Some Inequalities on Finsler Manifolds with Weighted Ricci Curvature Bounded Below

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 Poincare-Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a better lower bound for the first eigenvalue on the Finsler manifolds.

Automated summary

In this paper, we establish some important inequalities on Finsler manifolds under a lower weighted Ricci curvature bound. These include a relative volume comparison, an upper bound for volumes, a Bonnet-Myers type theorem, and a sharp Poincare-Lichnerowicz inequality.