Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below
We show that in any infinitesimally Hilbertian CD* (K,N)-space at almost every point there exists a Euclidean weak tangent, i.e., there exists a sequence of dilations of the space that converges to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian CD* (0,N)-spaces.
Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure
In a prior work of the first two authors with Savar´e, a new Riemannian notion of a lower bound for Ricci curvature in the class of metric measure spaces (X, d, m) was introduced, and the corresponding class of spaces was denoted by RCD(K,∞). This notion relates the CD(K, N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In this prior work the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) 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 measure…