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arXiv · 2512.17670

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

Lorenzo Maniscalco·Luciano Mari
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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.

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Lorenzo Maniscalco, Luciano Mari. 2025-12-19. Prescribing the mean curvature of an achronal hypersurface as a measure: the case of 3D spacetimes. https://arxiv.org/abs/2512.17670

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