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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 FROM AUTHOR]

We consider the Einstein–Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conformal boundary at future null infinity; see (Andersson and Chruściel 1994 Commun. Math. Phys.161 533–68). We work with initial data sets in a variety of regularity classes, primarily considering those data sets whose geometries are weakly asymptotically hyperbolic, as defined in (Allen et al 2015 arXiv:1506.03399). These metrics are C1,1 conformally compact, but not necessarily C2 conformally compact. In order to ensure that the data sets we construct are indeed shear-free, we make use of the conformally covariant traceless Hessian introduced in (Allen et al 2015 arXiv:1506.03399). We furthermore construct a class of initial data sets with weakly asymptotically hyerbolic metrics that may be only C0,1 conformally compact; these data sets are insufficiently regular to make sense of the shear-free condition. [ABSTRACT FROM AUTHOR]

In general relativity, an initial data set for an isolated gravitational system takes the form of a solution of the Einstein constraint equations which is asymptotically Euclidean on a specified end. Given a collection of N such data sets with a subregion of interest (bounded away from the specified end) chosen in each, we show that there exists a family of new initial data sets, each of which contains exact copies of each of the N chosen subregions, positioned in a chosen array in a single asymptotic end. These composite initial data sets model isolated, relativistic gravitational systems containing N chosen bodies in specified initial configurations. [ABSTRACT FROM AUTHOR]

We prove that if a stationary, real analytic, asymptotically flat vacuum black hole spacetime of dimension n [?] 4 contains a non-degenerate horizon with compact cross-sections that are transverse to the stationarity generating Killing vector field then, for each connected component of the black hole's horizon, there is a Killing field which is tangent to the generators of the horizon. For the case of rotating black holes, the stationarity generating Killing field is not tangent to the horizon generators and therefore the isometry group of the spacetime is at least two dimensional. Our proof relies on significant extensions of our earlier work on the symmetries of spacetimes containing a compact Cauchy horizon, allowing now for non-closed generators of the horizon. [ABSTRACT FROM AUTHOR]

We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the vacuum Einstein constraint equations. This result extends previous work which required the conformal metric to be in the negative Yamabe class, and required the mean curvature function to be nonzero. [ABSTRACT FROM AUTHOR]