Rigidity of the round cylinders in Ricci shrinkers

TL;DR

This paper proves that round cylinders are rigid in Ricci shrinkers, meaning any shrinker close to a round cylinder in the pointed-Gromov-Hausdorff sense must be isometric to it.

Contribution

It establishes a rigidity result for round cylinders within the space of Ricci shrinkers, showing local uniqueness under Gromov-Hausdorff closeness.

Findings

Round cylinders are rigid in Ricci shrinkers.

Any Ricci shrinker near a round cylinder is isometric to it.

Rigidity holds in the pointed-Gromov-Hausdorff topology.

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 $S^{n-1}\times \mathbb R$.