Our primary object of this paper is to give a representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space of curvature -1 in terms of their normal Gauss maps. For CMC (constant mean curvature) surfaces, we derive another representation formula in terms of their adjusted Gauss maps. These formulas are hyperbolic versions of the Kenmotsu representation formula for surfaces in the Euclidean 3-space. As an application, we give a construction of complete simply connected CMC surfaces embedded in the hyperbolic 3-space.
"Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space." J. Math. Soc. Japan 52 (4) 877 - 898, October, 2000. https://doi.org/10.2969/jmsj/05240877