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

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 $T{S}^2$, 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 $|k|\ge 3$.