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A semigroup approach to Finsler geometry: Bakry-Ledoux's isoperimetric inequality

Shin-ichi Ohta

Abstract

We develop the celebrated semigroup approach à la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenböck formula established by Sturm and the author. We show the $L^1$-gradient estimate on Finsler manifolds (under some additional assumptions in the noncompact case), which is equivalent to a lower weighted Ricci curvature bound and the improved Bochner inequality. As a geometric application, we prove Bakry-Ledoux's Gaussian isoperimetric inequality, again under some additional assumptions in the noncompact case. This extends Cavalletti-Mondino's inequality on reversible Finsler manifolds to non-reversible metrics, and improves the author's previous estimate, both based on the localization (also called needle decomposition) method.

A semigroup approach to Finsler geometry: Bakry-Ledoux's isoperimetric inequality

Abstract

We develop the celebrated semigroup approach à la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenböck formula established by Sturm and the author. We show the -gradient estimate on Finsler manifolds (under some additional assumptions in the noncompact case), which is equivalent to a lower weighted Ricci curvature bound and the improved Bochner inequality. As a geometric application, we prove Bakry-Ledoux's Gaussian isoperimetric inequality, again under some additional assumptions in the noncompact case. This extends Cavalletti-Mondino's inequality on reversible Finsler manifolds to non-reversible metrics, and improves the author's previous estimate, both based on the localization (also called needle decomposition) method.

Paper Structure

This paper contains 19 sections, 20 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Geometry and analysis on Finsler manifolds
  3. Finsler manifolds
  4. Uniform convexity and smoothness
  5. Weighted Ricci curvature
  6. Nonlinear Laplacian and heat flow
  7. Bochner--Weitzenböck formula
  8. Linearized semigroups and gradient estimates
  9. Linearized heat semigroups and their adjoints
  10. Improved Bochner inequality
  11. $L^1$-gradient estimate
  12. On the hypothesis \ref{['eq:hypo']}
  13. Weighted Riemannian case
  14. $\mathop{\mathrm{RCD}}\nolimits$-case
  15. Characterizations of lower Ricci curvature bounds
  16. ...and 4 more sections

Key Result

Theorem 1

Let $(M,F,\mathfrak{m})$ be complete and satisfy $\mathop{\mathrm{Ric}}\nolimits_{\infty} \ge K>0$, $\mathfrak{m}(M)=1$, $\mathsf{C}_F<\infty$ and $\mathsf{S}_F<\infty$. We also assume that holds for any global solution $(u_t)_{t \ge 0}$ to the heat equation with $u_0 \in \mathcal{C}^{\infty}_c(M)$ and any $t>0$. Then we have for all $\theta \in [0,1]$, where

Theorems & Definitions (30)

  • Theorem 1: Bakry--Ledoux's isoperimetric inequality
  • Definition 2.1: Finsler structures
  • Theorem 2.2: Euler's theorem
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5: Weighted Ricci curvature
  • Remark 2.6: $\mathbf{S}$-curvature
  • Definition 2.7: Reverse Finsler structures
  • Definition 2.8: Global solutions
  • Theorem 2.9
  • ...and 20 more