On the blow-up of four-dimensional Ricci flow singularities

Abstract

Abstract. In this paper we prove a conjecture by Feldman–Ilmanen–Knopf (2003) that the gradient shrinking soliton metric they constructed on the tautological line bundle over ℂℙ 1 $\mathbb {CP}^1$ 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.