Measure rigidity of Ricci curvature lower bounds

The measure contraction property, MCP for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space.We start our investigation from the Euclidean case by proving that if a positive Radon measure m over Rd is such that (Rd,|{dot operator}|,m) verifies a weaker variant of MCP, then its support spt(m) must be convex and m has to be absolutely continuous with respect to the relevant Hausdorff measure of spt(m). This result is then used as a starting point to investigate the rigidity of MCP in the metric framework.We introduce the new notion of reference measure for a metric space and prove that if (X, d, m) is essentially non-branching and verifies MCP, and μ is an essentially non-branching MCP reference measure for (spt(m), d), then m is absolutely continuous with respect to μ, on the set of points where an inversion plan exists. As a consequence, an essentially non-branching MCP reference measure enjoys a weak type of uniqueness, up to densities. We also prove a stability property for reference measures under measured Gromov-Hausdorff convergence, provided an additional uniform bound holds.In the final part we present concrete examples of metric spaces with reference measures, both in smooth and non-smooth setting. The main example will be the Hausdorff measure over an Alexandrov space. Then we prove that the following are reference measures over smooth spaces: the volume measure of a Riemannian manifold, the Hausdorff measure of an Alexandrov space with bounded curvature, and the Haar measure of the subRiemannian Heisenberg group.