We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent tend to its true value. We prove that the solutions of the sub-critical equations remain bounded which yields solutions of the constraint equation unless a certain limit equation admits a nontrivial solution. Finally, we give conditions which ensure that the limit equation admits no non-trivial solution. More exactly, these are the Christoffel symbols of the induced metric that can be considered as the magnetic part, the metric g is the analogue of the potential vector. 1 2. Preliminaries 2.1. The conformal method. A natural way to understand the constraint equations (1.1) and (1.2) is to consider the Hamiltonian constraint (1.1) as a scalar equation for the metric and the momentum constraint (1.2) as a vectorial equation for the second fundamental form K. As a consequence, to construct solutions (M, g, K) of the system (1.1)-(1.2), we will look for g in the conformal class of a given metric g, i.e. in the form g = φ κ g, where κ = 4 n−2 . In order to understand the structure of solutions of the momentum constraint (1.2), we decompose K as K = τ g + σ, where τ = 1 n tr g K is the mean curvature of the hypersurface M ⊂ M and σ is a symmetric traceless 2-tensor. The equation (1.2) then becomesThis equation still involves φ in the term g ik ∇ k σ ij . Setting σ = φ −2 σ, one obtains the following equation to be solved for σ:To solve this equation, one has to freeze some degrees of freedom of σ. We decompose σ as a sum σ = σ 0 + σ 1 of a particular solution σ 1 and a solution σ 0 of the homogeneous problemNote that a 2-tensor which is symmetric, traceless and divergence-free is called a TT-tensor. A construction of TT-tensors will be given in Corollary 3.3. As for σ 1 , it can be chosen as the traceless part of the Lie derivative of the metric in the direction of the dual of some 1-form ψ:

In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space H n . The graphs are considered as unbounded hypersurfaces of H n+1 which carry the induced metric and have an interior boundary. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article [22] concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu in [19] we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric. appeared on arXiv. In the first of these papers an Alexandrov-Fenchel type inequality for hypersurfaces in hyperbolic space is stated, which together with Proposition 4.1 implies the Penrose inequality (1) for graphs. Certain steps of the proof seem to need further clarification, for example the convergence of hypersurfaces to round spheres under the inverse mean curvature flow. However, combining with arguments of the second paper [7] the result should follow. Note also that a special case of [7, Theorem 2] follows from our formula (13) in Section 4.2.

We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyperbolic space in the intrinsic flat sense, if the limit of the mass along the sequence is zero.

Abstract. In this paper we introduce a new method for manufacturing harmonic morphisms from semi-Riemannian manifolds. This is employed to yield a variety of new examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their standard Riemannian metrics. We develop a duality principle and show how this can be used to construct the first known examples of harmonic morphisms from the non-compact Lie groups SLn(R), SU * (2n), Sp(n, R), SO * (2n), SO(p, q), SU(p, q) and Sp(p, q) equipped with their standard dual semi-Riemannian metrics.

We consider the several geometric inequalities in general relativity involving mass, area, charge, and angular momentum for asymptotically hyperboloidal initial data. We show how to reduce each one to the known maximal (or time symmetric) case in the asymptotically flat setting, whenever a geometrically motivated system of elliptic equations admits a solution.