Stability of ALE Ricci-flat manifolds under Ricci flow

TL;DR

This paper proves that ALE Ricci-flat manifolds that are linearly stable and integrable remain close and converge to a Ricci-flat metric under Ricci flow, extending stability results to non-compact manifolds.

Contribution

It establishes the dynamic stability of ALE Ricci-flat manifolds under Ricci flow, especially for ALE Calabi-Yau manifolds, by adapting Tian's approach.

Findings

ALE Ricci-flat manifolds are dynamically stable under Ricci flow

Integrability holds for ALE Calabi-Yau manifolds

Stability results extend to non-compact ALE manifolds

Abstract

We prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to $g$. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable.