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In this article we continue our effort to do a systematic development of the solution theory for conformal formulations of the Einstein constraint equations on compact manifolds with boundary. By building in a natural way on our recent work…
$[n+1]$-dimensional ($n\geq 3$) smooth Einsteinian spaces of Euclidean and Lorentzian signature are considered. The base manifold $M$ is supposed to be smoothly foliated by a two-parameter family of codimension-two-surfaces which are…
Let (M,g) be a smooth compact, n dimensional Riemannian manifold,with smooth n-1 dimensional boundary. We prove that the stable critical points of the mean curvature of the boundary generates solutions for a singularly perturbed elliptic…
This is the first in a series of articles on the numerical solution of Friedrich's conformal field equations for Einstein's theory of gravity. We will discuss in this paper why one should be interested in applying the conformal method to…
We consider the Einstein-Boltzmann system for massless particles in the Bianchi I space-time with scattering cross-sections in a certain range of soft potentials. We assume that the space-time has an initial conformal gauge singularity and…
We present new classes of exact solutions with noncommutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such…
The intimate relations between Einstein's equation, conformal geometry, geometric asymptotics, and the idea of an isolated system in general relativity have been pointed out by Penrose many years ago. A detailed analysis of the interplay of…
Given a closed Riemannian manifold $(M, g_M)$ of dimension $n \geq 3$, we prove the existence of a conformally compact Einstein metric $g_{+}$ defined on a collar neighborhood $M \times (0,1]$ whose conformal infinity is $[g_M]$.
In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the…
We give an alternate proof of one of the results given in [16] showing that initial data sets with boundary for the Einstein equations $(M, g, k)$ satisfying the dominant energy condition can be conformally deformed to the strict dominant…
We consider a system representing self-gravitating balls of dust in an expanding Universe. It is demonstrated that one can prescribe data for such a system at infinity and evolve it backward in time without the development of shocks or…
The Einstein constraint equations have been the subject of study for more than fifty years. The introduction of the conformal method in the 1970's as a parameterization of initial data for the Einstein equations led to increased interest in…
We prove the existence of a large class of initial data for the vacuum Einstein equations which possess a finite number of asymptotically Euclidean and asymptotically conformally cylindrical or periodic ends. Aside from being asymptotically…
By suitably re-scaling the conformal Einstein's equations we are able to apply recent results in the theory of PDE, and prove that they possess slow solutions in a future neighborhood of an initial surface reaching ${\cal I}^+$. The…
In this paper, we construct the first analytic examples of (3+1)-dimensional self-gravitating regular cosmic tube solutions which are superconducting, free of curvature singularities and with non-trivial topological charge in the…
We develop a Hamiltonian formulation of Bianchi type-I cosmological model in conformal gravity, i.e. the theory described by a Lagrangian which involves the quadratic curvature invariant constructed from the Weyl tensor, in four dimensions.…
We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded…
Given any asymptotically flat 3-manifold $(M,g)$ with smooth, non-empty, compact boundary $\Sigma$, the conformal conjecture states that for every $\delta>0$, there exists a metric $g' = u^4 g$, with $u$ a harmonic function, such that the…
Given a noncollapsing sequence of m-dimensional compact Einstein manifolds with a uniform energy bound, the Gromov-Hausdorff limit is a compact Einstein orbifold with at most finitely many singularities. Conversely, starting with a compact…
Original abstract: "We construct periodic solutions of nonlinear wave equations using analytic continuation. The construction applies in particular to Einstein equations, leading to infinite-dimensional families of time-periodic solutions…