Mathematische Zeitschrift

, Volume 271, Issue 1–2, pp 257–270 | Cite as

On Hamiltonian stationary Lagrangian spheres in non-Einstein Kähler surfaces

  • Ildefonso Castro
  • Francisco Torralbo
  • Francisco Urbano
Article

Abstract

Hamiltonian stationary Lagrangian spheres in Kähler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kähler surfaces given by the product Σ1 × Σ2 of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example, defined when the surfaces Σ1 and Σ2 are spheres, is unstable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abresch U., Rosenberg H.: A Hopf differential for constant mean curvature surfaces in \({\mathbb{S}^2\times\mathbb{R}}\) and \({\mathbb{H}^2\times\mathbb{R}}\). Acta Math. 193, 141–174 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Castro I., Chen B.-Y.: Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves. Tohoku Math. J. 58, 565–579 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Castro I., Urbano F.: Examples of unstable Hamiltonian-minimal Lagrangian tori in \({\mathbb{C}^2}\). Compositio Math. 111, 1–14 (1998)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Castro I., Urbano F.: Minimal Lagrangian surfaces in \({\mathbb{S}^2\times\mathbb{S}^2}\). Comm. Anal. Geom. 15, 217–248 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Eschenburg J.-H., Guadalupe I.V., Tribuzy R.A.: The fundamental equations of minimal surfaces in \({\mathbb{C}\mathbb{P}^2}\). Math. Ann. 270, 571–598 (1985)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Harvey R., Lawson H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hélein F., Romon P.: Hamiltonian stationary Lagrangian surfaces in \({\mathbb{C}^2}\). Comm. Anal. Geom. 10, 79–126 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Hélein F., Romon P.: Hamiltonian stationary tori in complex projective plane. Proc. Lond. Math. Soc. 90, 472–496 (2005)MATHCrossRefGoogle Scholar
  9. 9.
    Hoffman, D.A., Osserman, R.: The geometry of the generalized Gauss map. Mem. Am. Math. Soc. 236 (1980)Google Scholar
  10. 10.
    Lawlor G.: The angle criterion. Invent. Math. 95, 437–446 (1989)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Li P., Yau S.T.: A new conformal invariant and its applications to the Willmore conjecture and firt eigenvalue of compact surfaces. Invent. Math. 69, 269–291 (1982)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Oh Y.G.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212, 175–192 (1993)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Schoen R., Wolfson J.G.: Minimizing area among Lagrangian surfaces: the mapping problem. J. Differ. Geom. 58, 1–86 (2001)MathSciNetMATHGoogle Scholar
  14. 14.
    Simon L.: Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1, 281–326 (1993)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Ildefonso Castro
    • 1
  • Francisco Torralbo
    • 2
  • Francisco Urbano
    • 2
  1. 1.Departamento de MatemáticasUniversidad de JaénJaénSpain
  2. 2.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain

Personalised recommendations