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.