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

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.