Rigidity of the round cylinders in Ricci shrinkers

Abstract

In this paper, we prove that the round cylinders are rigid in the space of Ricci shrinkers. Namely, any Ricci shrinker that is sufficiently close to $S^{n-1} \times \mathbb{R}$ in the pointed-Gromov-Hausdorff topology must itself be isometric to $S^{n-1} \times \mathbb{R}$.