Conformal CMC-Surfaces in Lorentzian Space Forms*

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