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Richard Bamler: Lectures

May 28, 2021
TITLE: U(2)-invariant Ricci flows in dimension 4 and partial regularity theory for Ricci flows

In this talk I will present work of my former graduate student, Alexander Appleton, on cohomogeneity-1 Ricci flows in dimension 4 that are invariant under an isometric U(2)-action.

I will first show that there are certain U(2)-invariant, metrics that are asymptotic to S^3 / \mathbb{Z}_k, k \geq 2, and whose Ricci flow develops a Type II singularity. This singularity is caused by the contraction of a 2-sphere of self-intersection -k. If k=2, then the singularity is modeled on the Eguchi-Hanson metric. This is the first example of a finite-time singularity in Ricci flow whose blow-up limit is Ricci flat. Numerical evidence suggests that the singularities in the case k \geq 3 are modeled on a new family of non-collapsed, U(2)-invariant, steady solitons, which were also constructed by Appleton.

In the last part of the talk, I will relate Appleton’s results to a new compactness and partial regularity theory for Ricci flows. In the context of this theory, Appleton’s examples are optimal, as they exhibit a singular set with the lowest possible codimension.

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