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

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.

References in corpus (1)

• Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature