Geometric and Functional Inequalities on Finsler Manifolds

Abstract

In this paper, we give an (integrated) $p(>1)$-Bochner–Weitzenböck formula and the p-Reilly type formula on Finsler manifolds. As applications, we obtain the p-Poincaré inequality on an n-dimensional compact Finsler manifold without boundary or with convex boundary under the assumption that Ricci$_N\ge K$ for $N\in [n, \infty ]$ and $K\in \mathbb {R}$. In fact, we have the sharp p-Poincaré inequality on such a manifold. Based on this, we further prove the existence of two types of optimal $(\mathfrak p, \mathfrak q)$-Sobolev inequalities on compact Finsler manifolds with Ric$_N\ge K$ for $N\in [n, \infty )$ and $K\in \mathbb {R}$. In particular, when $K>0$, we establish the sharp $(\mathfrak p, 2)$-Sobolev inequality for $2\le \mathfrak p\le 2N/(N-2)$. With this, we obtain the sharp lower bound estimate for the first eigenvalue $\lambda _1$ of the Finsler Laplacian and characterize the manifold on which $\lambda _1$ reaches its extremum, generalizing the well known Lichnerowicz–Obata’s results (in: Geometrie des groupes de transformations, Dunod, Paris, 1958; J Math Soc Jpn 14:333–340, 1962).