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We study the stability and instability of ALE Ricci-flat metrics around whicha 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 authorsin a recent work and it has been proven that it satisfies a good enoughLojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flatmetrics in dimension larger than or equal to 5.

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