Siberian Mathematical Journal

, Volume 48, Issue 3, pp 395–407 | Cite as

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

  • D. A. Berdinsky
  • I. A. Taimanov
Article
Received:

Abstract

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.

Keywords

Heisenberg group surface of revolution isoperimetric problem Willmore functional 
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References

  1. 1.
    Berdinsky D. A. and Taimanov I. A., “Surfaces in three-dimensional Lie groups,” Siberian Math. J., 46, No. 6, 1005–1019 (2005).CrossRefMathSciNetGoogle Scholar
  2. 2.
    Taimanov I. A., “Modified Novikov-Veselov equation and differential geometry of surfaces,” Amer. Math. Soc. Transl., 179, No. 2, 133–151 (1997).MathSciNetGoogle Scholar
  3. 3.
    Taimanov I. A., “Two-dimensional Dirac operator and surface theory,” Russian Math. Surveys, 61, No. 1, 79–159 (2006).CrossRefMathSciNetGoogle Scholar
  4. 4.
    Abresch U. and Rosenberg H., “Generalized Hopf differentials,” Mat. Contemp., 28, 1–28 (2005).MATHMathSciNetGoogle Scholar
  5. 5.
    Figueroa C., Mercuri F., and Pedrosa R., “Invariant surfaces of the Heisenberg groups,” Ann. Math. Pura Appl., 177, 173–194 (1999).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Abresch U. and Rosenberg H., “The Hopf differential for constant mean curvature surfaces in S 2 × ℝ and H 2 × ℝ,” Acta Math., 193, 141–174 (2004).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fernández I. and Mira P., “A characterization of constant mean curvature surfaces in homogeneous 3-manifolds.” Available at http://arxiv.org (see math.DG/0512280).Google Scholar
  8. 8.
    Morgan F., “Regularity of isoperimetric hypersurfaces in Riemannian manifolds,” Trans. Amer. Math. Soc., 355, 5041–5052 (2003).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Schmidt E., “Der Brunn-Minkowskische Satz und sein Spiegeltheorem sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und hyperbolischen Geometrie,” Math. Ann., 120, 307–429 (1949).CrossRefGoogle Scholar
  10. 10.
    Tomter P., “Constant mean curvature surfaces in the Heisenberg group,” in: Proceedings of Symposia in Pure Mathematics, 1993, 53, No. 1, pp. 485–495.MathSciNetGoogle Scholar
  11. 11.
    Morgan F. and Johnson D., “Some sharp isoperimetric theorems for Riemannian manifolds,” Indiana Univ. Math. J., 49, 1017–1041 (2000).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pedrosa R. H. L., “The isoperimetric problem in spherical cylinders,” Ann. Global Anal. Geom., 26, 333–354 (2004).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • D. A. Berdinsky
    • 1
  • I. A. Taimanov
    • 1
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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