Title: 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.