Brussels-London 8: Lower bounds on Ricci curvature

Université libre de Bruxelles, 12/01/2016

Andrea Mondino

“Non-smooth spaces with Ricci curvature lower bounds”

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the ’80s and was pushed by Cheeger and Colding in the ’90s who investigated the structure of the spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can a give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, etc.), permitting to give further insights in the theory. In the seminar I will give an overview of the topic.

Fabio Cavalletti.

“Lévy-Gromov Isoperimetric inequality for metric measure spaces under lower Ricci curvature bounds”

By using an L^1 localization argument, we prove that in metric measure spaces satisfying lower Ricci curvature bounds (more precisely RCD^*(K,N) or more generally essentially non branching CD^*(K,N)) the classical Lévy-Gromov isoperimetric inequality holds with the associated rigidity and almost rigidity statements. (This is joint work with Andrea Mondio.)

Theo Sturm

“Super-Ricci flows of metric measure spaces”

A time-dependent family of Riemannian manifolds is a super-Ricci flow if 2 Ric + \partial_t g \ge 0. This includes all static manifolds of nonnegative Ricci curvature as well as all solutions to the Ricci flow equation. We extend this concept of super-Ricci flows to time-dependent metric measure spaces. In particular, we present characterizations in terms of dynamical convexity of the Boltzmann entropy on the Wasserstein space as well in terms of Wasserstein contraction bounds and gradient estimates. And we prove stability and compactness of super-Ricci flows under mGH-limits.