On the geometry of irreversible metric-measure spaces: Convergence, Stability and Analytic aspects

TL;DR

This paper investigates the convergence and stability of irreversible metric-measure spaces, introducing new methods to handle noncompact cases and demonstrating geometric inequalities specific to these structures.

Contribution

It develops a framework for analyzing Gromov-Hausdorff convergence in irreversible spaces, especially noncompact ones, and applies it to Finsler manifolds to reveal key differences from reversible cases.

Findings

Established convergence and stability results for irreversible spaces

Introduced a reversibility-dependent Gromov-Hausdorff topology

Proved geometric and functional inequalities on irreversible structures

Abstract

The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable -- reversibility depending -- Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures.