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 Σ≃B_1⊂ℝ^3 and the outgoing null hypersurface ℋ emanating from ∂Σ , 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.