[Paper Review] Constant mean curvature surfaces in hyperbolic 3-space via loop groups

Josef F. Dorfmeister, Jun-ichi Inoguchı+1 more|arXiv (Cornell University)|Aug 8, 2011

Geometric Analysis and Curvature Flows39 references3 citations

TL;DR

This paper presents a generalized Weierstrass-type representation using loop group methods to construct constant mean curvature (CMC) surfaces in hyperbolic 3-space 𝕍³ for the previously underexplored case of 0 ≤ H < 1. By leveraging the harmonic Gauss map and extended frames within the loop group framework, the authors establish a unified method to generate both minimal (H = 0) and non-minimal CMC surfaces (0 < H < 1), extending the DPW method to this non-Lawson case and proving the harmonicity of the oriented normal geodesic congruence as a key geometric characterization.

ABSTRACT

In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H &lt; 1$, $H = 1$, $H &gt; 1$. 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 $H eq 0$, respectively. These surface classes have been investigated intensively in the literature. For the case $0 \leq H &lt; 1$ 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 $0\leq H &lt; 1$. 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

Motivation & Objective

• To develop a systematic method for constructing constant mean curvature (CMC) surfaces in hyperbolic 3-space 𝕍³ for the case 0 ≤ H < 1, which lacks a Lawson correspondence to Euclidean space.
• To extend the DPW (Dorfmeister-Pedit-Wu) loop group method—previously applied to CMC surfaces in 𝕆³ and 𝕊³—to the hyperbolic setting for 0 ≤ H < 1.
• To establish a generalized Weierstrass-type representation that unifies the construction of minimal (H = 0) and non-minimal CMC surfaces (0 < H < 1) in 𝕍³.
• To prove that the oriented normal geodesic congruence of a CMC surface in 𝕍³ is a harmonic map into the space of oriented geodesics, enabling the use of loop group techniques.

Proposed method

• The method employs the DPW framework by associating CMC surfaces in 𝕍³ with harmonic maps into the space of oriented geodesics, identified with the Grassmannian Gr₁,₁(𝔼¹,³).
• The extended frame Φ is constructed via a potential form, and the generalized Gauss map is defined as ℒ = (Φ(e₀ + e₁)Φ*, Φ(e₀ − e₁)Φ*), mapping into the space of oriented geodesics.
• The harmonicity of the Gauss map is established by showing that the pullback of the canonical symplectic form vanishes, using the Legendrian property of the associated Legendre map F.
• The construction is validated by proving that the resulting map ℒ is Lagrangian and harmonic, ensuring integrability via the loop group dressing action.
• The method is applied to fronts by defining parallel surfaces in de Sitter 3-space 𝕊¹,² via the transformation f_q = cosh(q)f + sinh(q)n, which yields spacelike CMC immersions.
• The key equation for the parallel surface is given by ℒ_q = Φ diag(e^{q/2}, -e^{-q/2}) Φ*, showing that the extended frame generates a loop of spacelike CMC immersions.

Experimental results

Research questions

• RQ1 Can the DPW loop group method be extended to construct CMC surfaces in hyperbolic 3-space for the case 0 ≤ H < 1, where no Lawson correspondence exists?
• RQ2 Is the oriented normal geodesic congruence of a CMC surface in 𝕍³ harmonic, and does this allow for integrable surface construction via loop groups?
• RQ3 How can minimal surfaces (H = 0) and non-minimal CMC surfaces (0 < H < 1) be constructed simultaneously within a unified framework?
• RQ4 What is the role of the generalized Gauss map in identifying the integrable structure of CMC surfaces in 𝕍³?
• RQ5 Can the loop group method be adapted to produce spacelike CMC surfaces in de Sitter 3-space via parallel surfaces in 𝕍³?

Key findings

• The generalized Weierstrass-type representation via loop groups successfully constructs CMC surfaces in hyperbolic 3-space for the case 0 ≤ H < 1, which was previously difficult to access via classical methods.
• The oriented normal geodesic congruence of a CMC surface in 𝕍³ is shown to be a harmonic map into the space of oriented geodesics, satisfying the Ruh-Vilms property in this setting.
• The construction unifies minimal (H = 0) and non-minimal (0 < H < 1) CMC surfaces through a single loop group framework, with the extended frame Φ encoding the surface geometry.
• The map ℒ = (Φ(e₀ + e₁)Φ*, Φ(e₀ − e₁)Φ*) is proven to be Lagrangian and harmonic, confirming integrability and enabling the use of dressing transformations.
• The parallel surface f_q in de Sitter 3-space 𝕊¹,² is shown to be a spacelike CMC immersion with the same Hopf differential, and its construction is given explicitly via ℒ_q = Φ diag(e^{q/2}, -e^{-q/2}) Φ*.
• The method provides a complete integrable system for CMC fronts in 𝕍³ by associating them with harmonic Legendre maps into the unit tangent bundle of 𝕍³, with the Gauss map satisfying the harmonicity condition.