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

Lorenzo Maniscalco, Luciano Mari

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.

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

Abstract

We study the existence problem for achronal hypersurfaces in a globally hyperbolic spacetime, whose mean curvature is a prescribed -- possibly singular -- source, and whose boundary is a given smooth spacelike submanifold. Since 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 . Although most of our estimates hold in any dimension, recent counterexamples show that some of our conclusions fail in ambient dimension at least . The case of D-spacetimes is an open problem. Our theorems have application to Born-Infeld electrostatics in general static spacetimes.

Paper Structure

This paper contains 15 sections, 19 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Consequences of hypothesis \ref{['cauchy compatto']}, and approximation
  4. Some consequences of \ref{['cauchy compatto']}
  5. Approximating the mean curvature
  6. Outline of the proof of Theorem \ref{['main result']}
  7. Main estimates
  8. Energy bound (tilt function estimate)
  9. Removable singularities
  10. Second fundamental form estimate
  11. Higher integrability
  12. Proof of the Theorem \ref{['main result']}
  13. Local uniform integrability of $\{w_j\}$ in $\mathring\Sigma$
  14. Application to the Born-Infeld theory
  15. Proof of Proposition \ref{['prop_appendix_sec']}

Key Result

Theorem 1.6

Let $\,\overline{\!{M}}$ be a globally hyperbolic spacetime of dimension $2+1$ and let $\Sigma$ be a smooth, compact spacelike hypersurface satisfying cauchy compatto. Choose a splitting time function $\tau$ and let $\varphi \in C^\infty(\Sigma)$ be the height function of $\Sigma$. Consider a pair $ is valued and bounded in $L^2(\Sigma')$. Then, the Dirichlet problem has a weak solution $u\in\mat

Theorems & Definitions (47)

  • Remark 1.1
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 37 more