Super-Ricci Flows for Metric Measure Spaces

TL;DR

This paper introduces the concepts of super-Ricci flows and Ricci flows for evolving metric measure spaces, establishing their stability, compactness, and connections to entropy convexity and gradient estimates.

Contribution

It defines super-Ricci and Ricci flows in a synthetic setting, proves their stability and compactness, and links these notions to entropy convexity and the Bakry-Émery calculus.

Findings

Super-Ricci flows are stable under suitable convergence.

Families of super-Ricci flows are compact.

The paper establishes equivalences with gradient estimates.

Abstract

We introduce the notions of `super-Ricci flows' and `Ricci flows' for time-dependent families of metric measure spaces $(X,d_t,m_t)_{t\in I}$. The former property is proven to be stable under suitable space-time versions of mGH-convergence. Uniformly bounded families of super-Ricci flows are compact. In the spirit of the synthetic lower Ricci bounds of Lott-Sturm-Villani for static metric measure spaces, the defining property for super-Ricci flows is the `dynamic convexity' of the Boltzmann entropy ${\mathrm Ent}(.|m_t)$ regarded as a functions on the time-dependent geodesic space $({\mathcal P}(X),W_t)_{t\in I}$. For Ricci flows, in addition a nearly dynamic concavity of the Boltzmann entropy is requested. Alternatively, super-Ricci flows will be studied in the framework of the $\Gamma$-calculus of Bakry-\'Emery-Ledoux and equivalence to gradient estimates will be derived. For both notions of super-Ricci flows, also enforced versions involving an `upper dimension bound' $N$ will be presented.