The rigidity of $\mathbb{S}^3\times\mathbb{R}$ under ancient Ricci flow

TL;DR

This paper proves a rigidity result for four-dimensional nonnegatively curved Type I $ abla$-solutions, showing that if such a solution asymptotically resembles the standard cylinder, then it must be exactly the cylinder with its standard shrinking metric.

Contribution

It extends the neck-stability theorem to a class of four-dimensional solutions and establishes a rigidity theorem for solutions asymptotic to the standard cylinder.

Findings

Generalized neck-stability theorem for certain 4D solutions

Proved that solutions asymptotic to the standard cylinder are exactly the cylinder

Enhanced understanding of the structure of ancient Ricci flows in four dimensions

Abstract

In this paper we generalize the neck-stability theorem of Kleiner-Lott to a special class of four-dimensional nonnegatively curved Type I $\kappa$-solutions, namely, those whose asymptotic shrinkers are the standard cylinder $\mathbb{S}^3\times\mathbb{R}$. We use this stability result to prove a rigidity theorem: if a four-dimensional Type I $\kappa$-solution with nonnegative curvature operator has the standard cylinder $\mathbb{S}^3\times\mathbb{R}$ as its asymptotic shrinker, then it is exactly the cylinder with its standard shrinking metric.