Chinese Annals of Mathematics, Series B

, Volume 28, Issue 3, pp 299–310

Conformal CMC-Surfaces in Lorentzian Space Forms*

ORIGINAL ARTICLES

Abstract

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 \( \mathbb{H}^{1} {\left( {\frac{1} {{{\sqrt 2 }}}} \right)} \times \mathbb{H}^{1} {\left( {\frac{1} {{{\sqrt 2 }}}} \right)}\;{\text{in}}\;\mathbb{H}^{3}_{1} . \)

Keywords

Conformal geometry Willmore surfaces Lorentzian space 

2000 MR Subject Classification

53A30 53B30 

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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

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