Eguchi–Hanson Singularities in U(2)-Invariant Ricci Flow

Abstract

AbstractWe 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 $$TS^2$$ 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 $${\mathbb {R}}^4 /{\mathbb {Z}}_2$$ R 4 / Z 2 orbifold, (iii) the 4d Bryant soliton quotiented by $${\mathbb {Z}}_2$$ Z 2 , and (iv) the shrinking cylinder $${\mathbb {R}}\times {\mathbb {R}}P^3$$ R × 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 $$|k| \ge 3$$ | k | ≥ 3 .