Acta Mathematica Sinica

, Volume 22, Issue 6, pp 1603–1612 | Cite as

A Weierstrass Representation Formula for Minimal Surfaces in ℍ3 and ℍ2 × ℝ

ORIGINAL ARTICLES

Abstract

We give a general setting for constructing a Weierstrass representation formula for simply connected minimal surfaces in a Riemannian manifold. Then, we construct examples of minimal surfaces in the three dimensional Heisenberg group and in the product of the hyperbolic plane with the real line.

Keywords

Minimal immersions Weierstrass representation Heisenberg group Hyperbolic space 

MR (2000) Subject Classification

53C41 53A10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barbosa, J. L. M., Colares, A. G.: Minimal surfaces in ℝ3, Lecture Notes in Mathematics, 1195, Springer-Verlag, Berlin, 1986Google Scholar
  2. 2.
    Gray, A.: Modern differential geometry of curves and surfaces with Mathematica, Second edition, CRC Press, Boca Raton, FL, 1998Google Scholar
  3. 3.
    Bekkar, M.: Exemples de surfaces minimales dans l’espace de Heisenberg. Rend. Sem. Fac. Sci. Univ. Cagliari, 61, 123–130 (1991)MATHMathSciNetGoogle Scholar
  4. 4.
    Bekkar, M., Sari, T.: Surfaces minimales réglées dans l’espace de Heisenberg H 3. Rend. Sem. Mat. Univ. Politec. Torino, 50, 243–254 (1992)MATHMathSciNetGoogle Scholar
  5. 5.
    Caddeo, R., Piu, P., Ratto, A.: SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces. Manuscripta Math., 87, 1–12 (1995)MATHMathSciNetGoogle Scholar
  6. 6.
    Figueroa, C. B., Mercuri, F., Pedrosa, R. H. L.: Invariant surfaces of the Heisenberg groups. Ann. Mat. Pura Appl., 177(4), 173–194 (1999)MATHMathSciNetGoogle Scholar
  7. 7.
    Piu, P., Sanini, A.: One-parameter subgroups and minimal surfaces in the Heisenberg group. Note Mat., 18, 143–153 (1998)MATHMathSciNetGoogle Scholar
  8. 8.
    Scott, D. P.: Minimal surfaces in the Heisenberg group, preprintGoogle Scholar
  9. 9.
    Nelli, B., Rosenberg, H.: Minimal surfaces in ℍ2 × ℝ. Bull. Braz. Math. Soc. (N.S.), 33, 263–292 (2002)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kokubu, M.: Weierstrass representation for minimal surfaces in hyperbolic space. Tohoku Math. J., 49, 367–377 (1997)MATHMathSciNetGoogle Scholar
  11. 11.
    Koszul, J. L., Malgrange, B.: Sur certaines fibrées complexes. Arch. Math., 9, 102–109 (1958)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Eells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86, 109–160 (1964)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Eells, J., Lemaire, L.: Selected topics in harmonic maps, AMS regional conference series in mathematics, 50, 1983Google Scholar
  14. 14.
    Inoguchi, J., Kumamoto, T., Ohsugi, N., Suyama, Y.: Differential geometry of curves and surfaces in 3-dimensional homogeneous spaces I and II. Fukuoka Univ. Sci. Rep., 29 and 30, (1999 and 2000)Google Scholar
  15. 15.
    Figueroa, C. B.: Geometria das subvariedades do grupo de Heisenberg, Ph.d Thesis, CampinasGoogle Scholar
  16. 16.
    Rosenberg, H.: Minimal surfaces in \( \Bbb M ^{2} \times \Bbb R \). Illinois J. Math., 46, 1177–1195 (2002)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Francesco Mercuri
    • 1
  • Stefano Montaldo
    • 2
  • Paola Piu
    • 2
  1. 1.Departamento de MatematicaC.P. 6065, IMECC, UNICAMPCampinas, SPBrazil
  2. 2.Università degli Studi di CagliariDipartimento di MatematicaCagliari

Personalised recommendations