Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

TL;DR

This paper investigates the stability and instability of ALE Ricci-flat metrics under Ricci flow, utilizing a specialized Perelman-type functional and Lojasiewicz inequalities to analyze their behavior in higher dimensions.

Contribution

It introduces a new functional, mbda_{ ext{ALE}}, and demonstrates its effectiveness in studying the stability of ALE Ricci-flat metrics, extending previous work to higher dimensions.

Findings

Established conditions for stability and instability of ALE Ricci-flat metrics.

Proved mbda_{ ext{ALE}} satisfies a Lojasiewicz inequality near integrable metrics.

Extended stability analysis to dimensions  and above.

Abstract

We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman's $\lambda$-functional adapted to the ALE situation and denoted $\lambda_{\operatorname{ALE}}$. This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.