Abstract
We prove positivity of energy for a class of asymptotically locally hyperbolic manifolds in dimensions . The result is established by first proving deformation-of-mass-aspect theorems in dimensions . Our positivity results extend to the case n = 3 when more stringent conditions are imposed.
Highlights
An interesting global invariant of asymptotically hyperbolic manifolds is provided by the total mass, for general conformal boundaries at infinity, or the total energy-momentum vector when the conformal structure at conformal infinity is that of a round sphere [6, 8, 20]
These objects provide a generalisation of the Arnowitt-Deser-Misner (ADM) energy-momentum, which is defined for asymptotically flat manifolds, to the asymptotically hyperbolic case
While there are sharp positivity results for the ADM mass in all dimensions [19], the asymptotically hyperbolic case is still poorly understood
Summary
An interesting global invariant of asymptotically hyperbolic manifolds is provided by the total mass, for general conformal boundaries at infinity, or the total energy-momentum vector when the conformal structure at conformal infinity is that of a round sphere [6, 8, 20] (compare [1, 9]). Theorem 4.1 provides the following minor improvement of the Andersson-Cai-Galloway theorem, keeping in mind that their hypothesis of mass aspect of constant sign implies that the energymomentum vector is timelike (see Section 2 for terminology): Theorem 1.5 Let (M n, g), 4 ≤ n ≤ 7, be a manifold with scalar curvature R[g] ≥ −n(n − 1) with a metric which is smoothly conformally compactifiable with spherical conformal infinity. We note that examples of metrics with constant negative scalar curvature and with a null or spacelike energy-momentum vector on a (non-complete) asymptotically hyperbolic manifold have been constructed by Cortier in [10].
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