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Maximal surfaces in the Lorentzian Heisenberg group

David Brander, Shimpei Kobayashi

TL;DR

This work develops a DPW loop-group framework for spacelike maximal surfaces in the Lorentzian Heisenberg group $Nil^3_1$, linking them to nowhere-holomorphic harmonic maps into $S^2$ and to a spinor Dirac formalism. It provides precise singularity criteria in terms of the Gauss map $g$ and the Abresch-Rosenberg differential $B dz^2$, and then shows how to realize and classify cuspidal edges, swallowtails, and cuspidal cross-caps through DPW potentials and Cauchy data. A simplified normalization yields concrete, testable conditions on $B$ for degeneracy and for the three singularity types, while the DPW construction offers a practical route to generate and control singularities. The paper also connects maximal surfaces to associated CMC surfaces in $\,R^3$ via the equatorial-curve construction and proves an application: a regular spacelike maximal disc with null boundary must have at least two cuspidal cross-cap singularities on its boundary, highlighting geometric constraints on boundary behavior.

Abstract

The 3-dimensional Heisenberg group can be equipped with three different types of left-invariant Lorentzian metric, according to whether the center of the Lie algebra is spacelike, timelike or null. Using the second of these types, we study spacelike surfaces of mean curvature zero. These surfaces with singularities are associated with harmonic maps into the 2-sphere. We show that the generic singularities are cuspidal edge, swallowtail and cuspidal cross-cap. We also give the loop group construction for these surfaces, and the criteria on the loop group potentials for the different generic singularities. Lastly, we solve the Cauchy problem for harmonic maps into the 2-sphere using loop groups, and use this to give a geometric characterization of the singularities. We use these results to prove that a regular spacelike maximal disc with null oundary must have at least two cuspidal cross-cap singularities on the boundary.

Maximal surfaces in the Lorentzian Heisenberg group

TL;DR

This work develops a DPW loop-group framework for spacelike maximal surfaces in the Lorentzian Heisenberg group , linking them to nowhere-holomorphic harmonic maps into and to a spinor Dirac formalism. It provides precise singularity criteria in terms of the Gauss map and the Abresch-Rosenberg differential , and then shows how to realize and classify cuspidal edges, swallowtails, and cuspidal cross-caps through DPW potentials and Cauchy data. A simplified normalization yields concrete, testable conditions on for degeneracy and for the three singularity types, while the DPW construction offers a practical route to generate and control singularities. The paper also connects maximal surfaces to associated CMC surfaces in via the equatorial-curve construction and proves an application: a regular spacelike maximal disc with null boundary must have at least two cuspidal cross-cap singularities on its boundary, highlighting geometric constraints on boundary behavior.

Abstract

The 3-dimensional Heisenberg group can be equipped with three different types of left-invariant Lorentzian metric, according to whether the center of the Lie algebra is spacelike, timelike or null. Using the second of these types, we study spacelike surfaces of mean curvature zero. These surfaces with singularities are associated with harmonic maps into the 2-sphere. We show that the generic singularities are cuspidal edge, swallowtail and cuspidal cross-cap. We also give the loop group construction for these surfaces, and the criteria on the loop group potentials for the different generic singularities. Lastly, we solve the Cauchy problem for harmonic maps into the 2-sphere using loop groups, and use this to give a geometric characterization of the singularities. We use these results to prove that a regular spacelike maximal disc with null oundary must have at least two cuspidal cross-cap singularities on the boundary.
Paper Structure (24 sections, 15 theorems, 121 equations, 6 figures)

This paper contains 24 sections, 15 theorems, 121 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Surfaces with harmonic Gauss maps
  4. Surfaces with singularities
  5. Outline of this article
  6. Left invariant metrics on the Heisenberg group
  7. Left invariant Riemannian metrics
  8. Left invariant Lorentzian metrics
  9. Maximal spacelike surfaces in Nil3
  10. Spinor representation for spacelike conformal immersions in Nil3
  11. Generalized maximal surfaces
  12. Numeric Examples
  13. Generic Singularities for Maximal Surfaces in Nil3
  14. Characterization of Standard Singularities
  15. A simplified characterization
  16. ...and 9 more sections

Key Result

Theorem 3.1

The pair of generating spinors $\{\psi_1, \psi_2\}$ satisfies the following nonlinear Dirac equation, where Here $H$, $e^{u}$ and $h$ are the mean curvature, the conformal factor and the support function for $f$ respectively. Moreover, together with the second fundamental form, the vector $\tilde{\psi} = (\psi_1, \psi_2)$ satisfies the following system where Here the function $e^{w/2}$ is the

Figures (6)

  • Figure 1: Top: Portions of a single constant mean curvature surface of revolution, with three different spatial orientations. Bottom: the corresponding maximal surfaces in ${\hbox{Nil}^3_1}$, in the same order. (Example \ref{['example0']}.)
  • Figure 2: The solutions of Example \ref{['example1']}. Left: harmonic maps; middle: corresponding maximal surface in ${\hbox{Nil}^3_1}$; right: corresponding CMC surfaces in Euclidean space;
  • Figure 3: Maximal surfaces with rotational symmetry (Example \ref{['example2']}), computed from normalized potentials with the displayed functions $(a(z),b(z))$.
  • Figure 4: Top: CMC $1/2$ surfaces in ${\mathbb E}^3$. Bottom: the corresponding maximal surfaces in ${\hbox{Nil}^3_1}$, with, in order, cuspidal edge, swallowtail, and cuspidal cross-cap singularities. (Example \ref{['singexample']}).
  • Figure 5: Solutions to the geometric Cauchy problem using Theorem \ref{['singcauchythm2']}. See Example \ref{['examplegcp']}. Top: swallowtail. Bottom: cuspidal cross-cap.
  • ...and 1 more figures

Theorems & Definitions (39)

  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • proof
  • Definition 3.7: Generalized spacelike maximal surfaces
  • Remark 3.8
  • Example 3.9
  • ...and 29 more