Geometriae Dedicata

, Volume 60, Issue 3, pp 301–315 | Cite as

Conformal geometry of surfaces in Lorentzian space forms

  • L. J. Alĺas
  • B. Palmer
Article

Abstract

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.

Mathematics Subject Classifications (1991)

53A30 53C50 

Key words

conformal geometry Lorentzian space forms Willmore surfaces Gauss map 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [K]
    Kuiper, N. H.: On conformally-flat spaces in the large, Ann. Math., 50 (1949), 916–924.Google Scholar
  2. [M]
    Morrey, C.: Multiple Integrals in the Calculus of Variations, Springer, Berlin, Heidelberg; New York, 1966.Google Scholar
  3. [P]
    Palmer, B.: Spacelike constant mean curvature surfaces in pseudo-Riemannian space forms, Ann. Global Anal. Geom. 8 (1990), 217–226.Google Scholar
  4. [P1]
    Palmer, B.: The conformal Gauss map and the stability of Willmore surfaces, Ann. Global Anal. Geom. 9 (1991), 305–317.Google Scholar
  5. [S]
    Smale, S.: On the Morse index theorem, J. Math. Mech., 14 (1965), 1049–1055.Google Scholar
  6. [T]
    Thomsen, G.: Über Konforme Geometrie I, Grundlagen der Konformen Flächentheorie, Abh. Math. Sem. Hamburg 11 (1923), 31–56.Google Scholar
  7. [W]
    Wolf, J. A.: Spaces of Constant Curvature, Publish or Perish, Boston, 1974.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • L. J. Alĺas
    • 1
  • B. Palmer
    • 2
  1. 1.Departamento de MatemáticasUniversidad de MurciaEspinardo, MurciaSpain
  2. 2.Department of Mathematical SciencesUniversity of DurhamDurhamEngland

Personalised recommendations