On the geometry of metric measure spaces

Szymon M. Walczak - Lodz

Resumo:

In 1951, D. Alexandrov introduced the notion of the lower curvature bounds for metric spaces in terms of comparison properties for geodesic triangles. These bounds are equivalent to lower bounds for sectional curvature for Riemannian manifolds, and are stable under Gromov-Hausdorff convergence.

In the lecture, the generalization of lower Ricci curvature bounds in the framework of metric measure spaces will be presented [1]. At the beginning, the lecture will contain very brief introduction to L^2-Wasserstein space. A complete and separable metric D on the family of all isomorphim classes on normalized metric measure spaces will be also presented. Next, a notion of lower curvature bounds for metric measure spaces (M, d, m) based on convexity properties of the relative entropy Ent(.|m) with respect to the reference measure m will be studied. Finally, the connection between lower Ricci curvature bounds for Riemannian manifolds and lower curvature bounds for metric measure spaces, as well as volume growth estimates will be presented.

[1] K.-T. Sturm, On the geometry of metric measure spaces, Acta Math. 196 (2006), 65-177.

[2] J. Lott & C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. 169 (2009). 903-991.

© Szymon M. Walczak.