On Weak Super Ricci Flow through Neckpinch

TL;DR

This paper extends the concept of Ricci flow to metric measure spaces, introducing weak super Ricci flows and characterizing their behavior through neckpinch singularities and convex cost functions.

Contribution

It defines weak super Ricci flows in the context of metric measure spaces and characterizes their behavior during neckpinch singularities, linking smooth and weak flow continuations.

Findings

Weak super Ricci flow is characterized by the single point pinching phenomenon.

Spacetime is a refined weak super Ricci flow if the flow is smooth with possibly singular final time.

The notion of weak super Ricci flow applies to metric measure spaces with convex cost functions.

Abstract

In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion components. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form $\{x\} \times \sphere^n$. We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.