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

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 $K\mathrm{\ne }0$ 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.