Tohoku Mathematical Journal
2024
December
SECOND SERIES VOL. 76, NO. 4

Tohoku Math. J.
76 (2024), 561-575

Title A RIGIDITY RESULT OF SPECTRAL GAP ON FINSLER MANIFOLDS AND ITS APPLICATION

Author Cong Hung Mai

(Received September 20, 2022, revised March 31, 2023)
Abstract. We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\mathrm{Ric}_{\infty} \geq K > 0$. 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.

Mathematics Subject Classification. Primary 53C60; Secondary 53C24.
Key words and phrases. Finsler manifold, weighted Ricci curvature, sharp spectral gap, needle decomposition, logarithmic Sobolev inequality, isoperimetric inequality.

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