Eguchi-Hanson singularities in U(2)-invariant Ricci flow

2019/03/24 by Appleton, Alexander · 3 citations
#53C44 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.09936

Abstract

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, 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 orbifold, (iii) the 4d Bryant soliton quotiented by ℤ₂, and (iv) the shrinking cylinder ℝ × ℝ P³. 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.

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• Ricci flow and the scalar curvature rigidity of Einstein manifolds
• Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery
• On Ricci Solitons with Isoparametric Potential Functions

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