Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularityfree, vacuum space-times which are stationary in a neighborhood of i 0 ; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global I ; we prove existence of initial data for many black holes which are exactly Kerr -or exactly Schwarzschild -both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to r −m terms, for any fixed m, and with multipole moments freely prescribable within certain ranges. * Partially supported by a Polish Research Committee grant 2 P03B 073 24; email piotr@ gargan.math.univ-tours.fr † Visiting Scientist. Permanent address: Département de Mathématiques, Faculté des Sciences, Parc de Grandmont, F37200 Tours, France. ‡ Partially supported by the ACI program of the French Ministry of Research; email delay@ gargan.math.univ-tours.fr Lemma 2.3 Suppose that dim M ≥ 2, then P * ′ (x, ξ) is injective for ξ = 0.Proof: We define a linear map α from the space S 2 of two-covariant symmetric tensors into itself by the formula α(S) = S − (tr S)g .(2.6)Let ξ = 0, if (Y, N ) is in the kernel of P * ′ (x, ξ) then α(ξ (i Y j) ) = 0 , so that ξ (i Y j) = 0, and Y = 0. It follows that α(ξ i ξ j )N = 0 , which implies N = 0. ✷ The lemma implies:2 See Appendix A for the definitions of the function spaces we use.one is led toWe have thus showed that for C 2 compactly supported vector fields we have 16) and it should be clear that this remains true for vector fields which are only differentiable once. To continue, we use (2.16) with Y replaced with φψY ; the hypothesis that Ric (g) ∈ φ −2 L ∞ φ allows us to write |b(φψY )| + 2α(∇ (i (φψY j) )) L 2 + ψY L 2 ≥ c ∇(φψY ) L 2 .We have 2α(∇ (i (φY j) )) L 2 ψ = 2α(ψ∇ (i (φY j) )) L 2

We construct non-trivial vacuum space-times with a global I + . The construction proceeds by proving extension results for initial data sets across compact boundaries, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon. * Supported in part by a grant of the Polish Research Foundation KBN. † Supported in part by the ACI program of the French Ministry of Research. 1 Here we mean the metric induced by the Schwarzschild metric on the usual t = 0 hypersurface in Schwarzschild space-time; we will make such an abuse of terminology throughout.

We prove that the area of sections of future event horizons in spacetimes satisfying the null energy condition is non-decreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic space-time and there exists a conformal completion with a "Hregular" I + ; 3) the horizon is a black hole event horizon in a space-time which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends * Supported in part by KBN grant # 2 P03B 130 16. E-mail : Chrusciel@Univ-Tours.fr † Current adress: Department of Mathematics, Royal Institute of Technology, S-10044 Stockholm. E-mail : Delay@gargan.math.Univ-Tours.fr ‡ Supported in part by NSF grant # DMS-9803566. E-mail : galloway@math.miami.edu § Supported in part by DoD Grant # N00014-97-1-0806 E-mail : howard@math.sc.edu 1 a theorem of Hawking, in which piecewise smoothness of the event horizon seems to have been assumed. We prove smoothness or analyticity of the relevant part of the event horizon when equality in the area inequality is attained -this has applications to the theory of stationary black holes, as well as to the structure of compact Cauchy horizons. In the course of the proof we establish several new results concerning the differentiability properties of horizons.