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

We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound Ric∞ ≥ 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.