17 feb 2005

Abstract.

KARL-THEODOR STURM (University of Bonn)

``On the Geometry of Measure Metric Spaces''

Giovedi' 17.2 alle 15.00, nell'Aula Dini della SNS (in piazza del Castelletto, di fronte all'ingresso della biblioteca di Scienze)

ABSTRACT. We study the space of normalized metric measure spaces $(M,d,m)$ and introduce a complete separable metric $D$ on it. This metric has a natural interpretation in terms of mass transportation. It turns out that the family of normalized metric measure spaces with doubling constant $\le C$ is closed under $D$-convergence. Moreover, the subfamily of spaces with diameter $\le R$ is compact.

Furthermore, we introduce and analyze curvature bounds for metric measure spaces $(M,d,m)$, based on convexity properties of the relative entropy $Ent(.
m)$. For Riemannian manifolds, $Curv(M,d,m) \ge K$ if and only if $Ric_M(v,v) \ge K
v
^2$ for all $v \in TM$. Our lower curvature bounds are stable under $D$-convergence.