On weak formulations of (super) Ricci flows

TL;DR

The paper introduces weak formulations of Ricci flows using metrics and measures, enabling extension to singular settings, and characterizes solutions via saturation conditions.

Contribution

It provides new weak formulations of Ricci flows based solely on metrics and measures, applicable to singular cases, and characterizes solutions through saturation conditions.

Findings

Two characterizations of smooth Ricci flow solutions using metrics and measures.

Weak formulations that extend to singular settings.

Saturation conditions ensure the flow inequality becomes an equality.

Abstract

We present two characterizations of smooth compact Ricci flow solutions solely in terms of metrics and measures (one of them only works under positive scalar curvature along the flow); thus, provide weak formulations that are generalized to the singular setting in a straightforward manner. These formulations are achieved by weakly formulating super Ricci flows and imposing a saturation condition (solely in terms of metric and measure) to ensure the super Ricci flow inequality is an equality.