Synthetic approaches to Ricci flows

TL;DR

The work develops and connects synthetic notions of Ricci flow for time-dependent metric measure spaces, rooted in heat flow behavior, optimal transport, and volume growth. It presents two main frameworks—weak and rough Ricci flows—based on entropy convexity and heat-flow expansion, and extends them with $N$-dimensional refinements to capture upper bounds and dimensional effects; it further introduces minimal super-Ricci flows via Gaussian-volume slopes to recover classical Ricci flow in smooth settings and to analyze singular spaces. The paper provides extensive examples (cones, suspensions, doubling, gluing, and Gaussian weights) to illustrate when these synthetic flows realize genuine Ricci flow or its variants, and it collects novel characterizations of SRFs that unify dynamic convexity, heat-contractivity, and gradient/Harnack-type inequalities. Collectively, the results advance a robust, non-smooth framework for Ricci flow with potential applications to spaces with singularities and to the analysis of volume-based curvature functionals. The insights offer a toolkit for extending geometric analysis, enabling characterizations of curvature-driven evolutions beyond classical smooth manifolds.

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.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Corollary 2.6
• Theorem 2.7
• Theorem 2.8
• Proposition 3.1
• Theorem 3.2