Communications in Analysis and Geometry

Volume 30 (2022)

Number 10

A semigroup approach to Finsler geometry: Bakry–Ledoux’s isoperimetric inequality

Pages: 2347 – 2387



Shin-Ichi Ohta (Department of Mathematics, Osaka University, Osaka, Japan; and RIKEN Center for Advanced Intelligence Project (AIP), Tokyo, Japan)


We develop the celebrated semigroup approach à la Bakry et al. on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner–Weitzenböck 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.

This work was supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 15K04844, 19H01786.

Received 19 September 2018

Accepted 1 July 2020

Published 29 September 2023