Tohoku Mathematical Journal
2021
December
SECOND SERIES VOL. 73, NO. 4

Tohoku Math. J.
73 (2021), 539-564

Title REMARKS ABOUT SYNTHETIC UPPER RICCI BOUNDS FOR METRIC MEASURE SPACES

Author Karl-Theodor Sturm

(Received March 23, 2020, revised July 13, 2020)
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 Wasserstein 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_2\big(\hat P_t\delta_x,\hat P_t\delta_y\big)\end{eqnarray*} in terms of the mean value of the Bakry-Émery 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.

Mathematics Subject Classification. Primary 53C23; Secondary 49Q20.
Key words and phrases. Synthetic Ricci bounds, metric measure space, upper Ricci bounds, singular spaces.

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