On Type I Singularities in Ricci flow

TL;DR

This paper investigates the structure of Type I singularities in Ricci flow, proving convergence to gradient shrinking solitons, and establishing regularity results and volume vanishing properties at singular times.

Contribution

It introduces unified notions of singular sets, proves their equivalence, and extends convergence results to nontrivial solitons, advancing understanding of singularity formation in Ricci flow.

Findings

Singular sets for Type I Ricci flows coincide.

Blow-ups around singular points converge to gradient shrinking solitons.

Volume of finite-volume singular sets vanishes at singular time.

Abstract

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber. As a by-product we conclude that the volume of a finite-volume singular set vanishes at the singular time. We also define a notion of density for Type I Ricci flows and use it to prove a regularity theorem reminiscent of White's partial regularity result for mean curvature flow.