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

TL;DR

This paper investigates the existence and regularity of achronal hypersurfaces with prescribed mean curvature in 3D spacetimes, allowing for singular sources and null regions, with applications to electrostatics.

Contribution

It establishes a general existence and regularity theorem for such hypersurfaces in 3D spacetimes, addressing cases with singular mean curvature sources.

Findings

Proves existence and regularity of hypersurfaces in 3D spacetimes

Shows limitations of results in higher dimensions, especially at 5D and above

Applies results to Born-Infeld electrostatics in static spacetimes

Abstract

We study the existence problem for achronal hypersurfaces $M \hookrightarrow \overline{M}$ 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 $4$D-spacetimes is an open problem. Our theorems have application to Born-Infeld electrostatics in general static spacetimes.