Real forms of complex surfaces of constant mean curvature

TL;DR

This paper classifies all real form surfaces derived from complex constant mean curvature immersions, revealing seven classes called integrable surfaces, characterized by harmonic Gauss maps into specific symmetric spaces, unified through a generalized Weierstrass representation.

Contribution

It classifies all real form surfaces of complex CMC immersions based on real forms of a loop algebra, identifying seven classes called integrable surfaces with harmonic Gauss maps.

Findings

Seven classes of integrable surfaces identified.

All integrable surfaces characterized by harmonic Gauss maps.

Unified description via generalized Weierstrass representation.

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.