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Complete CMC hypersurfaces in Minkowski (n+1)-space

Francesco Bonsante, Andrea Seppi, Peter Smillie

Abstract

We prove that any regular domain in Minkowski space is uniquely foliated by spacelike constant mean curvature (CMC) hypersurfaces. This completes the classification of entire spacelike CMC hypersurfaces in Minkowski space initiated by Choi and Treibergs. As an application, we prove that any entire surface of constant Gaussian curvature in 2+1 dimensions is isometric to a straight convex domain in the hyperbolic plane.

Complete CMC hypersurfaces in Minkowski (n+1)-space

Abstract

We prove that any regular domain in Minkowski space is uniquely foliated by spacelike constant mean curvature (CMC) hypersurfaces. This completes the classification of entire spacelike CMC hypersurfaces in Minkowski space initiated by Choi and Treibergs. As an application, we prove that any entire surface of constant Gaussian curvature in 2+1 dimensions is isometric to a straight convex domain in the hyperbolic plane.

Paper Structure

This paper contains 16 sections, 43 theorems, 50 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. Causality and Entire hypersurfaces
  3. Domains of dependence and regular domains
  4. CMC hypersurfaces
  5. Asymptotics
  6. Uniqueness
  7. Existence
  8. Barrier argument
  9. Outline of the proof of Proposition \ref{['pr:extec']}
  10. CMC foliations
  11. Continuous foliations
  12. Analyticity of the foliation
  13. Application in dimension 2 + 1
  14. Hyperbolic surfaces and correspondence by the normal flow
  15. Minimal Lagrangian maps
  16. ...and 1 more sections

Key Result

Theorem 1

Given any regular domain $\mathcal{D}$ in ${\mathbb R}^{n,1}$ and any $H>0$, there exists a unique entire hypersurface $\Sigma\subset{\mathbb R}^{n,1}$ of constant mean curvature $H$ such that the domain of dependence of $\Sigma$ is $\mathcal{D}$. Moreover, as $H$ varies in $(0,+\infty)$, the entire

Figures (3)

  • Figure 1: The two dimensional trough $T$, whose domain of dependence is a wedge.
  • Figure 2: The relative positions of $p_0$ and $q_0$.
  • Figure 3: An isometric boost between two spacelike hyperplanes.

Theorems & Definitions (78)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Proposition 1.1: bon_smillie_seppi
  • Proposition 1.2: Bonsante
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • ...and 68 more