The Mass of an Asymptotically Hyperbolic Manifold with a Non-compact Boundary
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Annales Henri Poincar´ e
The Mass of an Asymptotically Hyperbolic Manifold with a Non-compact Boundary S´ergio Almaraz
and Levi Lopes de Lima
Abstract. We define a mass-type invariant for asymptotically hyperbolic manifolds with a non-compact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application, we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space. Mathematics Subject Classification. Primary 53C21; Secondary 53C27, 53C80.
1. Introduction Given a non-compact Riemannian manifold (M, g) of dimension n ≥ 3 arising as the (time-symmetric) initial data set for a solution (M , g) of Einstein field equations in dimension n + 1, standard physical reasoning suggests the existence of a geometric invariant defined in terms of the asymptotic behavior of the underlying metric at spatial infinity. Roughly speaking, it is assumed that in the asymptotic region (M, g) converges to some reference space (N, b), which by its turn is required to propagate to a static solution (i.e. a solution displaying a time-like vector field whose orthogonal distribution is integrable), and the mass invariant, which is denoted by m(g,b) and should be interpreted as the total energy of the isolated gravitational system modelled by (M , g), is designed so as to capture the coefficient of the leading term in the asymptotic expansion of g around b. In particular, the important question arises as to S. Almaraz has been partially supported by CNPq/Brazil Grant 309007/2016-0 and CAPES/Brazil Grant 88881.169802/2018-01, and L. de Lima has been partially supported by CNPq/Brazil Grant 312485/2018-2. Both authors have been partially supported by FUNCAP/CNPq/PRONEX Grant 00068.01.00/15.
S. Almaraz, and L. L. de Lima
Ann. Henri Poincar´e
whether, under a suitable dominant energy condition, the invariant in question satisfies the positive mass inequality m(g,b) ≥ 0,
(1.1)
with equality taking place if and only if (M, g) = (N, b) isometrically. The classical example is the asymptotically flat case, where the reference space is (Rn , δ), the Euclidean space endowed with the standard flat metric δ. Here, m(g,δ) is the so-called ADM mass and it has been conjectured that the corresponding positive mass inequality holds true whenever the scalar curvature Rg of g is non-negative. After previous contributions by Schoen–Yau [38] if n ≤ 7 and by Witten and Bartnik [5,41] in the spin case, the conjecture has at last been settled in independent contributions by Schoen–Yau [36] and Lohkamp [27]. Partly motivated by the so-called AdS/CFT correspondence, which in the Euclidean semi-classical limit highlights Einstein metrics with negative scalar curvature, recently there has been much interest in studying similar invariants for non-compact Riemannian manifolds whose geometry at infinity approaches some referenc
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