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

2025/12/19 by Lorenzo Maniscalco, Maniscalco, Lorenzo, Luciano Mari +1
Mathematics · #Geometric Analysis and Curvature Flows #Advanced Operator Algebra Research #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.2512.17670

Abstract

We study the existence problem for achronal hypersurfaces M \hookrightarrow 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.

Citations

• A sharp gradient estimate and W^2,q regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space
• The electrostatic Born-Infeld equations with integrable charge densities
• Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space
• Hypersurfaces with Light-Like Points in a Lorentzian Manifold II
• Isolated singularities of the prescribed mean curvature equation in\n Minkowski 3-space
• Surfaces with light-like points in Lorentz-Minkowski 3-space with applications
• Geometric analysis of Lorentzian distance function on spacelike hypersurfaces
• The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space ł³
• Maximal surfaces with singularities in Minkowski space

Related

• Existence of Constant Mean Curvature Hypersurfaces in Asymptotically Flat Spacetimes
• The global geometry of surfaces with prescribed mean curvature in\n \ℝ³
• Complete constant mean curvature spacelike hypersurfaces in the\n Einstein-de Sitter spacetime
• Uniformly Area Expanding Flows in Spacetimes
• Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime