<Emphasis Type="Italic">H</Emphasis>-Surfaces with Arbitrary Topology in Hyperbolic 3-Space

We show that any open orientable surface can be properly embedded in \(\mathbb {H}^3\) as a constant mean curvature H-surface for \(H\in 0,1)\). 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 \(\mathbb {H}^3\) as a minimal surface.