Geometric Analysis and Topology Seminar

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The isoperimetric problem on spaces with curvature bounded from below

Speaker: Gioacchino Antonelli, Scuola Normale Superiore

Location: Online

Videoconference link: https://nyu.zoom.us/j/99465062960

Date: Wednesday, December 1, 2021, 11 a.m.

Synopsis:

Abstract: I will discuss the existence of isoperimetric regions on spaces with curvature bounded from below.
When the space is compact, the existence for every volume is established through an application of the direct method of Calculus of Variations. In the noncompact case, part of the mass could be lost at infinity in the minimization process.
Nevertheless, the lost mass can be recovered in isoperimetric regions sitting in limits at infinity of the space. Following this heuristics, and building on top of results by Ritoré--Rosales and Nardulli, I will show a generalized existence result for the isoperimetric problem on Riemannian manifolds with Ricci curvature bounded from below and a uniform bound from below on the volumes of unit balls.
The main novelty is the use of the synthetic theory of curvature bounds to describe in a natural way where the mass is lost at infinity.
I will use the generalized existence result to prove new existence criteria for the isoperimetric problem on manifolds with nonnegative Ricci curvature. In particular, I will show that on a complete manifold with nonnegative sectional curvature and Euclidean volume growth at infinity, isoperimetric regions exist for every sufficiently large volume.
Eventually, I will describe some examples, and some sharp concavity results for the isoperimetric profile function.
This talk is based on papers and ongoing collaborations with E. Bruè, M. Fogagnolo, S. Nardulli, E. Pasqualetto, M. Pozzetta, and D. Semola.