Journal of Differential Geometry

Lorentzian Einstein metrics with prescribed conformal infinity

Alberto Enciso and Niky Kamran

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove a local well-posedness theorem for the $(n+1)$-dimensional Einstein equations in Lorentzian signature, with initial data $(\widetilde{g},K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\widehat{g}$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an $n$-dimensional asymptotically hyperbolic Riemannian manifold $(M, \widetilde{g})$ such that the conformally rescaled metric $x^2 \widetilde{g}$ (with $x$ a boundary defining function) extends to the closure $\overline{M}$ of $M$ as a metric of class $C^{n-1} (\overline{M})$ which is also poly-homogeneous of class $C^p_{\mathrm{polyhom}} (\overline{M})$. Likewise we assume that the conformally rescaled symmetric $(0, 2)$-tensor $x^ 2 K$ extends to $\overline{M}$ as a tensor field of class $C^{n-1} (\overline{M})$ which is polyhomogeneous of class $C^{p-1}_{\mathrm{polyhom}} (\overline{M})$. We assume that the initial data $(\widetilde{g}, K)$ 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 $p \geqslant 2q + r_n$, with $q$ a positive integer, then there is $T \gt 0$, depending only on the norms of the initial and boundary data, such that the Einstein equations (1.1) has a unique (up to a diffeomorphism) solution $g$ on $(-T, T) \times M$ with the above initial and boundary data, which is such that $x^2 g \in C^{n-1} ((-T, T) \times \overline{M}) \; \cap \; C^q_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$. Furthermore, if $x^2 \widetilde{g} , x^2 K$ are polyhomogeneous of class $C^{\infty}$ and $\widehat{g}$ is in $C^{\infty} ((-T_0, T_0) \times \partial \overline{M})$, then $x^2 g$ is in $C^{\infty}_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$.

Article information

Source
J. Differential Geom., Volume 112, Number 3 (2019), 505-554.

Dates
Received: 13 July 2016
First available in Project Euclid: 16 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1563242472

Digital Object Identifier
doi:10.4310/jdg/1563242472

Mathematical Reviews number (MathSciNet)
MR3981296

Zentralblatt MATH identifier
07088293

Citation

Enciso, Alberto; Kamran, Niky. Lorentzian Einstein metrics with prescribed conformal infinity. J. Differential Geom. 112 (2019), no. 3, 505--554. doi:10.4310/jdg/1563242472. https://projecteuclid.org/euclid.jdg/1563242472


Export citation