Title: Concavity of isoperimetric profiles and applications to geometry
Abstract: In this talk, I shall discuss two results that show how the isoperimetric structure of a space is connected to its geometry.
First, I will present a sharp and rigid spectral generalization of the Bishop-Gromov volume comparison theorem. The proof of this result builds on a concavity property of an unequally weighted isoperimetric profile on the manifold. I will discuss how this volume estimate has been recently used by L. Mazet, following contributions by O. Chodosh, C. Li, P. Minter, and D. Stryker, to settle a well-known open problem in the theory of minimal surfaces: the stable Bernstein problem in R^n, with n<=6.
Second, I will show a sharp concavity property of the isoperimetric profile of noncompact manifolds with Ricci lower bounds. Although the statement is set in the smooth context, its proof relies on tools from non-smooth geometry that have been developed in recent years. I will explain how this concavity result interplays with the existence of isoperimetric regions in spaces with lower curvature bounds.