Synthetic approaches to Ricci flows

TL;DR

This paper reviews various synthetic Ricci flow concepts applicable to metric measure spaces, highlighting their equivalence to classical Ricci flow in smooth cases and discussing their features through examples.

Contribution

It provides a comprehensive comparison of different synthetic Ricci flow notions based on heat flow, optimal transport, and volume behavior, clarifying their relationships and applications.

Findings

Different synthetic Ricci flow notions are equivalent to classical Ricci flow in smooth cases.

The paper illustrates features of these notions through diverse examples.

Synthetic approaches extend Ricci flow concepts to non-smooth spaces.

Abstract

We review different notions of synthetic Ricci flow that apply to time-dependent families of metric measure spaces and which are based on properties of the heat flow, ideas from optimal transport, and the asymptotic behaviour of volumes. Each notion equivalently characterises (weighted) Ricci flow for smooth families of weighted Riemannian manifolds. We discuss the features of the different notions on various examples.