Geometry and analysis of groups and manifolds

Isoperimetry and lower bounds on the curvature through the lens of nonsmooth geometry

speaker: Gioacchino Antonelli (Scuola Normale Superiore)

abstract: The isoperimetric profile $I$ is the function that associates to each volume the infimum of the perimeter of sets with that volume. In this talk I will prove the following, for $n\geq 2$. On a nonnegatively Ricci curved $n$-dimensional smooth Riemannian manifold $I^{\frac{n}{n-1}}$ is concave. In the compact case the proof uses the existence of isoperimetric regions and the first, and second variation of the perimeter. See, e.g., Bavard--Pansu, and Bayle. In the noncompact case isoperimetric regions might fail to exist for some volumes and might escape at infinity in isoperimetric sets in possibly nonsmooth spaces. Here, nonsmooth geometry naturally enters into play. Similar results hold for nonsmooth spaces with (possibly negative) synthetic lower bounds on Ricci curvature. The results presented come from several collaborations.

timetable:
Wed 28 Jun, 10:30 - 11:00, Aula Dini
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