Synthetic notions of Ricci flow for metric measure spaces

TL;DR

This paper introduces new synthetic definitions of Ricci flow for time-dependent metric measure spaces using optimal transport concepts, characterizing smooth Ricci flows and exploring their relations in non-smooth settings.

Contribution

It develops multiple synthetic notions of Ricci flow based on entropy convexity and transport costs, extending the theory to non-smooth metric measure spaces.

Findings

Characterizes smooth Ricci flows via synthetic properties.

Establishes relations between different synthetic notions.

Extends Ricci flow concepts to non-smooth spaces.

Abstract

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces.