A Łojasiewicz inequality for ALE metrics

Abstract

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. ContentsALE 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

Spinors and mass on weighted manifolds

“…The renormalized Perelman functional is the correct modification of Perelman's λ-functional (for closed manifolds) to AE manifolds: it has the crucial property that Ricci flow, ∂ t g = −2Ric, is its gradient flow, as shown in [DO1]. Thus equality (0.8) implies that a Ricci flow on an AE manifold with nonnegative scalar curvature is the gradient flow of the weighted mass (see Corollary 2.20):…”

“…If (M n , g) admits a Witten spinor ψ, then testing the right-hand-side of the above equation with u = |ψ| gives that λ ALE (g) ≤ 0, by Kato's inequality, |∇|ψ||≤ |∇ψ|. As mentioned in the Introduction, Ricci flow is the gradient flow of λ ALE on AE manifolds and λ ALE has various advantages over the ADM mass in the context of Ricci flow; see the Introduction and also [DO1].…”

“…The additional assumptions R ≥ 0, R ∈ L 1 (M, g) ensure the existence of an (unweighted) Witten spinor ψ [LP]. Further, the assumption |∇f |= O(r −(n−1) ) is satisfied if R f = 0; see [DO1,Prop. (2.2)].…”

“…Proof. By [DO1,(2.3)], there exists a strictly positive minimizer w = e −f /2 of (2.16) with w − 1 ∈ C 2,α −τ (M ) satisfying −4∆w + R w = 0 and w → 1 at infinity. Integration by parts gives…”

“…Proof. Since m f (g) = −λ ALE (g), equation (2.21) follows from the formula for the first variation of λ ALE , which can be found in [DO1,Prop. 2.3 and 3.13].…”