A \L{}ojasiewicz inequality for ALE metrics

TL;DR

This paper introduces a new functional for ALE metrics inspired by Perelman's $$-functional, proves its properties, and establishes a Lojasiewicz-Simon inequality, advancing understanding of Ricci-flat ALE metrics and their stability.

Contribution

It defines a novel _{ ext{ALE}} functional, proves its analyticity and monotonicity, and develops a general scheme for Lojasiewicz-Simon inequalities on non-compact manifolds.

Findings

_{ ext{ALE}} is analytic near Ricci-flat ALE metrics.

Monotonicity of _{ ext{ALE}} along Ricci flow.

Establishment of a Lojasiewicz-Simon inequality for _{ ext{ALE}}.

Abstract

We introduce a new functional inspired by Perelman's $\lambda$-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted $\lambda_{\operatorname{ALE}}$. Its expression includes a boundary term which turns out to be the ADM-mass. We prove that $\lambda_{\operatorname{ALE}}$ is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to $\lambda_{\operatorname{ALE}}$ around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations.