Geometriae Dedicata

, Volume 60, Issue 3, pp 301-315

First online:

Conformal geometry of surfaces in Lorentzian space forms

  • L. J. AlĺasAffiliated withDepartamento de Matemáticas, Universidad de Murcia
  • , B. PalmerAffiliated withDepartment of Mathematical Sciences, University of Durham

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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