A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature

TL;DR

This paper refines the analysis of singularities in Ricci flow by classifying singular points and establishing local curvature blow-up rates, with applications to flows with bounded scalar curvature.

Contribution

It introduces local definitions of singular point types and proves that curvature must blow up at least at a Type I rate near singularities, extending previous global results.

Findings

Singular points are classified into Type I and Type II only.

Curvature tensors blow up at least at a Type I rate near singular points.

Results apply to Ricci flows with bounded scalar curvature.

Abstract

We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result of Enders, Topping and the first author that relied on a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum's result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. Finally, we show some applications of the theory to Ricci flows with bounded scalar curvature.