Real forms of complex surfaces of constant mean curvature

Abstract

It is known that complex constant mean curvature ({\sc CMC} for short) immersions in ${\mathbb{C}}^3$ are natural complexifications of {\sc CMC}-immersions in ${\mathbb{R}}^3$. In this paper, conversely we consider {\it real form surfaces} of a complex {\sc CMC}-immersion, which are defined from real forms of the twisted $\mathfrak{sl}(2,\mathbb{C})$ loop algebra $\Lambda \mathfrak{sl}(2,\mathbb{C}{)}_{\sigma }$, and classify all such surfaces according to the classification of real forms of $\Lambda \mathfrak{sl}(2,\mathbb{C}{)}_{\sigma }$. There are seven classes of surfaces, which are called {\it integrable surfaces}, and all integrable surfaces will be characterized by the (Lorentz) harmonicities of their Gauß maps into the symmetric spaces $S^2$, $H^2$, $S^{1,1}$ or the 4-symmetric space $SL(2,\mathbb{C})/U(1)$. We also give a unification to all integrable surfaces via the generalized Weierstraß type representation.