Isoperimetric inequality on Finsler metric measure manifolds with non-negative weighted Ricci curvature

TL;DR

This paper establishes a sharp isoperimetric inequality on Finsler manifolds with non-negative weighted Ricci curvature, linking volume entropy, Cheeger constants, and eigenvalues of the Finsler Laplacian.

Contribution

It introduces new inequalities involving volume entropy and Cheeger constants on Finsler manifolds, extending classical results to this broader geometric setting.

Findings

Proves a sharp isoperimetric inequality involving volume entropy.

Establishes a Cheeger-Buser type inequality for the first eigenvalue of Finsler Laplacian.

Connects geometric and spectral properties on Finsler manifolds.

Abstract

In this paper, we define the volume entropy and the second Cheeger constant and prove a sharp isoperimetric inequality involving the volume entropy on Finsler metric measure manifolds with non-negative weighted Ricci curvature ${\rm Ric}_{\infty}$. As an application, we prove a Cheeger-Buser type inequality for the first eigenvalue of Finsler Laplacian by using the volume entropy and the second Cheeger constant.