Differential properties of the isoperimetric profile on manifolds and the role of nonsmooth geometry
Speaker: Marco Pozzetta, Universita di Napoli Federico II
Location: Warren Weaver Hall 1302
Date: Thursday, November 3, 2022, 11 a.m.
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 the 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 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. Brue, M. Fogagnolo,
S. Nardulli, E. Pasqualetto, and D. Semola.