Hostname: page-component-5d84bcc8dc-wwlbm Total loading time: 0 Render date: 2026-08-12T15:56:29.958Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal surfaces in the Lorentzian Heisenberg group

Part of: Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  29 July 2025

DAVID BRANDER  and
SHIMPEI KOBAYASHI
DAVID BRANDER
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Building 324, DK-2800, Kgs. Lyngby, Denmark e-mail: dbra@dtu.dk
SHIMPEI KOBAYASHI
Affiliation:
Department of Mathematics, Hokkaido University, Nishi 8-Chome Kita 10-Jou Kita-Ku, Sapporo 060-0810, Japan e-mail: shimpei@math.sci.hokudai.ac.jp
Get access Rights & Permissions [Opens in a new window]

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 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.

MSC classification

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53C43: Differential geometric aspects of harmonic maps

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 179 , Issue 3 , November 2025 , pp. 579 - 608
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

*

Partially supported by Independent Research Fund Denmark, grant 9040-00196B.

**

Partially supported by Kakenhi 22K03265.

References

Berdinskii, D. and Taimanov, I.. Surfaces in three-dimensional Lie groups. Siberian Math. J. 46 (2005), 1005–1019.CrossRefGoogle Scholar
Bobenko, A. I.. Surfaces in terms of 2 by 2 matrices. Old and new integrable cases. In Harmonic Maps and Integrable Systems. Aspects Math. E23 (Vieweg, 1994), pp. 83–127.CrossRefGoogle Scholar
Brander, D.. Spherical surfaces. Experiment. Math. 25(3) (2016), 257–272.CrossRefGoogle Scholar
Brander, D. and Dorfmeister, J. F.. Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) (2015), XIV:1–31.Google Scholar
Cartier, S.. Surfaces des espaces homogènes de dimension 3. PhD. thesis. Université Paris-Est (2011).Google Scholar
Cintra, A., Mercuri, F. and Onnis, I.. Minimal surfaces in Lorentzian Heisenberg group and Damek-Ricci spaces via the Weierstrass representation. J. Geom. Physics 121 (2017), 396–412.CrossRefGoogle Scholar
Daniel, B.. The Gauss map of minimal surfaces in the Heisenberg group. Int. Math. Res. Not. 2011(3) (2011), 674–695.Google Scholar
Dorfmeister, J., Inoguchi, J. and Kobayashi, S.-P.. A loop group method for minimal surfaces in the three-dimensional Heisenberg group. Asian J. Math. 20 (2016), 409–448.CrossRefGoogle Scholar
Dorfmeister, J., Pedit, F. and Wu, H.. Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6 (1998), 633–668.CrossRefGoogle Scholar
Figueroa, C.. Weierstrass formula for minimal surfaces in Heisenberg group. Pro Mathematica 13 (1999), 71–85.Google Scholar
Figueroa, C.. On the Gauss map of a minimal surface in the Heisenberg group. Mat. Contemp. 33 (2007), 139–156.Google Scholar
Fujimori, S., Saji, K., Umehara, M. and Yamada, K.. Singularities of maximal surfaces. Math. Z. 259 (2008), 827–848.CrossRefGoogle Scholar
Hoffman, D. A., Osserman, R. and Schoen, R.. On the Gauss map of complete surfaces of constant mean curvature in ${\bf R}^3$ and ${\bf R}^4$ . Comment. Math. Helv. 57 (1982), 519–531.CrossRefGoogle Scholar
Ishikawa, G.. Singularities of frontals. Adv. Stud. Pure Math. 78 (2018), 55–106.CrossRefGoogle Scholar
Kenmotsu, K.. Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann. 245(2) (1979), 89–99.CrossRefGoogle Scholar
Kiyohara, H. and Kobayashi, S.. Timelike minimal surfaces in the three-dimensional Heisenberg group. J. Geom. Anal. 32(8):Paper No. 225, 2022.CrossRefGoogle Scholar
Kokubu, M., Rossman, W., Saji, K., Umehara, M. and Yamada, K.. Singularities of flat fronts in hyperbolic space. Pacific J. Math. 221 (2005), 303–351.CrossRefGoogle Scholar
Lee, H.. Maximal surfaces in Lorentzian Heisenberg space. Differential Geom. Appl. 29 (2011), 73–84.CrossRefGoogle Scholar
Lira, J. H., Melo, M. and Mercuri, F.. A Weierstrass representation for minimal surfaces in 3-dimensional manifolds. Results. Math. 60 (2011), 311–323.CrossRefGoogle Scholar
Milnor, J.. Curvatures of left invariant metrics on Lie groups. Adv. Math. 21 (1976), 293–329.CrossRefGoogle Scholar
Rahmani, N. and Rahmani, S.. Lorentzian geometry of the Heisenberg group. Geom. Dedicata 118 (2006), 133–140.CrossRefGoogle Scholar
Rahmani, S.. Métriques de Lorentz sur les groupes de Lie unimodulaires, de dimension trois. J. Geom. Phys. 9 (1992), 295–302.CrossRefGoogle Scholar
Umehara, M. and Yamada, K.. Maximal surfaces with singularities in minkowski space. Hokkaido Math. J. 35 (2006), 13–40.CrossRefGoogle Scholar
Wu, H.. A simple way for determining the entials for harmonic maps. Ann. Global Anal. Geom. 17(2) (1999), 189–199.CrossRefGoogle Scholar