On Finsler metric measure manifolds with integral weighted Ricci curvature bounds

In this paper, we study deeply geometric and topological properties of Finsler metric measure manifolds with the integral weighted Ricci curvature bounds. We first establish the Laplacian comparison theorem, the Bishop-Gromov-type volume comparison theorem and the relative volume comparison theorem on such Finsler manifolds. Then we obtain a volume growth estimate and Gromov pre-compactness for Finsler metric measure manifolds under the integral weighted Ricci curvature bounds. Furthermore, we prove the local Dirichlet isoperimetric constant estimate on Finsler metric measure manifolds with integral weighted Ricci curvature bounds. As applications of the Dirichlet isoperimetric constant estimates, we get the first Dirichlet eigenvalue estimate and a gradient estimate for harmonic functions.