Lorentzian Einstein metrics with prescribed conformal infinity

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 $C^\infty$.