Measure rigidity of Ricci curvature lower bounds

The measure contraction property, $\mathsf{MCP}$ 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 $\mathsf{m}$ over $\mathbb{R}^{d}$ is such that $(\mathbb{R}^{d},|\cdot |, \mathsf{m})$ verifies a weaker variant of $\mathsf{MCP}$, then its support $\text{spt}(\mathsf{m})$ must be convex and $\mathsf{m}$ has to be absolutely continuous with respect to the relevant Hausdorff measure of $\text{spt}(\mathsf{m})$. This result is then used as a starting point to investigate the rigidity of $\mathsf{MCP}$ in the metric framework. We introduce the new notion of $reference \ measure$ for a metric space and prove that if $(X,\mathsf{d},\mathsf{m})$ is essentially non-branching and verifies $\mathsf{MCP}$, and $\mu$ is an essentially non-branching $\mathsf{MCP}$ reference measure for $(\text{spt}(\mathsf{m}), \mathsf{d})$, then $\mathsf{m}$ is absolutely continuous with respect to $\mu$, on the set of points where an inversion plan exists. As a consequence, an essentially non-branching $\mathsf{MCP}$ 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.