Let (M, g) be a compact Riemannian manifold on which a tracefree and divergence-free σ ∈ W 1,p and a positive function τ ∈ W 1,p , p > n, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data σ and τ . We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on 1-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which in particular hold on a dense set of metrics g for the C 0 -topology.

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.

Abstract. In this paper we study the extent to which conformally compact asymptotically hyperbolic metrics may be characterized intrinsically. Building on the work of the first author in [6], we prove that decay of sectional curvature to −1 and decay of covariant derivatives of curvature outside an appropriate compact set yield Hölder regularity for a conformal compactification of the metric. In the Einstein case, we prove that the estimate on the sectional curvature implies the control of all covariant derivatives of the Weyl tensor, permitting us to strengthen our result.

ABSTRACT. In this short note, we give a construction of solutions to the Einstein constraint equations using the well known conformal method. Our method gives a result similar to the one in [15,16,24], namely existence when the so called TT-tensor σ is small and the Yamabe invariant of the manifold is positive. The method we describe is however much simpler than the original method and allows easy extensions to several other problems. Some non-existence results are also considered.