On the blow-up of four dimensional Ricci flow singularities

TL;DR

This paper proves a conjecture that a specific Ricci soliton is the limit of blow-ups of a Type I Ricci flow singularity on a closed 4-manifold, revealing that such limits can lack non-negative Ricci curvature.

Contribution

It confirms a conjecture linking Ricci solitons to singularity blow-ups and shows limits can have negative Ricci curvature, expanding understanding of Ricci flow singularities.

Findings

The constructed Ricci soliton is the limit of blow-ups of a Type I singularity.

Limits of blow-ups can have negative Ricci curvature.

Confirmed the conjecture by Feldman-Ilmanen-Knopf.

Abstract

In this paper we prove a conjecture by Feldman-Ilmanen-Knopf in \cite{FIK} that the gradient shrinking soliton metric they constructed on the tautological line bundle over $\CP^1$ is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four dimensional manifolds do not necessarily have non-negative Ricci curvature.