Graphs with constant mean curvature in the 3-hyperbolic space

Abstract

In this work we will deal with disc type surfaces of constant mean curvature in the three dimensional hyperbolic space which are given as graphs of smooth functions over planar domains. From the various types of graphs that could be defined in the hyperbolic space we consider in particular the horizontal and the geodesic graphs. We proved that if the mean curvature is constant, then such graphs are equivalent in the following sense: suppose that M is a constant mean curvature surface in the 3-hyperbolic space such that M is a geodesic graph of a function ρ that is zero at the boundary, then there exist a smooth function f, that also vanishes at the boundary, such that M is a horizontal graph of f. Moreover, the reciprocal is also true.