Perelman's lambda-functional and the stability of Ricci-flat metrics

TL;DR

This paper introduces a novel approach using Perelman's lambda-functional to analyze the stability of compact Ricci-flat metrics, providing new stability criteria, inequalities, and insights into Ricci flow behavior.

Contribution

It develops a new method based on lambda-functional to study Ricci-flat metric stability, including criteria for local maximizers and gradient inequalities.

Findings

Ricci-flat metrics are local maximizers of lambda under certain conditions

Lambda satisfies a Lojasiewicz-Simon gradient inequality

Ricci flow stability and instability results are established

Abstract

In this article, we introduce a new method (based on Perelman's lambda-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (A) a Ricci-flat metric is a local maximizer of lambda in a C^2,alpha-sense iff its Lichnerowicz Laplacian is nonpositive, (B) lambda satisfies a Lojasiewicz-Simon gradient inequality, (C) the Ricci flow does not move excessively in gauge directions. As consequences, we obtain a rigidity result, a new proof of Sesum's dynamical stability theorem, and a dynamical instability theorem.