Concentration and relevant properties of Finsler metric measure manifolds
Xinyue Cheng, Yalu Feng
TL;DR
The paper develops a systematic concentration of measure framework for irreversible Finsler metric measure manifolds, linking concentration to observable diameter, isoperimetric profiles, and the first eigenvalue. It builds foundational machinery for irreversible metric spaces (including Finsler Laplacian and weighted Ricci curvature) and then derives broad concentration results, including normal, exponential, and moment-type inequalities, specialized to Finsler geometry. It establishes fundamental connections between observable diameter and concentration, and between isoperimetric inequalities and concentration, under curvature-dimension-type conditions. A highlight is the exponential concentration bound derived from the first eigenvalue on closed Finsler manifolds, along with a Cheng-type upper bound for the first eigenvalue, extending classical spectral-concentration relationships to non-reversible Finsler settings.
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.