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 Gau{\ss} 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 Weierstra{\ss} type representation.