Remarks about Synthetic Upper Ricci Bounds for Metric Measure Spaces

Abstract

We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along certain Wasserstein geodesics which is stable under convergence of mm-spaces. And we prove that a related characterization is equivalent to an asymptotic lower bound on the growth of the Wasseretein distance between heat flows. For weighted Riemannian manifolds, the crucial result will be a precise uniform two-sided bound for \begin{eqnarray*}\frac{d}{dt}\Big|_{t=0}W\big(\hat P_t\delta_x,\hat P_t\delta_y\big)\end{eqnarray*} in terms of the mean value of the Bakry-Emery Ricci tensor $\mathrm{Ric}+\mathrm{Hess}\, f$ along the minimizing geodesic from $x$ to $y$ and an explicit correction term depending on the bound for the curvature along this curve.