Skip to main content
SpringerLink
Log in

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S 4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them the interesting duality theorem holds. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.

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. Bryant R. A duality theorem for Willmore surfaces. J Diff Geom, 20: 23–53 (1984)

    MATH  Google Scholar 

  2. Burstall F, Ferus D, Leschke K, et al. Conformal geometry of surfaces in S 4 and quaternions. In: Lecture Notes in Mathematics 1772. Berlin: Springer, 2002

    MATH  Google Scholar 

  3. Ejiri N. Willmore surfaces with a duality in S n(1). Proc London Math Soc (3), 57(2): 383–416 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Montiel S. Willmore two spheres in the four-sphere. Trans Amer Math Soc, 352(10): 4469–4486 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Musso E. Willmore surfaces in the four-sphere. Ann Global Anal Geom, 8(1): 21–41 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Alias L J, Palmer B. Conformal geometry of surfaces in Lorentzian space forms. Geometriae Dedicata, 60: 301–315 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Deng Y J, Wang C P. Time-like Willmore surfaces in Lorentzian 3-space. Sci China Ser A-Math, 49(1): 75–85 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Nie C X. Conformal geometry of hypersurfaces and surfaces in Lorentzian space forms. Dissertation, Peking University, 2006

  9. Ma X. Adjoint transforms of Willmore surfaces in S n. Manuscripta Math, 120: 163–179 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ma X. Isothermic and S-Willmore surfaces as solutions to a problem of Blaschke. Results in Math, 48: 301–309 (2005)

    MATH  Google Scholar 

  11. Cahen M, Kerbrat Y. Domaines symetriques des quacriques projectives. J Math Pureset Appl, 62: 327–348 (1983)

    MATH  MathSciNet  Google Scholar 

  12. Burstall F, Pedit F, Pinkall U. Schwarzian derivatives and flows of surfaces. Contemporary Mathematics 308, 39–61. Providence, RI: Amer Math Soc, 2002

    Google Scholar 

  13. Wang C P. Moebious geometry of submanifolds in S n. Manuscripta Math, 96(4): 517–534 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ma X. Willmore surfaces in S n: transforms and vanishing theorems. dissertation, Technischen Universität Berlin, 2005. http://edocs.tu-berlin.de/diss/2005/ma_xiang.pdf

  15. Ejiri N. Willmore surfaces with a duality in S n(1). Proc London Math Soc (3), 57(2): 383–416 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Leitner F. Twistorial construction of spacelike surfaces in Lorentzian 4-manifolds. Geometry and Topology of Submanifolds X. Singapore: World Scientific, 1999, 113–135

    Google Scholar 

  17. Lawson H B. Complete minimal surfaces in S 3. Ann of Math, 92(2): 335–374 (1970)

    Article  MathSciNet  Google Scholar 

  18. Chern S S. On the minimal immersions of the two-sphere in a space of constant curvature. Problems in Analysis, 27–40. Princeton, NJ: Princeton Univ Press, 1970

    Google Scholar 

  19. Song Y P, Wang C P. Classification of Laguerre minimal surfaces. Chin Ann Math Ser B (to appear)

  20. Ma X, Wang P. Spacelike isothermic surfaces in 4-dimensional Lorentzian space forms. Results Math (to appear)

  21. Dusssan M P. Spacelike isothermic surfaces and Grassmannian Systems. Tsukuba J Math, 30(1): 81–102 (2006)

    MathSciNet  Google Scholar 

  22. Zuo D, Chen Q, Cheng Y. Darboux Transformations for Space-like isothermic surfaces in ℝm,1. Commun Theor Phys, 41: 816–820 (2004)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Xiang Ma & Peng Wang

Authors
  1. Xiang Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiang Ma.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10771005)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ma, X., Wang, P. Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms. Sci. China Ser. A-Math. 51, 1561–1576 (2008). https://doi.org/10.1007/s11425-008-0052-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-008-0052-9

Keywords

MSC(2000)

Search

Navigation