Conformal CMC-Surfaces in Lorentzian Space Forms*

Let ℚ3 be the common conformal compactification space of the Lorentzian space forms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{R}^{3}_{1} ,\mathbb{S}^{3}_{1} \;{\text{and}}\;\mathbb{H}^{3}_{1} $$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{R}^{3}_{1} ,\mathbb{S}^{3}_{1} \;{\text{and}}\;\mathbb{H}^{3}_{1} $$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{R}^{3}_{1} $$\end{document}, or K = 0 and x is conformally equivalent to the surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{H}^{1} {\left( {\frac{1} {{{\sqrt 2 }}}} \right)} \times \mathbb{H}^{1} {\left( {\frac{1} {{{\sqrt 2 }}}} \right)}\;{\text{in}}\;\mathbb{H}^{3}_{1} . $$\end{document}