Geometric analysis and comparison geometry on weighted manifolds

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During this research period, I study the studies on the rigidity of sharp spectral gaps in the setting of Finsler manifolds. I investigated the rigidity problem for the sharp spectral gap on Finsler manifolds and obtained the diffeomorphic splitting (or isometric splitting in the particular class of Berwald spaces). By analogous techniques to the Riemannian case, I also exhibit the rigidity results of logarithmic Sobolev and Bakry-Ledoux isoperimetric inequalities via needle decomposition for reversible Finsler manifolds. This splitting phenomenon is comparable to the Cheeger-Gromoll type splitting theorem by Ohta.