Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure

Abstract:

In a prior work of the first two authors with Savaré, a new Riemannian notion of a lower bound for Ricci curvature in the class of metric measure spaces $(X,\textsf {d},\mathbf {m})$ was introduced, and the corresponding class of spaces was 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 this prior work 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. In the above-mentioned work only finite reference measures $\mathbf {m}$ have been considered. The goal of this paper is twofold: on one side we extend these results to general $\sigma$-finite spaces, and on the other we remove a technical assumption that appeared in the earlier work 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.