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 $T{S}^2$, 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 $$\mathbb{R}$^4 /$\mathbb{Z}$_2$orbifold,(iii)the4dBryantsolitonquotientedby$$\mathbb{Z}$_2$,and(iv)theshrinkingcylinder$$\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 $\mathrm{\mid }k\mathrm{\mid }\ge 3$.