Comparison Theorems in Finsler Geometry With Weighted Curvature Bounds and Related Results

Abstract

Abstract. We first extend the notions of weighted curvatures, including the weighted flag curvature and the weighted Ricci curvature, for a Finsler manifold with given volume form. Then we establish some basic comparison theorems for Finsler manifolds with various weighted curvature bounds. As applications, we obtain some McKean type theorems for the first eigenvalue of Finsler manifolds, some results on weighted curvature and fundamental group for Finsler manifolds, as well as an estimation of Gromov simplicial norms for reversible Finsler manifolds.

Uniqueness of subsolutions to the heat equation on Finsler measure spaces

“…By Theorem 3.1 and the assumption, we have for any and . Since Ric and , by Theorem 5.4 in [Wu], where C is a positive constant depending on N and . Note that by the assumption.…”

Li-Yau Estimates for a Nonlinear Parabolic Equation on Manifolds

“…B. Wu [20] defined the weighted flag curvature when k = n and B. Shen [16] completed this concept for any k.…”

Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds

“…We could say that Lichnerowicz-Obata type estimate [15], Li-Yau-Zhong-Yang type estimate [16], and Mckean type estimate [9,10,11] for both positive and negative Ricci curvature are the most well known work in this subject.…”