Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
Transactions of the AMS
In prior work of the first two authors with Savaré, a new Riemannian notion of lower bound for Ricci curvature
in the class of metric measure spaces $(X,d,m)$ was introduced, and the corresponding class of spaces denoted by
$RCD(K,\infty)$. This notion relates the $CD(K,N)$ theory of Sturm and Lott-Villani, in the case $N=\infty$, to the Bakry-Emery approach. In the aforementioned paper, the $RCD(K,\infty)$ property is defined in three equivalent ways and several properties of
$RCD(K,\infty)$ spaces, including the regularization properties of the heat flow,
the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. But
only finite reference measures $m$ have been considered. The goal of this paper is twofold: on one side we extend these results to general $\sigma$-finite spaces, on the other we remove a technical assumption concerning a strengthening of the $CD(K,\infty)$ condition. This more general class of spaces
includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the
pointed metric measure limits of manifolds with lower Ricci curvature bounds.
The paper is available on the
cvgmt preprint server.