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On the isoperimetry of manifolds with Ricci lower bounds: sharp concavity and applications

Marco Pozzetta
University of Naples Federico II
Thursday, June 13, 2024 - 14:00 to 15:00
The isoperimetric problem on a Riemannian manifold aims at minimizing the measure of the boundary, called perimeter, among subsets having a given volume. Minimizers of the problem are called isoperimetric sets, and the isoperimetric profile is the one-variable function assigning to any volume the infimum of the problem.

The aim of the talk is to give an overview on the isoperimetric problem on Riemannian manifolds having Ricci curvature bounded from below and to discuss some selected results.
Focusing on the case of manifolds with nonnegative Ricci curvature, we will first present the sharp concavity property enjoyed by the isoperimetric profile function. Next, we will discuss applications to several aspects of the variational problem on noncompact manifolds with nonnegative Ricci curvature and maximal volume growth: sharp and rigid isoperimetric inequalities, existence of isoperimetric sets, generic uniqueness and stability of isoperimetric sets. Time permitting, we will also briefly discuss the proof of the concavity property of the isoperimetric profile.
Proofs crucially exploit the theory of metric measure spaces with lower Ricci bounds. Fundamental tools are given by a decomposition result for minimizing sequences for the isoperimetric problem and a generalized notion of constant mean curvature for boundaries of isoperimetric sets in nonsmooth spaces with lower Ricci bounds.
The talk is based on a series of works in collaboration with G. Antonelli, E. Bruè, M. Fogagnolo, S. Nardulli, E. Pasqualetto, D. Semola, and I. Y. Violo.


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