相关论文: Rough Solutions of the Einstein Constraint Equatio…
We continue the study of the Einstein constraint equations on compact manifolds with boundary initiated by Holst and Tsogtgerel. In particular, we consider the full system and prove existence of solutions in both the near-CMC and…
We construct solutions with prescribed asymptotics to the Einstein constraint equations using a cut-off technique. Moreover, we give various examples of vacuum asymptotically flat manifolds whose center of mass and angular momentum are…
The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's…
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…
We find new classes of exact solutions of the initial momentum constraint for vacuum Einstein's equations. Considered data are either invariant under a continuous symmetry or they are assumed to have the exterior curvature tensor of a…
It is well-known that solutions to the conformal formulation of the Einstein constraint equations are unique in the cases of constant mean curvature (CMC) and near constant mean curvature (near-CMC). However, the new far-from-constant mean…
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 Einstein constraint…
The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic…
In this short note, we give a construction of solutions to the Einstein constraint equations using the well known conformal method. Our method gives a result similar to the one in [15, 16, 24], namely existence when the so called TT-tensor…
We consider the Einstein constraints on asymptotically euclidean manifolds $M$ of dimension $n \geq 3$ with sources of both scaled and unscaled types. We extend to asymptotically euclidean manifolds the constructive method of proof of…
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…
In this paper, we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, and obtain the local well-posedness result in $H^s, s>2$. The novelty of our approach lies in that, without…
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…
The effectiveness of the hyperbolic relaxation method for solving the Einstein constraint equations numerically is studied here on a variety of compact orientable three-manifolds. Convergent numerical solutions are found using this method…
This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is…
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…
This paper produces explicit strongly Hermitian Einstein-Maxwell solutions on the smooth compact $4$-manifolds that are $S^2$-bundles over compact Riemann surfaces of any genus. This generalizes the existence results by C. LeBrun in…
In this paper, we study the coupled Einstein constraint equations on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. This is physically well-motivated by standard cosmological…
We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the…
This paper contains a classification of all 3-dimensional manifolds with constant scalar curvature $S \not= 0$ that carry a non-trivial solution of the Einstein-Dirac equation.