Seminar Analysis and Mathematical Physics: Gioacchino Antonelli (Scuola Normale Superiore di Pisa)

The isoperimetric problem on spaces with curvature bounded from below

In this talk I will discuss the isoperimetric problem on spaces with curvature bounded from below. I will mainly deal with complete non-compact Riemannian manifolds, but most of the techniques described are metric in nature and the results could be extended to the case of metric measure spaces with synthetic bounds from below on the Ricci tensor, namely RCD spaces. 
When the space is compact, the existence of isoperimetric regions for every volume is established through a simple 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. Such a 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 state 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 in such an approach is the use of the synthetic theory of curvature bounds to describe in a rather natural way where the mass is lost at infinity. Later, I will use the latter described 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 big volume. Time permitting, I will describe some forthcoming works and some open problems. 
This talk is based on several papers and ongoing collaborations with E. Bruè, M. Fogagnolo, S. Nardulli, E. Pasqualetto, M. Pozzetta, and D. Semola.

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