open access

https://hdl.handle.net/2115/95856

Maximal surfaces in the Lorentzian Heisenberg group

英語

2025

Cambridge University Press

Mathematical Proceedings of the Cambridge Philosophical Society

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 characterisation of the singularities. We use these results to prove that a regular spacelike maximal disc with null boundary must have at least two cuspidal cross-cap singularities on the boundary.

学術雑誌論文

Accepted Manuscript

This article has been published in a revised form in Mathematical Proceedings of the Cambridge Philosophical Society https://doi.org/10.1017/S0305004125101412. This version is free to view and download for private research and study only. Not for re-distribution or re-use. ©2025 copyright holder.

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