Prescribing the mean curvature of an achronal hypersurface as a measure: the case of 3D spacetimes

We study the existence problem for achronal hypersurfaces M ,→ M in a globally hyperbolic spacetime, whose mean curvature is a prescribed – possibly singular – source, and whose boundary is a given smooth spacelike submanifold. Since M is allowed to go null somewhere, the mean curvature prescription is to be understood in the distributional sense. We prove a general existence and regularity theorem for surfaces in ambient dimension 3. Although most of our estimates hold in any dimension, recent counterexamples show that some of our conclusions fail in ambient dimension at least 5. The case of 4D-spacetimes is an open problem. Our theorems have application to Born-Infeld electrostatics in general static spacetimes.