Skip to main content
SpringerLink
Log in

Minimal translation surfaces in the Heisenberg group Nil3

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

A translation surface in the Heisenberg group Nil3 is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group Nil3.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abresch U., Rosenberg H.: Generalized Hopf differentials. Mat. Contemp. 28, 1–28 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Berdinskiĭ D.A., Taĭmanov I.A.: Surfaces in three-dimensional Lie groups. Sib. Math. J. 46(6), 1005–1019 (2005)

    Article  Google Scholar 

  3. Daniel B.: The Gauss map of minimal surfaces in the Heisenberg group. Int. Math. Res. Notices 3, 674–695 (2011)

    MathSciNet  Google Scholar 

  4. Daniel, B., Hauswirth, L., Mira, P.: Constant mean curvature surfaces in homogeneous 3-manifolds. Lectures Notes of the 4th KIAS Workshop on Differential Geometry. Constant Mean Curvature Surfaces in Homogeneous Manifolds. Seoul (2009)

  5. Dillen F., Goemans W., Van de Woestyne I.: Translation surfaces of Weingarten type in 3-space. Proc. Conf. RIGA 2008(Bull. Transilv. Univ. Brasov. 15(50), 1–14 (2008)

    Google Scholar 

  6. Figueroa C.B.: On the Gauss map of a minimal surface in the Heisenberg group. Mat. Contemp. 33, 139–156 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Figueroa C.B., Mercuri F., Pedrosa R.H.L.: Invariant Surfaces of the Heisenberg Groups. Annali di Matematica pura ed applicata (IV). 177, 173–194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fernández, I., Mira, P.: Constant mean curvature surfaces in 3-dimensional Thurston geometries. (to appear in Proceedings on the ICM 2010, Hyderabad) arXiv:1004.4752v1 [math.DG] (2010)

  9. Goemans, W., Van de Woestyne, I.: Translation surfaces with vanishing second Gaussian curvature in Euclidean and Minkowski 3-space. In: Dillen, F., Van de Woestyne, I. (eds.) Proceedings of the conference Pure and Applied Differential Geometry, PADGE 2007, 123–131

  10. Inoguchi J.: Flat translation invariant surfaces in the 3-dimensional Heisenberg group. J. Geom. 82, 83–90 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Inoguchi J.: Minimal surfaces in the 3-dimensional Heisenberg group. Diff. Geom. Dyn. Syst. 10, 163–169 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Inoguchi J., Kumamoto T., Ohsugi N., Suyama Y.: Differential geometry of curves and surfaces in 3-dimensional homogeneous spaces, II. Fukuoka Univ. Sci. Reports 30(1), 17–47 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Inoguchi, J., Munteanu, M.I.: Minimal translation surfaces in the hyperbolic 3-space. (in preparation)

  14. López, R., Munteanu, M.I.: Minimal translation surfaces in Sol3 (to appear in J. Math. Soc. Japan)

  15. Meeks III, W.H., Pérez, J.: Constant mean curvature surfaces in metric lie groups. Contemp. Math. (to appear)

  16. Munteanu M.I., Nistor A.I.: On the geometry of the second fundamental form of translation surfaces in \({\mathbb{E}^3}\). Houston J. Math. 37(4), 1087–1102 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Scherk H.F.: Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen. J. Reine Angew. Math. 13, 185–208 (1835)

    Article  MATH  Google Scholar 

  18. Thurston W.: Three-dimensional geometry and topology. Princeton Math. Ser. 35, Princeton University Press, Princeton, NJ (1997)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata, 990-8560, Japan

    Jun-ichi Inoguchi

  2. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Rafael López

  3. Department of Mathematics, Michigan State University, Wells Hall, East Lansing, MI, 48824-1029, USA

    Marian-Ioan Munteanu

  4. Faculty of Mathematics, Al.I. Cuza University of Iasi, Bd. Carol I, n. 11, 700506, Iasi, Romania

    Marian-Ioan Munteanu

Authors
  1. Jun-ichi Inoguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rafael López
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marian-Ioan Munteanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marian-Ioan Munteanu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Inoguchi, Ji., López, R. & Munteanu, MI. Minimal translation surfaces in the Heisenberg group Nil3 . Geom Dedicata 161, 221–231 (2012). https://doi.org/10.1007/s10711-012-9702-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-012-9702-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation