Concentration and relevant properties of Finsler metric measure manifolds

TL;DR

This paper explores the concentration phenomena in Finsler metric measure manifolds, linking them to isoperimetric inequalities, eigenvalues, and observable diameter, thereby extending the concentration theory in Finsler geometry.

Contribution

It systematically studies concentration properties in Finsler manifolds and establishes new relationships with geometric and spectral invariants, including a Cheng type eigenvalue estimate.

Findings

Established links between concentration and isoperimetric inequalities.

Derived a Cheng type upper bound for the first eigenvalue.

Extended concentration theory to irreversible Finsler spaces.

Abstract

In this paper, we study systematically the concentration properties of Finsler metric measure manifolds. We establish the relationships between the concentration properties and the observable diameter, isoperimetric inequalities and the first eigenvalue. In particular, as an application, we derive a Cheng type upper bound estimate for the first closed eigenvalue via the concentration property. The researches in this paper enrich and extend the concentration theory in Finsler geometry, even in irreversible metric measure spaces.