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

TL;DR

This paper classifies weakly complete constant negative Gaussian curvature surfaces in hyperbolic space using holomorphic quadratic differentials, establishing a loop group method and spectral deformation techniques.

Contribution

It introduces a novel loop group approach and spectral deformation framework for classifying constant Gaussian curvature surfaces in hyperbolic space.

Findings

Classifies surfaces with -1<K<0 via holomorphic quadratic differentials.

Establishes a loop group method for K>-1 and K≠0 surfaces.

Shows a spectral deformation links these surfaces to harmonic maps.

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.