# Metric measure spaces with Riemannian Ricci curvature bounded from below

**submitted**
**year:** 2011

**abstract:** In this paper we introduce a synthetic notion of
Riemannian Ricci bounds from below for metric measure spaces
$(X,d,m)$ which is stable under measured Gromov-Hausdorff
convergence and rules out Finsler geometries. It can be given in
terms of an enforcement of the Lott, Sturm and Villani geodesic
convexity condition for the entropy coupled with the linearity of
the heat flow. Besides stability, it enjoys the same tensorization,
global-to-local and local-to-global properties. In these spaces,
that we call $RCD(K,\infty)$ spaces, we prove that the heat flow
(which can be equivalently characterized either as the flow
associated to the Dirichlet form, or as the Wasserstein gradient
flow of the entropy) satisfies Wasserstein contraction estimates and
several regularity properties, in particular Bakry-Emery estimates
and the $L^\infty-{\rm Lip}$ Feller regularization. We also prove
that the distance induced by the Dirichlet form coincides with
$d$, that the local energy measure has density given by the
square of Cheeger's relaxed slope and, as a consequence, that the
underlying Brownian motion has continuous paths. All these results
are obtained independently of PoincarÃ© and doubling assumptions on
the metric measure structure and therefore apply also to spaces
which are not locally compact, as the infinite-dimensional ones.

The paper is available on the
cvgmt preprint server.