, Volume 28, Issue 3, pp 299-310
Date: 30 Apr 2007

Conformal CMC-Surfaces in Lorentzian Space Forms*

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

*Project supported by the National Natural Science Foundation of China (No. 10125105) and the Research Fund for the Doctoral Program of Higher Education.