Abstract
We study the stability and instability of ALE Ricci-flat metrics around which a Łojasiewicz inequality is satisfied for a variation of Perelman’s $$\lambda $$ functional adapted to the ALE situation and denoted $$\lambda _{{\text {ALE}}}$$ . This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Łojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.
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