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

We study the existence problem for achronal hypersurfaces $M \hookrightarrow \overline{M}$ 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.

💡 Research Summary

The paper addresses the problem of constructing achronal hypersurfaces with prescribed mean curvature in a globally hyperbolic spacetime, allowing the prescribed curvature to be a possibly singular Radon measure. While classical results (Bartnik, Simon, Gerhardt) required the mean curvature function to be at least continuous and bounded, this work relaxes those conditions dramatically, motivated by applications such as the Born‑Infeld model of nonlinear electrostatics where charge distributions are naturally modeled by measures (including Dirac deltas).

The authors focus on 2 + 1 dimensional spacetimes (three‑dimensional manifolds with Lorentzian metric). They fix a smooth compact spacelike boundary Σ and consider the class Y(Σ) of achronal hypersurfaces M that are graphs of Lipschitz functions u over Σ, satisfying the weak spacelike condition |Du|≤1 almost everywhere. By choosing a global time function τ, the spacetime is written as M = ℝ × S with metric (\bar g = \bar\alpha^2(-d\tau^2 + \bar\sigma)). The graph map (F_u(x) = (u(x),\pi(x))) embeds Σ into M, and the induced metric on Σ depends on u.

The prescribed mean curvature is encoded via a pair (ρ, X). Here ρ is a map from the function space Y(Σ) to the space of finite signed Radon measures on Σ, continuous with respect to the C⁰‑topology; X is a continuous vector field on the domain D(Σ). The weak formulation of the prescribed‑mean‑curvature (PMC) equation reads \