Calculus of Variations and Geometric Measure Theory

G. Antonelli - E. Pasqualetto - M. Pozzetta - D. Semola

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

created by pozzetta1 on 09 Aug 2022



Inserted: 9 aug 2022
Last Updated: 9 aug 2022

Year: 2022

ArXiv: 2208.03739 PDF

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


This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on $N$-dimensional $\mathrm{RCD}(K,N)$ 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.

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