Some inequalities on Finsler manifolds with weighted Ricci curvature bounded below

TL;DR

This paper derives key geometric inequalities for Finsler manifolds with lower bounds on weighted Ricci curvature, including volume comparison, diameter bounds, and eigenvalue estimates, extending classical Riemannian results to Finsler geometry.

Contribution

It establishes new inequalities under weighted Ricci curvature bounds on Finsler manifolds, including volume comparison, Bonnet-Myers type theorems, and sharp eigenvalue bounds.

Findings

Volume of Finsler manifolds is bounded above under curvature conditions.

Diameter bounds analogous to Bonnet-Myers theorem are obtained.

First eigenvalue has a sharp lower bound derived from Bochner inequality.

Abstract

We establish some important inequalities under a lower weighted Ricci curvature bound on Finsler manifolds. Firstly, we establish a relative volume comparison of Bishop-Gromov type. As one of the applications, we obtain an upper bound for volumes of the Finsler manifolds. Further, when the S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet-Myers type on Finsler manifolds. Finally, we obtain a sharp Poincar\'{e}-Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a sharp lower bound for the first eigenvalue on the Finsler manifolds.