Geometric analysis and comparison geometry on weighted manifolds

在此研究期间，我研究了Finsler流形背景下尖锐谱隙的刚性问题。研究了Finsler流形上尖锐谱隙的刚性问题，得到了其微分同胚分裂（或在特定的Berwald空间中等距分裂）。通过与黎曼情况类似的技术，我还通过可逆Finsler流形的针分解展示了对数Sobolev和Bakry-Ledoux等周不等式的刚性结果。这种分裂现象可与Ohta的Cheeger-Gromoll型分裂定理相比较。

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.