Maximal surfaces in the Lorentzian Heisenberg group

TL;DR

This paper investigates maximal spacelike surfaces with singularities in a Lorentzian version of the Heisenberg group, classifying their singularities, constructing them via loop groups, and analyzing boundary conditions.

Contribution

It introduces a loop group framework for maximal surfaces in the Lorentzian Heisenberg group and characterizes their singularities and boundary behaviors.

Findings

Classified generic singularities as cuspidal edge, swallowtail, and cuspidal cross-cap.

Developed a loop group construction and criteria for singularities.

Proved boundary conditions for maximal discs with null boundary.

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.