates and isoperimetric inequalities for both reversible and non-reversible Finsler manifolds, based on a nonlinear Laplacian and heat semigroup.

Contribution

It develops a nonlinear semigroup framework for Finsler manifolds, proving gradient estimates and isoperimetric inequalities that generalize previous results to non-reversible metrics.

Findings

Established $L^1$-gradient estimates under certain conditions

Proved Bakry-Ledoux's Gaussian isoperimetric inequality for Finsler manifolds

Extended inequalities to non-reversible Finsler metrics

Abstract

We develop the celebrated semigroup approach \`a la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenb\"ock formula established by Sturm and the author. We show the $L^1$-gradient estimate on Finsler manifolds (under some additional assumptions in the noncompact case), which is equivalent to a lower weighted Ricci curvature bound and the improved Bochner inequality. As a geometric application, we prove Bakry-Ledoux's Gaussian isoperimetric inequality, again under some additional assumptions in the noncompact case. This extends Cavalletti-Mondino's inequality on reversible Finsler manifolds to non-reversible metrics, and improves the author's previous estimate, both based on the localization (also called needle decomposition) method.