Calculus of Variations and Geometric Measure Theory

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

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

created by antonelli on 14 Jan 2022
modified by pozzetta1 on 09 Aug 2022



Inserted: 14 jan 2022
Last Updated: 9 aug 2022

Year: 2022

ArXiv: 2201.04916 PDF

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


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. Many other consequences of these results are explored in a companion paper by the authors.

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