Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M, [γ]) is non-negative, we prove that there exists a conformal metric on M with non-negative scalar curvature and whose boundary ∂M has either positive or zero constant inner mean curvature. In the spin case, we make use of a previous estimate obtained by X. Zhang and the authors for the Dirac operator of the induced metric on ∂M. As a consequence, we generalize and simplify the proof of the result by Andersson and Dahl in [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom.16 (1998) 1–27] about the rigidity of the hyperbolic space when the prescribed conformal infinity ∂M is a round sphere. We also provide non-existence results for conformally compact ALH spin metrics when ∂M is conformal to a Riemannian manifold with special holonomy.
On scalar curvature rigidity of vacuum static spaces
