  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 $\mathbb H^3$ with $K>-1$ but $K \neq 0$ 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 $\mathbb H^2$ for $-1< K<0$ or $\mathbb S^2$ for $K>0$, respectively.
