H-Surfaces with Arbitrary Topology in Hyperbolic 3-Space

We show that any open orientable surface can be properly embedded in H-3 as a constant mean curvature H-surface for H epsilon [0, 1). We obtain this result by proving a version of the bridge principle at infinity for H-surfaces. We also show that any open orientable surface can be nonproperly embedded in H-3 as a minimal surface.