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Polygonal surfaces in pseudo-hyperbolic spaces

Alex Moriani

Abstract

A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of them. Polygonal surfaces are characterized by finiteness of their total curvature and by asymptotic flatness. They have parabolic type and polynomial quartic differential. Our result relies on a comparison between three ideal boundaries associated with a maximal surface, corresponding to three distinct distances naturally defined on the maximal surface.

Polygonal surfaces in pseudo-hyperbolic spaces

Abstract

A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of them. Polygonal surfaces are characterized by finiteness of their total curvature and by asymptotic flatness. They have parabolic type and polynomial quartic differential. Our result relies on a comparison between three ideal boundaries associated with a maximal surface, corresponding to three distinct distances naturally defined on the maximal surface.
Paper Structure (84 sections, 67 theorems, 148 equations, 7 figures)

This paper contains 84 sections, 67 theorems, 148 equations, 7 figures.

Table of Contents

  1. Introduction
  2. General context
  3. Main theorem
  4. Related works
  5. Minimal surfaces in euclidean spaces
  6. Minimal surfaces in $\mathbf{H}^2\times\mathbb{R}$
  7. CMC cuts in Minkowski space
  8. $\mathrm{SL}_3(\mathbb{R})$ and affine spheres in $\mathbb{R}^3$
  9. $\mathrm{SL}_2(\mathbb{R})\times\mathrm{SL}_2(\mathbb{R})$ and lightlike polygons in $\mathbf{Ein}^{1,{1}}$
  10. $\mathrm{Sp}(4,\mathbb{R})$ and lightlike polygons in $\mathbf{Ein}^{1,{2}}$
  11. $G_2$ and almost complex curves in $\mathbf{S}^{2,4}$
  12. Quasiperiodic surfaces in $\mathbf{H}^{2,{n}}$
  13. Overview of the proof
  14. The ideal boundary
  15. The space boundary
  16. ...and 69 more sections

Key Result

Theorem 1.2.1

Let $\Sigma$ be a maximal surface in $\mathbf{H}^{2,{n}}$, and denote by $g$ its induced metric. The following are equivalent: Moreover, if the conditions are satisfied, and denoting by $N+4$ the number of vertices of $\partial^\mathrm{s}\Sigma$, the Riemann surface $(\Sigma,[g])$ is of parabolic type, has polynomial quartic differential of degree $N$ and the total curvature of $\Sigma$ equals $-

Figures (7)

  • Figure 1: Illustration of the behavior of geodesics in a Barbot surface with initial velocity between two singular directions (in red) at two different points.
  • Figure 2: Space-horoball (in gray) in a Barbot surface, associated to a vertex, with the regular direction $\xi$.
  • Figure 3: A semi-positive loop $\Lambda$ with a Barbot crown coinciding with near a vertex.
  • Figure 4: Shape of space-horoballs in a maximal surface. On the left the situation where $p$ is a positive point, on the right the situation where $p$ is a vertex.
  • Figure 5: Construction of a regular geodesic ray emanating from $x$.
  • ...and 2 more figures

Theorems & Definitions (146)

  • Theorem 1.2.1: Main Theorem
  • Theorem : Choi--Treibergs, Han--Tam--Treibergs--Wan
  • Corollary 1.3.1
  • Remark 1.3.2
  • Theorem 1.4.1
  • Remark 1.4.2
  • Theorem 1.4.3
  • Theorem 1.4.4
  • Theorem 1.4.5
  • Theorem 1.4.6
  • ...and 136 more