Weak null singularities in general relativity

We construct a class of spacetimes (without symmetry assumptions) satisfying the vacuum Einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2-sphere. The metric of these spacetimes extends continuously beyond the singularities while the Christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries. The construction shows moreover that the singularities are stable in a suitable sense. These singularities are stronger than the impulsive gravitational spacetimes considered by Luk and Rodnianski, and conjecturally they are present in the interior of generic black holes arising from gravitational collapse.