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Foliation of null cones by surfaces of constant spacetime mean curvature near MOTS

Ben Lambert, Julian Scheuer

Abstract

Marginally Outer Trapped Surfaces (MOTS) in spacetimes are well-known to indicate the existence of black holes. Using flow techniques, we prove that a neighbourhood of a stable MOTS in a null cone may be foliated by hypersurfaces of constant spacetime mean curvature. We also provide methods to construct prescribed spacetime mean curvature surfaces within null cones.

Foliation of null cones by surfaces of constant spacetime mean curvature near MOTS

Abstract

Marginally Outer Trapped Surfaces (MOTS) in spacetimes are well-known to indicate the existence of black holes. Using flow techniques, we prove that a neighbourhood of a stable MOTS in a null cone may be foliated by hypersurfaces of constant spacetime mean curvature. We also provide methods to construct prescribed spacetime mean curvature surfaces within null cones.
Paper Structure (7 sections, 24 theorems, 182 equations)

This paper contains 7 sections, 24 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Basic notions of null geometry
  3. Evolution equations
  4. Evolution equations for general speeds
  5. Evolution equations for PMCF
  6. Spacetime CMC foliations near a MOTS and proof of \ref{['thm:FoliationExistence']}
  7. The prescribed mean curvature problem and proof of \ref{['thm:PrescribedMeanCurvature']}

Key Result

Theorem 1.1

Let $n\geq 2$ and $\bar{N}$ be a null cone on a stable MOTS $\Sigma_{0}\subset \bar{N}$ in a spacetime $(\bar{M}^{n+2},\bar{g})$. Define Then the following statements hold.

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark
  • Remark
  • Theorem 1.2
  • Remark
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • ...and 50 more