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

Abstract

We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound ${\text{Ric}}_{\infty }\ge K>0$. 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.

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