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

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 TS² 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 R⁴ /Z₂ R 4 / Z 2 orbifold, (iii) the 4d Bryant soliton quotiented by Z₂ Z 2 , and (iv) the shrinking cylinder R× RP³ 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| ≥ 3 | k | ≥ 3 .