Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

TL;DR

This paper investigates sharp isoperimetric inequalities and asymptotic properties on non-collapsed spaces with lower Ricci bounds, providing new insights even in classical smooth manifold settings.

Contribution

It establishes new sharp and rigid isoperimetric comparison theorems and asymptotic properties for ${ m RCD}(K,N)$ spaces, extending classical results to a synthetic setting.

Findings

New sharp isoperimetric comparison theorems

Almost regularity results in terms of isoperimetric profile

Enhanced functional inequalities

Abstract

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on $N$-dimensional ${\rm RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements are new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.