# Geometry and Geometric Analysis Working Group

#### (DAY AND ROOM CHANGE) Differential properties of the isoperimetric profile on manifolds and the role of nonsmooth geometry

**Speaker:**
Marco Pozzetta, Università di Napoli Federico II

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, November 3, 2022, 11 a.m.

**Synopsis:**

(NOTE: This seminar will now be on Thursday, November 3rd, as part of the Analysis Seminar)

The isoperimetric problem on smooth Riemannian manifolds aims at minimizing the perimeter among sets having a fixed volume, and a natural assumption for the study of the problem is a lower bound on the Ricci curvature. The isoperimetric profile is the function assigning to any volume the infimum of the problem.

The main result we present is the validity of sharp second-order differential inequalities on the isoperimetric profile of manifolds with Ricci bounded below, regardless of existence of isoperimetric sets. We will then discuss several applications.

The proof of the differential properties of the isoperimetric profile is based on a mass decomposition result for minimizing sequences.

Such decomposition necessarily involves the study of the isoperimetric problem settled on some nonsmooth ambient spaces related to the original manifold; this makes the final result dependent on an analysis carried out on isoperimetric sets in a nonsmooth framework.

More generally, the results presented hold on \(\mathrm{RCD}\) metric measure spaces endowed with Hausdorff measure, which are spaces having Ricci curvature bounded from below in a generalized sense.

The talk is based on works in collaboration with G. Antonelli, E. Bruè, M. Fogagnolo, S. Nardulli, E. Pasqualetto, and D. Semola.