Ricci Flow recovering from pinched discs

الأصل · EN

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 Rᵖ⁺¹ × Sq. This gives a picture of Ricci flow through a singularity, in which a neighborhood of the manifold changes topology from Dᵖ⁺¹ × Sq to Sᵖ × Dq⁺¹ (through the cone over Sᵖ × Sq.) 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.