  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.
