A Weierstrass-Kenmotsu Type Representation for CMC $0\le H<1$ in \$\mathbb{H}^3(-1)$

TL;DR

This paper introduces a Weierstrass-Kenmotsu type representation for conformal immersions of constant mean curvature surfaces in hyperbolic 3-space with mean curvature between 0 and 1, using flat SL(2,C) connections.

Contribution

It develops a new representation method linking flat SL(2,C) connections to CMC surfaces in hyperbolic space, generalizing previous models and providing explicit formulas.

Findings

Explicit formula for mean curvature in terms of spectral parameter

Local construction of CMC surfaces from rank-one (1,0)-forms

Connection with existing representations and gauge transformations

Abstract

We develop a Weierstrass-Kenmotsu type representation for conformal immersions of constant mean curvature $0\le H<1$ in hyperbolic $3$-space $\HH$. The construction is based on the Hermitian model of $\HH$, a balanced spectral deformation, and Iwasawa splitting of $\SL$. We show that such immersions arise locally from a rank-one $(1,0)$-form $\eta$ and a constant complex parameter $\lambda\in\C^*$ through a flat $\SL$-connection of the form \[ S^{-1}dS=\eta-\lambda\,\eta^*, \] with mean curvature \[ H=\frac{1-|\lambda|^2}{1+|\lambda|^2}. \] Conversely, every conformal CMC immersion with $0\le H<1$ is locally obtained from such flat rank-one data. We establish an explicit correspondence with the representation of Aiyama and Akutagawa via a gauge transformation, and interpret the construction in terms of Kokubu's adjusted normal Gauss map. We further discuss the role of the flatness condition, present simple local and cylindrical model examples, and outline aspects of monodromy and numerical implementation within this framework.