LAUSR

We prove positive mass theorems for asymptotically hyperbolic and asymptotically locally hyperbolic Riemannian manifolds with black-hole-type boundaries. An interesting global invariant of asymptotically hyperbolic general relativistic initial data sets is the energy-momentum vector m ≡ (mμ) [5, 6, 13] (compare [1, 7]). It is known that m is timelike future pointing or vanishing [3,9] for conformally compactifiable manifolds with a connected conformal boundary at infinity with spherical topology and satisfying a natural lower bound on the scalar curvature. The object of this note is to point out how to generalise this result to manifolds with a compact boundary and with several ends in dimensions 3 ≤ n ≤ 7, using the results of [3, 8], without assuming that the manifold M carries a compatible spin structure: Theorem 1. Let (M,h) be a conformally compact n-dimensional, 3 ≤ n ≤ 7, asymptotically locally hyperbolic manifold with boundary. Assume that the scalar curvature of M satisfies R(h) ≥ −n(n−1), and that the boundary has mean curvature H ≤ n−1 with respect to the normal pointing into M . Then, the energy-momentum vector m of every spherical component of the conformal boundary at infinity of (M,h) is future causal. ∗Preprint UWThPh 2021-9 †email: [email protected]; http://homepage.univie.ac.at/piotr.chrusciel ‡email: [email protected] 1 ar X iv :2 10 7. 05 60 3v 1 [ gr -q c] 1 2 Ju l 2 02 1 Remark 2. Neither the boundary ∂M , nor the conformal boundary at infinity of M , need to be connected. Our sign convention for the mean curvature is such that the round sphere in Euclidean space has positive mean curvature with respect to the outward pointing normal. In order to avoid ambiguities, some definitions are in order. We say that a Riemannian manifold (M,h) is conformally compact if there exists a compact manifold with boundary M̂ such that the following holds: First, we allow M to have a boundary, which is then necessarily compact. Next, M is the interior of M̂ , whose boundary is the union of the boundary of M and of a number of new boundary components, at least one, which form the conformal boundary at infinity. Next, there exists on M̂ a smooth function Ω ≥ 0 which is positive on M , and which vanishes precisely on the new boundary components of M̂ , with dΩ nowhere vanishing there. Finally, the tensor field Ωh extends to a smooth metric on M̂ . An asymptotically locally hyperbolic (ALH) metric is a metric with all sectional curvatures approaching minus one as the conformal boundary at infinity is approached. An asymptotically hyperbolic (AH) metric is an ALH metric with spherical conformal boundary at infinity. Remark 3. For simplicity our definitions require smoothness up-to-boundary of the conformally rescaled metric, though much weaker conditions suffice for the result. Example 4. An example to keep in mind is provided by the space part of the Birmingham-Kottler metrics, g = V −2dr2 + rh−k , V 2 = r + k − 2m rn−2 . (1) where h−k is an Einstein metric on an (n − 1) dimensional manifold with scalar curvature equal to k(n− 1)(n− 2). The field of normals to the level sets of r equals N = V ∂r, hence H = 1 √ det g ∂i( √ det gN ) = V rn−1 ∂r(r n−1) = (n− 1) √ 1 + kr−2 − 2mr−n . (2) When k = 1 we find H > (n − 1) (with respect to the outward normal) for all sufficiently large r regardless of the sign of m. When m < 0 the range of H covers the interval (n − 1,∞) when r runs over (0,∞), which shows that the condition on H in Theorem 1 is optimal. In [4, Theorem 1.1] we proved non-negativity of mass for ALH manifolds whose closure has topology [0, 1] × (Sn−1/Γ), where Γ is a finite subgroup of SO(n − 1). In particular it was assumed that both conformal infinity and the boundary have