相关论文: Trilinear compensated compactness and Burnett's co…

We prove Burnett's conjecture in general relativity when the metrics satisfy a generalized wave coordinate condition, i.e., suppose $\{g_n\}_{n=1}^\infty$ is a sequence of Lorentzian metrics (in arbitrary dimensions $d \geq 3$) satisfying a…

In 1989, Burnett conjectured that, under appropriate assumptions, the limit of highly oscillatory solutions to the Einstein vacuum equations is a solution of the Einstein--massless Vlasov system. In a recent breakthrough, Huneau--Luk…

We review recent mathematical results concerning the high-frequency solutions to the Einstein vacuum equations and the limits of these solutions. In particular, we focus on two conjectures of Burnett, which attempt to give an exact…

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$. We will assume that the isotropy $H$-module $\mathfrak {g/h}$ has a simple spectrum, i.e. irreducible submodules are…

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor…

We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a compact manifold with boundary. We use order relations on appropriate Banach spaces to derive weak solution generalizations…

We present an explicit upper bound on the number of isolated homogeneous Einstein metrics on compact homogeneous spaces whose isotropy representations consist of pairwise inequivalent irreducibles. This is the BKK bound of the corresponding…

We construct low regularity solutions of the vacuum Einstein constraint equations on compact manifolds. On 3-manifolds we obtain solutions with metrics in $H^s$ where $s>3/2$. The constant mean curvature (CMC) conformal method leads to a…

On vacuum spacetimes of general dimension, we study the linearized Einstein vacuum equations with a spatially compactly supported and (necessarily) divergence-free source. We prove that the vanishing of appropriate charges of the source,…

The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein's field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B…

We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a…

We prove that given any smooth metric $\gamma$ and smooth positive function $H$ on $S^{2}$, there is a constant $\lambda > 0$, depending on $(\gamma, H)$, and an asymptotically flat solution $(M, g, u)$ of the static vacuum Einstein…

We prove a global well-posedness and asymptotic convergence theorem for the \((3+1)\)-dimensional vacuum Einstein equations with positive cosmological constant \(\Lambda\) on globally hyperbolic spacetimes \(\widetilde M \cong M \times…

This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein's constraints equations. We establish existence theorems for…

In this paper, we study the past asymptotics of $(2+1)$-dimensional solutions to the Einstein scalar-field Vlasov system which are close to Friedman-Lema\^itre-Robertson-Walker spacetimes on an initial hypersurface diffeomorphic to a closed…

We discuss the linearization of Einstein equations in the presence of a cosmological constant, by expanding the solution for the metric around a flat Minkowski space-time. We demonstrate that one can find consistent solutions to the…

We study the nonlinear stability of the $(3+1)$-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the…

We consider general initial data for the Einstein scalar-field system on a closed $3$-manifold $(M,\gamma)$ which is close to data for a Friedman-Lema\^itre-Robertson-Walker solution with homogeneous scalar field matter and a negative…

Compensated compactness is an important method used to solve nonlinear PDEs. A simple formulation of a compensated compactness problem is to ask for conditions on a set $\mathcal{K}\subset M^{m\times n}$ such that $$ \lim_{n\rightarrow…

The aim of this article is to construct initial data for the Einstein equations on manifolds of the form R n+1 x T m , which are asymptotically flat at infinity, without assuming any symmetry condition in the compact direction. We use the…