Super-Ricci flows for metric measure spaces

Abstract

We introduce the notions of ‘super-Ricci flows’ and ‘Ricci flows’ for time-dependent families of metric measure spaces (X,dt,mt)t∈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 Ent(.|mt) regarded as a functions on the time-dependent geodesic space (P(X),Wt)t∈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 Γ-calculus of Bakry–Émery–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.