Related papers: Limit equation for vacuum Einstein constraints wit…
In a space-time $M$ with a Killing vector field $\xi^a$ which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of $\xi^a$ gives a 3-dimension space $S$. Besides the symmetry-reduced action from that…
We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary…
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…
We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions which is constraint-preserving and sufficiently general to include…
The solutions of vacuum Einstein's field equations, for the class of Riemannian metrics admitting a non Abelian bidimensional Lie algebra of Killing fields, are explicitly described. They are parametrized either by solutions of a…
We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean…
In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In…
Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated…
Motivated by recent accelerating cosmological model, we derive the solutions to vacuum Einstein equation in $(d+1)$-dimensional Minkowski space with $n$-dimensional hyperbolic manifold. The conditions of accelerating expansion are given in…
A Hilbert manifold structure is described for the ADM phase space of asymptotically flat initial data $(g,\pi)$ with local $H^2\times H^1$ Sobolev regularity. Solutions of the constraint equations form a Hilbert submanifold. A regularized…
In this paper, we prove a far-from-CMC result similar to the ones obtained by Holst, Nagy, Tsogtgerel and Maxwell for the conformal Einstein-scalar field constraint equations on compact Riemannian manifolds with positive (modified) Yamabe…
A review of the treatment of boundaries in general relativity is presented with the emphasis on application to the formulations of Einstein's equations used in numerical relativity. At present, it is known how to treat boundaries in the…
We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically…
Certain features associated with the symmetry reduction of the vacuum Einstein equations by two commuting, space-like Killing vector fields are studied. In particular, the discussion encompasses the equations for the Gowdy $T^3$ cosmology…
We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that…
The Harmonic Einstein equation is the vacuum Einstein equation supplemented by a gauge fixing term which we take to be that of DeTurck. For static black holes analytically continued to Riemannian manifolds without boundary at the horizon…
Specifying boundary conditions continues to be a challenge in numerical relativity in order to obtain a long time convergent numerical simulation of Einstein's equations in domains with artificial boundaries. In this paper, we address this…
When the vacuum Einstein equations are cast in the form of hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold M\Sigma\ of riemannian metrics on a Cauchy hypersurface \Sigma. As in every lagrangian…
We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound…
We prove the existence of solutions to the conformal Einstein-scalar constraint system of equations for closed compact Riemannian manifolds in the positive case. Our results apply to the vacuum case with positive cosmological constant and…