Chinese Annals of Mathematics, Series B

, Volume 28, Issue 3, pp 299–310

Conformal CMC-Surfaces in Lorentzian Space Forms*



Let ℚ3 be the common conformal compactification space of the Lorentzian space forms \( \mathbb{R}^{3}_{1} ,\mathbb{S}^{3}_{1} \;{\text{and}}\;\mathbb{H}^{3}_{1} \). We study the conformal geometry of space-like surfaces in ℚ3. It is shown that any conformal CMC-surface in ℚ3 must be conformally equivalent to a constant mean curvature surface in \( \mathbb{R}^{3}_{1} ,\mathbb{S}^{3}_{1} \;{\text{and}}\;\mathbb{H}^{3}_{1} \). We also show that if x : M → ℚ3 is a space-like Willmore surface whose conformal metric g has constant curvature K, then either K = −1 and x is conformally equivalent to a minimal surface in \( \mathbb{R}^{3}_{1} \), or K = 0 and x is conformally equivalent to the surface \( \mathbb{H}^{1} {\left( {\frac{1} {{{\sqrt 2 }}}} \right)} \times \mathbb{H}^{1} {\left( {\frac{1} {{{\sqrt 2 }}}} \right)}\;{\text{in}}\;\mathbb{H}^{3}_{1} . \)


Conformal geometry Willmore surfaces Lorentzian space 

2000 MR Subject Classification

53A30 53B30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alias, L. J. and Palmer, B., Conformal geometry of surfaces in Lorentzian space forms, Geometriae Dedicata, 60, 1996, 301–315MATHMathSciNetGoogle Scholar
  2. 2.
    Babich, M. and Bobenko, A., Willmore tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. J., 72, 1993, 151–185MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blaschke, W., Vorlesungen ¨uber Differentialgeometrie, Vol. 3, Springer, Berlin, 1929Google Scholar
  4. 4.
    Bryant, R. L., A duality theorem for Willmore surfaces, J. Diff. Geom., 20, 1984, 23–53MATHMathSciNetGoogle Scholar
  5. 5.
    Cahen, M. and Kerbrat, Y., Domaines symetriques des quadriques projectives, J. Math. Pures et Appl., 62, 1983, 327–348MATHMathSciNetGoogle Scholar
  6. 6.
    Deng, Y. J. and Wang, C. P., Willmore surfaces in Lorentzian space, Sci. China Ser. A, 35, 2005, 1361–1372Google Scholar
  7. 7.
    Hertrich-Jeromin, U., Introduction to M¨obius Differential Geometry, London Math. Soc. Lecture Note Series, Vol. 300, Cambridge University Press, Cambridge, 2003Google Scholar
  8. 8.
    Hertrich-Jeromin, U. and Pinkall, U., Ein Beweis der Willmoreschen Vermutung f¨ur Kanaltori, J. Reihe Angew. Math., 430, 1992, 21–34MATHMathSciNetGoogle Scholar
  9. 9.
    Ma, X. and Wang, C. P., Willmore surfaces of constant Moebius curvature, Annals of Global Analysis and Geometry, to appearGoogle Scholar
  10. 10.
    O'Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, New York, 1983Google Scholar
  11. 11.
    Pinkall, U., Hopf tori in S 3, Invent. Math., 81, 1985, 379–386MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pinkall, U. and Sterling, I., Willmore surfaces, Math. Intell., 9, 1987, 38–43MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.Lab of Mathematics and Applied MathematicsSchool of Mathematical Sciences, Peking UniversityBeijingChina

Personalised recommendations