Remarks about Synthetic Upper Ricci Bounds for Metric Measure Spaces

TL;DR

This paper explores various characterizations of synthetic upper Ricci bounds in metric measure spaces, focusing on heat flow, entropy, and optimal transport, and establishes new stability and equivalence results, especially for weighted Riemannian manifolds.

Contribution

It introduces a new characterization of synthetic upper Ricci bounds via entropy semiconcavity and proves their equivalence to bounds on Wasserstein distance growth for heat flows, with explicit bounds for weighted manifolds.

Findings

Entropy semiconcavity characterization is stable under convergence.

Asymptotic Wasserstein distance growth bounds are equivalent to synthetic Ricci bounds.

Explicit bounds relate Ricci curvature and Hessian of weight functions along geodesics.

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.