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

Weakly complete constant Gaussian curvature $-1<K<0$ surfaces will be classified in terms of holomorphic quadratic differentials. For this purpose, we will first establish a loop group method for constant Gaussian curvature surfaces in H³ with $K>-1$ but K ≠ 0 by using harmonicities of Lagrangian and Legendrian Gauss maps. Then we will show that a spectral parameter deformation of the Lagrangian harmonic Gauss map gives a harmonic map into H² for $-1< K<0$ or S² for $K>0$, respectively.