Eguchi-Hanson singularities in U(2)-invariant Ricci flow

TL;DR

This paper demonstrates that four-dimensional Ricci flows with certain symmetries can develop singularities modeled on the Eguchi-Hanson space, identifying possible blow-up limits and introducing new types of Type II singularities.

Contribution

It establishes the formation of Eguchi-Hanson modeled singularities in Ricci flow and classifies all possible blow-up limits in this setting.

Findings

Type II singularity modeled on Eguchi-Hanson develops in finite time

Possible blow-up limits include Eguchi-Hanson space, flat orbifold, Bryant soliton quotient, and shrinking cylinder

New family of Type II singularities caused by collapsing self-intersecting two-spheres

Abstract

We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthermore, we show that for these Ricci flows the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat $\mathbb{R}^4 /\mathbb{Z}_2$ orbifold, (iii) the 4d Bryant soliton quotiented by $\mathbb{Z}_2$, and (iv) the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$. As a byproduct of our work, we also prove the existence of a new family of Type II singularities caused by the collapse of a two-sphere of self-intersection $|k| \geq 3$.