Integrable Deformations and Stability of the Ricci Flow

TL;DR

This paper proves the stability of Ricci flow near certain Ricci-flat metrics using a new approach based on integrability and orthogonality properties, with results in weighted spaces and connections to existing $L^p$ stability theorems.

Contribution

It introduces a simplified proof of Ricci flow stability near integrable Ricci-flat ALE metrics by leveraging an equivalence between integrability and an orthogonality property.

Findings

Established stability of Ricci flow near Ricci-flat ALE metrics.

Connected integrability with an orthogonality property of the Ricci-DeTurck tensor.

Extended results to recover known $L^p$-stability theorems.

Abstract

We provide a comparatively simple proof of the dynamical stability of Ricci flow near a linearly stable Ricci-flat ALE metric with integrable deformations. Our proof relies on the equivalence between integrability and an "almost-orthogonality" property of the Ricci-DeTurck tensor, allowing us to analyze the latter directly. We obtain our main results in weighted Holder spaces and then show how to recover the $L^p$-stability theorems of Deruelle-Kroncke and Kroncke-Petersen.