Lorentzian Einstein metrics with prescribed conformal infinity

TL;DR

This paper proves local well-posedness for the Einstein equations with asymptotically anti-de Sitter boundary conditions, establishing existence, uniqueness, and regularity of solutions given suitable initial and boundary data.

Contribution

It provides a rigorous framework for solving Einstein equations with prescribed conformal infinity, including regularity and compatibility conditions, extending previous results to more general boundary data.

Findings

Existence of solutions under specified boundary conditions

Uniqueness of solutions up to diffeomorphism

Regularity and polyhomogeneity of solutions

Abstract

We prove a local well-posedness theorem for the (n+1)-dimensional Einstein equations in Lorentzian signature, with initial data $(\tilde g, K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\hat g$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an n-dimensional asymptotically hyperbolic Riemannian manifold $(M,\tilde g)$ such that the conformally rescaled metric $x^2 \tilde g$ (with $x$ a boundary defining function) extends to the closure $\bar M$ of $M$ as a metric of class $C^{n-1}$ which is also polyhomogeneous of class $C^{p}$ on $\bar M$. Likewise we assume that the conformally rescaled symmetric (0,2)-tensor $x^{2}K$ extends to the closure as a tensor field of class $C^{n-1}$ which is polyhomogeneous of class $C^{p-1}$. We assume that the initial data $(\tilde g, K)$ satisfy the Einstein constraint equations and also that the boundary datum is of class $C^p$ on $\partial M\times (-T_0,T_0)$ and satisfies a set of natural compatibility conditions with the initial data. We then prove that there exists an integer $r_n$, depending only on the dimension n, such that if $p \geq 2q+r_n$, with $q$ a positive integer, then there is $T>0$, depending only on the norms of the initial and boundary data, such that the Einstein equations have a unique (up to a diffeomorphism) solution $g$ on $(-T,T)\times M$ with the above initial and boundary data, which is such that $x^2g$ is of class $C^{n-1}$ and polyhomogeneous of class $C^q$. Furthermore, if $x^2\tilde g$ and $x^2K$ are polyhomogeneous of class $C^\infty$ and $\hat g$ is in $C^\infty$, then $x^2g$ is polyhomogeneous of class $C^\infty$.