A classification of constant Gaussian curvature surfaces in the three-dimensional hyperbolic space

TL;DR

The paper classifies weakly complete constant Gaussian curvature surfaces in hyperbolic 3-space with $-1<K<0$ by establishing a loop-group framework tied to holomorphic quadratic differentials. It proves a Ruh-Vilms-type correspondence: constant $K>-1$ is equivalent to the harmonicity of Legendrian and Lagrangian Gauss maps and to holomorphicity of the Klotz differential, and it connects these data to harmonic maps into $\mathbb H^2$ or $\mathbb S^2$ via a spectral-parameter deformation. The main result is a bijection between weakly complete CGC surfaces and holomorphic quadratic differentials on $\mathbb D$ or $\mathbb C$ (mod Möbius). The authors also establish existence results for complete and equivariant complete CGC surfaces using bounded holomorphic data, highlighting the geometric and analytic interplay between CGC surfaces, harmonic maps, and loop-group methods.

Abstract

We classify weakly complete constant Gaussian curvature $-1<K<0$ surfaces in the hyperbolic three-space in terms of holomorphic quadratic differentials. For this purpose, we first establish a loop group method for constant Gaussian curvature surfaces with $K>-1$ and $K \neq 0$ via the harmonicity of the Lagrangian and Legendrian Gauss maps. We then show that a spectral parameter deformation of the Lagrangian harmonic Gauss map gives a harmonic map into the hyperbolic two-space for $-1< K<0$ or the two-sphere for $K>0$, respectively. Consequently, weakly complete constant Gaussian curvature surfaces with $-1 < K <0$ are in one-to-one correspondence with holomorphic quadratic differentials on the unit disk or the complex plane.

Theorems & Definitions (35)

• Theorem A
• Theorem B
• Theorem C
• Remark 1.1
• Remark 1.2
• Definition 1.1
• Remark 1.3
• Proposition 2.1
• proof
• proof : Proof of Theorem \ref{['thm:Ruh-Vilms']}