A Lojasiewicz inequality for ALE metrics

Abstract

Translate Article

We introduce a new functional inspired by Perelman's λ-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted λ ALE . Its expression includes a boundary term which turns out to be the ADM-mass. We prove that λ ALE is defined and analytic on convenient neighborhoods of Ricci-flat 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 Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to λ ALE around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations. ALE 3. A functional defined on a C 2,α τ -neighborhood of Ricci-flat ALE metrics 4. Energy estimates on the potential function 5. Fredholm properties of the Lichnerowicz operator in weighted spaces 6. Properties of λ ALE in the integrable case 7. A Lojasiewicz inequality for λ ALE : the general case 8. Deformation of Ricci-flat ALE metrics, scalar curvature and mass Appendix A. Real analytic maps between Banach spaces Appendix B. Divergence-free gauging of asymptotically conical spaces Appendix C. First and second variations of geometric quantities References