  In this paper we study the spacelike-characteristic Cauchy problem for the
Einstein vacuum equations. We prove that given initial data on a maximal
compact spacelike hypersurface $\Sigma \simeq \overline{B(0,1)} \subset
\mathbb{R}^3$ and the outgoing null hypersurface $\mathcal{H}$ emanating from
$\partial \Sigma$, the time of existence of a solution to the Einstein vacuum
equations is controlled by low regularity bounds on the initial data at the
level of curvature in $L^2$. The proof uses the bounded $L^2$ curvature theorem
by Klainerman, Szeftel and Rodnianski, the extension procedure for the
constraint equations by Czimek, Cheeger-Gromov theory in low regularity
developed by Czimek, the canonical foliation on null hypersurfaces in low
regularity by Czimek and Graf, and global elliptic estimates for spacelike
maximal hypersurfaces.
