Ricci Flow recovering from pinched discs

2017/04/21 by Timothy Carson, Carson, Timothy
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.1704.06385

Abstract

We construct smooth solutions to Ricci flow starting from a class of singular metrics and give asymptotics for the forward evolution. The singular metrics heal with a set of points (of codimension at least three) coming out of the singular point. We conjecture that these metrics arise as final-time limits of Ricci flow encountering a Type-I singularity modeled on ℝ^p+1 × S^q. This gives a picture of Ricci flow through a singularity, in which a neighborhood of the manifold changes topology from D^p+1 × S^q to S^p × D^q+1 (through the cone over S^p × S^q.) We work in the class of doubly-warped product metrics. We also briefly discuss some possible smooth and non-smooth forward evolutions from other singular initial data.

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