Some important applications of improved Bochner inequality on Finsler manifolds

TL;DR

This paper uses an improved Bochner inequality to derive key geometric inequalities on Finsler manifolds with positive weighted Ricci curvature, including sharp Poincaré-Lichnerowicz, logarithmic Sobolev, and volume estimates.

Contribution

It introduces an enhanced Bochner inequality approach to establish fundamental inequalities on Finsler manifolds with positive Ricci curvature.

Findings

Proved a sharp Poincaré-Lichnerowicz inequality.

Provided a new proof for the logarithmic Sobolev inequality.

Estimated the volume of geodesic balls.

Abstract

We establish some important inequalities under the condition that the weighted Ricci curvature $\mathrm{Ric}_{\infty}\geq K$ for some constant $K >0$ by using improved Bochner inequality and its integrated form. Firstly, we obtain a sharp Poincar\'{e}-Lichnerowicz inequality. Further, we give a new proof for logarithmic Sobolev inequality. Finally, we obtain an estimate of the volume of geodesic balls.