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 $TS^2$, 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 $|k| \ge 3$.

Funding

NFS(DMS-1344991)