Constant mean curvature surfaces in hyperbolic 3-space via loop groups

Abstract

In hyperbolic 3-space ${\mathbb{H}}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0\le H<1$, $H=1$, $H>1$. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space ${\mathbb{E}}^3$ with H=0 and $H\ne 0$, respectively. These surface classes have been investigated intensively in the literature. For the case $0\le H<1$ there is no Lawson correspondence in Euclidean space and there are relatively few publications. Examples have been difficult to construct. In this paper we present a generalized Weierstraß type representation for surfaces of constant mean curvature in ${\mathbb{H}}^3$ with particular emphasis on the case of mean curvature $0\le H<1$. In particular, the generalized Weierstraß type representation presented in this paper enables us to construct simultaneously minimal surfaces (H=0) and non-minimal constant mean curvature surfaces ($0<H<1$).

37 pages, 4 figures. v3: Various typos fixed. v4: Proposition D.1 has been fixed

References in corpus (4)

• Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms
• Harmonic maps in unfashionable geometries
• Twistors, 4-symmetric spaces and integrable systems
• Gauss maps of constant mean curvature surfaces in three-dimensional homogeneous spaces

Cited by in corpus (1)

• Real forms of complex surfaces of constant mean curvature