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On Finsler metric measure manifolds with integral weighted Ricci curvature bounds

Xinyue Cheng, Yalu Feng

TL;DR

The work extends the theory of integral Ricci curvature bounds to Finsler metric measure manifolds by introducing integral weighted Ricci curvature notions and proving core geometric–analytic estimates. By developing Laplacian and volume comparison theorems under these bounds, it derives volume growth and Gromov pre-compactness results, and establishes a Dirichlet isoperimetric constant estimate. This, in turn, yields quantitative control of the first Dirichlet eigenvalue and gradient bounds for harmonic functions on Finsler spaces with nonreversible geometry. The results significantly broaden the analytic toolkit for non-Riemannian settings, enabling precise geometric and spectral conclusions under integral curvature constraints.

Abstract

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 Laplacian comparison theorem, Bishop-Gromov type volume comparison theorem and relative volume comparison theorem on such Finsler manifolds. Then we obtain a volume growth estimate and Gromov pre-compactness 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 first Dirichlet eigenvalue estimate and a gradient estimate for harmonic functions.

On Finsler metric measure manifolds with integral weighted Ricci curvature bounds

TL;DR

The work extends the theory of integral Ricci curvature bounds to Finsler metric measure manifolds by introducing integral weighted Ricci curvature notions and proving core geometric–analytic estimates. By developing Laplacian and volume comparison theorems under these bounds, it derives volume growth and Gromov pre-compactness results, and establishes a Dirichlet isoperimetric constant estimate. This, in turn, yields quantitative control of the first Dirichlet eigenvalue and gradient bounds for harmonic functions on Finsler spaces with nonreversible geometry. The results significantly broaden the analytic toolkit for non-Riemannian settings, enabling precise geometric and spectral conclusions under integral curvature constraints.

Abstract

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 Laplacian comparison theorem, Bishop-Gromov type volume comparison theorem and relative volume comparison theorem on such Finsler manifolds. Then we obtain a volume growth estimate and Gromov pre-compactness 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 first Dirichlet eigenvalue estimate and a gradient estimate for harmonic functions.
Paper Structure (5 sections, 21 theorems, 191 equations)

This paper contains 5 sections, 21 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integral weighted Ricci curvature
  4. Dirichlet isoperimetric constant estimate
  5. Applications

Key Result

Theorem 2.2

(OHTAOS2) For $u \in C^{\infty}(M)$, we have on $M_{u}$. Moreover, for $u \in H_{\mathrm{loc}}^{2}(M) \bigcap C^1(M)$ with $\Delta u \in H_{\mathrm{loc}}^{1}(M)$, we have for all bounded functions $\phi \in H_{0}^{1}(M) \bigcap L^{\infty}(M)$. Here $\left\|\cdot \right\|_{{\rm HS}(\nabla u)}$ denotes the Hilbert-Schmidt norm with respect to $g_{\nabla u}$.

Theorems & Definitions (38)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • Theorem 3.5
  • ...and 28 more