Geometry Seminar: G. Antonelli (Scuola Normale Superiore)

Geometry Seminar (Geometric Analysis)

Speaker: Gioacchino Antonelli (Scuola Normale Superiore di Pisa)

Title: The isoperimetric problem on spaces with curvature bounded from below.

Abstract: In this talk I will deal with the isoperimetric problem on spaces with curvature bounded from below. 

When the space is compact, the existence of isoperimetric regions 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. 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 structure result for minimizing sequences for the isoperimetric problem on Riemannian manifolds with Ricci curvature bounded from below and with a uniform bound from below on the volumes of unit balls. The novelty in such an approach is the use of the synthetic theory of curvature bounds to describe in a natural way where the mass is lost at infinity. 

Later, I will use the latter 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. Finally, I will describe some examples and some forthcoming works. 

This talk is based on several papers and ongoing collaborations with E. Bruè, M. Fogagnolo, S. Nardulli, E. Pasqualetto, M. Pozzetta, and D. Semola.