We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with suitable corresponding conditions on the extrinsic curvature. In the compact setting a mild nondegeneracy condition is required. In the final section of the paper, we list a number ways this construction may be used to produce new types of vacuum spacetimes.

Complete, conformally flat metrics of constant positive scalar curvature on the complement of k k points in the n n -sphere, k ≥ 2 k \ge 2 , n ≥ 3 n \ge 3 , were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension k k . For a generic set of nearby conformal classes the moduli space is shown to be a k k -dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.

Abstract. We examine the space of surfaces in R 3 which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space M k of all such surfaces with k ends (where surfaces are identified if they differ by an isometry of R 3 ) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has no L 2 −nullspace we prove that M k is locally the quotient of a real analytic manifold of dimension 3k − 6 by a finite group (i.e. a real analytic orbifold), for k ≥ 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension of M k is independent of the topology of the underlying punctured Riemann surface to which Σ is conformally equivalent. These results also apply to hypersurfaces of H n+1 with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.

We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.

We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.

Abstract. We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.

Introduction and statement of the resultsAmongst the recent developments in the study of embedded complete minimal and constant mean curvature surfaces in R 3 is the realization that these objects are far more robust and flexible than is apparent from their Weierstrass representations. Our aim in this paper is to prove a 'gluing theorem', which states roughly that if two (appropriate) constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration (in the Hausdorff topology), but which is a topological connected sum of the two surfaces. We shall explain what we mean by appropriate, or at least give our preliminary interpretation of it, in the next paragraph. Throughout this paper, the acronym CMC shall mean a surface with constant mean curvature equal to one (or minus one depending on the orientation).The simplest context for our result is when we are given two orientable, immersed, compact CMC surfaces, Σ 1 and Σ 2 , with nonempty boundary. Suppose that we have applied a rigid motion to each of these surfaces so that 0 ∈ Σ 1 ∩ Σ 2 and T 0 Σ 1 = T 0 Σ 2 is the x y-plane. (These surfaces may intersect elsewhere, but that is irrelevant for our considerations.) We now define the orientation on these surfaces so that at 0 the oriented unit normal ν 1 of Σ 1 equals (0, 0, 1), while the oriented unit normal ν 2 of Σ 2 equals (0, 0, −1). Let us assume that with this orientation the two surfaces have the same mean curvature H 0 (so either H 0 = 1 or H 0 = −1 for both of the surfaces). We shall prove that there is a 'geometric connected sum' of these two surfaces, which may be thought of as a desingularization of this configuration. Moreover, the boundary of this desingularization will be the union of the boundaries of the Σ i , each possibly transformed by a small rigid motion.In order to state this first result rigorously, we make the following definition:Definition 1 A compact CMC surface Σ with boundary is said to be nondegenerate if there are no Jacobi fields on Σ which vanish on ∂Σ. Namely, if w : Σ −→ R is a C 2,α solution ofthen w = 0. Here A Σ is the second fundamental form of Σ.Theorem 1 (Connected sum theorem) Let Σ 1 and Σ 2 be two compact, smooth, immersed, orientable, nondegenerate CMC surfaces with boundary. Assume that these surfaces are positioned and oriented as above and have the same mean curvature H 0 . Then there exist an ε 0 > 0 and a one-parameter family of surfaces S ε , for ε ∈ (0, 4. The dilated surface ε −1 S ε converges in the C ∞ topology on any compact set to a catenoid with vertical axis.

Abstract. We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.