REAL FORMS OF COMPLEX SURFACES OF CONSTANT MEAN CURVATURE

It is known that complex constant mean curvature (CMC for short) immersions in C-3 are natural complexifications of CMC-inimersions in I. In this paper, conversely we consider real form surfaces of a complex CMC-immersion, which are defined from real forms of the twisted sl(2, C) loop algebra Lambda sl(2, C)(sigma) and classify all such surfaces according to the classification of real forms of Lambda sl(2, C)(sigma). There are seven classes of surfaces, which are called 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,S-1 or the 4-symmetric space SL(2, C)/U(1). We also give a unification to all integrable surfaces via the generalized Weierstrass type representation.