Accepted Paper

Inserted: 14 jan 2022
Last Updated: 7 oct 2023

Journal: Annales Scientifiques de l'École Normale Supérieure
Year: 2022

This is the first of two companion papers originally appeared in a joint version in arXiv:2201.04916v1. The second companion paper is arXiv:2208.03739

Abstract:

This paper studies sharp isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called $N$-dimensional $\mathrm{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences. Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies.

Keywords: isoperimetric problem, RCD space, isoperimetric profile, Lower Ricci bounds