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We consider 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…
We show that asymptotically hyperbolic initial data satisfying smallness conditions in dimensions $n\ge 3$, or fast decay conditions in $n\ge 5$, or a genericity condition in $n\ge 9$, can be deformed, by a deformation which is supported…
We consider non-linear gravitational models with a multidimensional warped product geometry. Particular attention is payed to models with quadratic scalar curvature terms. It is shown that for certain parameter ranges, the extra dimensions…
The existence of the initial value constraints means that specifying initial data for the Einstein equations is non-trivial. The standard method of constructing initial data in the asymptotically flat case is to choose an asymptotically…
We demonstrate that in constructing asymptotically flat vacuum initial data sets in General Relativity via the conformal method, certain asymptotic structures may be prescribed a priori through the specified seed data, including the ADM…
In this paper we continue earlier investigations of evolutionary formulations of the Einstein vacuum constraint equations originally introduced by R\'{a}cz. Motivated by the strong evidence from these works that the resulting vacuum initial…
We investigate the evolution of geometric invariants, as defined by Michel \cite{Michel}, in the context of asymptotically hyperboloidal initial data sets. Our focus lies on the charges of energy and linear momentum, and we study their…
We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a…
When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of…
In this paper, we continue our investigations of R\'acz's parabolic-hyperbolic formulation of the Einstein vacuum constraints. Our previous studies of the asymptotically flat setting provided strong evidence for unstable asymptotics which…
We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers…
We consider a characteristic initial value problem for a class of symmetric hyperbolic systems with initial data given on two smooth null intersecting characteristic surfaces. We prove existence of solutions on a future neighborhood of the…
Hyperbolic monopole solutions can be obtained from circle-invariant ADHM data if the curvature of hyperbolic space is suitably tuned. Here we give explicit ADHM data corresponding to axial hyperbolic monopoles in a simple, tractable form,…
We show that the ADM mass and momentum are geometric invariants of asymptotically flat initial data sets
The gauge/string dualities have drawn attention to a class of variational problems on a boundary at infinity, which are not well defined unless a certain boundary term is added to the classical action. In the context of supergravity in…
We provide a Hilbert manifold structure {\`a} la Bartnik for the space of asymptotically hyperbolic initial data for the vacuum constraint equations. The adaptation led us to prove new weighted Poincar{\'e} and Korn type inequalities for AH…
We study the utilization of conformal compactification within the conformal approach to solving the constraints of general relativity for asymptotically flat initial data. After a general discussion of the framework, particular attention is…
We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has…
The asymptotic structure of AdS spacetimes in the context of General Relativity coupled to the Maxwell field in three spacetime dimensions is analyzed. Although the fall-off of the fields is relaxed with respect to that of Brown and…
We review and announce recent results on the asymptotic behavior of asymptotically Euclidean relativistic initial data sets and asymptotic foliations thereof. In particular, we discuss the geometrization of asymptotic flatness and of…