Ricci Flow Emerging from Rotationally Symmetric Degenerate Neckpinches

TL;DR

This paper constructs Ricci flow solutions originating from type-II singularities, demonstrating that curvature decreases at the same rate it previously blew up, marking the first such example.

Contribution

It provides the first known Ricci flow solutions starting from a type-II singularity, expanding understanding of singularity formation and evolution.

Findings

Curvature decreases at the same rate it blew up

Constructs solutions from singular initial data

First example of Ricci flow from a type-II singularity

Abstract

In previous work, Angenent, Isenberg, and Knopf created type-II Ricci flow neckpinch singularities. In this paper we construct solutions to Ricci flow whose initial data is the singular metric resulting from these singularities. We show in particular that the curvature decreases at the same rate at which it blew up. This is the first example of Ricci flow starting from a type-II singularity.