A Łojasiewicz inequality for ALE metrics
Abstract
We introduce a new functional inspired by Perelman's $\lambda$functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted $\lambda_{\operatorname{ALE}}$. Its expression includes a boundary term which turns out to be the ADMmass. We prove that $\lambda_{\operatorname{ALE}}$ is defined and analytic on convenient neighborhoods of Ricciflat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricciflat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a LojasiewiczSimon inequality on noncompact manifolds and prove that it applies to $\lambda_{\operatorname{ALE}}$ around Ricciflat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricciflat deformations.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.09937
 Bibcode:
 2020arXiv200709937D
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Mathematics  Analysis of PDEs
 EPrint:
 63 pages, no figure