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

In this article, we introduce a new method (based on Perelman’s λ-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 λ in a C2,α-sense if and only if its Lichnerowicz Laplacian is nonpositive, (b) λ satisfies a Łojasiewicz-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.