A rigidity result of spectral gap on Finsler manifolds and its application

TL;DR

This paper establishes a rigidity theorem for the spectral gap on Finsler manifolds with positive Ricci curvature, showing that equality implies a splitting structure and deriving related inequalities as corollaries.

Contribution

It proves a new rigidity result linking spectral gap equality to manifold splitting in Finsler geometry, extending classical results to this broader setting.

Findings

Equality in the spectral gap implies manifold splitting.

Rigidity results for logarithmic Sobolev inequalities.

Rigidity results for Bakry-Ledoux isoperimetric inequalities.

Abstract

We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\text{Ric}_{\infty} \geq K > 0$. Our main results show that if the equality holds, the manifold necessarily admits a diffeomorphic splitting (or isometric splitting in the particular class of Berwald spaces). This splitting phenomenon is comparable to the Cheeger-Gromoll type splitting theorem by Ohta. We also obtain the rigidity results of logarithmic Sobolev and Bakry-Ledoux isoperimetric inequalities via needle decomposition as corollaries.