Translator Disclaimer
2021 Remarks about synthetic upper Ricci bounds for metric measure spaces
Karl-Theodor Sturm
Tohoku Math. J. (2) 73(4): 539-564 (2021). DOI: 10.2748/tmj.20200717

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.

Citation

Download Citation

Karl-Theodor Sturm. "Remarks about synthetic upper Ricci bounds for metric measure spaces." Tohoku Math. J. (2) 73 (4) 539 - 564, 2021. https://doi.org/10.2748/tmj.20200717

Information

Published: 2021
First available in Project Euclid: 22 December 2021

Digital Object Identifier: 10.2748/tmj.20200717

Subjects:
Primary: 53C23
Secondary: 49Q20

Keywords: metric measure space , singular spaces , Synthetic Ricci bounds , upper Ricci bounds

Rights: Copyright © 2021 Tohoku University

JOURNAL ARTICLE
26 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.73 • No. 4 • 2021
Back to Top