Lower bounds for the scalar curvatures of Ricci flow singularity models

TL;DR

This paper applies Bamler's advanced Ricci flow theory to establish lower bounds on scalar curvatures of singularity models, notably deriving a quadratic decay bound for 4D steady solitons.

Contribution

It provides new lower bounds for scalar curvatures in Ricci flow singularity models using Bamler's recent theoretical developments.

Findings

Quadratic decay lower bound for scalar curvature in 4D steady solitons

Application of Bamler's theory to singularity models

Advancement in understanding scalar curvature behavior near singularities

Abstract

In a series of papers, Bamler [Bam20a,Bam20b,Bam20c] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity theory analogous to Cheeger--Colding theory. In this paper we give an application of his theory to lower bounds for the scalar curvatures of singularity models for Ricci flow. In the case of $4$-dimensional non-Ricci-flat steady soliton singularity models, we obtain as a consequence a quadratic decay lower bound for the scalar curvature.