The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface \(\Sigma \simeq \overline{B_1} \subset {{\mathbb {R}}}^3\) and the outgoing null hypersurface \({{\mathcal {H}}}\) emanating from \({\partial }\Sigma \), we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in \(L^2\). The proof uses the bounded \(L^2\) curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.