Journal of the
Korean Mathematical Society JKMS

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.

Keywords: Finsler manifold, weighted flag curvature, weighted Ricci curvature, first eigenvalue, fundamental group

MSC numbers: Primary 53C60; Secondary 53B40