Differential Geometry Seminar: Eguchi-Hanson singularities in U(2)-invariant Ricci flow

Seminar | February 25 | 3:10-4 p.m. | 939 Evans Hall

 Alexander Appleton, UC Berkeley

 Department of Mathematics

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 in our setup blow-up limits at larger scales are isometric to either (i) the flat $\mathbb R^4 /\mathbb Z_2$ orbifold, (ii) a rotationally symmetric, positively curved, asymptotically cylindrical ancient orbifold Ricci flow on $\mathbb R^4/\mathbb Z_2$, or (iii) the shrinking soliton on $\mathbb R \times \mathbb R P^3$. 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 $|k| \geq 3$.