Ricci Flow recovering from pinched discs

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 $\mathbb{R}^{p+1} \times S^q$. This gives a picture of Ricci flow through a singularity, in which a neighborhood of the manifold changes topology from $D^{p+1} \times S^{q}$ to $S^p \times D^{q+1}$ (through the cone over $S^p \times S^q$.) 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.