On an Enneper-Weierstrass-type representation of constant Gaussian curvature surfaces in $3$-dimensional hyperbolic space

For all $k\in]0,1[$, we construct a canonical bijection between the space of ramified coverings of the sphere and the space of complete immersed surfaces in $3$-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to $k$. We show, furthermore, that this bijection restricts to a homeomorphism over each stratum of the space of ramified coverings of the sphere.