Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition

TL;DR

This paper establishes new sharp isoperimetric inequalities on Riemannian manifolds with probability measures under curvature, dimension, and diameter bounds, identifying extremal model spaces for various parameter regimes.

Contribution

It introduces sharp isoperimetric inequalities for manifolds with generalized Ricci curvature, dimension, and diameter bounds, and characterizes extremal model spaces across all parameter ranges.

Findings

Recovered classical inequalities for positive curvature.

Identified new model spaces for negative or infinite curvature cases.

Provided sharp bounds for all parameter combinations.

Abstract

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov--L\'evy and Bakry--Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly positively bounded from below, these model spaces are the $n$-sphere and Gauss space, corresponding to generalized dimension being $n$ and $\infty$, respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural one parameter family of model spaces is required, nevertheless yielding a sharp result.