$H$-Surfaces with Arbitrary Topology in Hyperbolic 3-Space

Abstract

In this paper, we show that any open orientable surface S can be properly embedded in H^3 as a minimizing H-surface for any 0<=H<1. We obtained this result by proving a version of the bridge principle at infinity for H-surfaces. We also show that any open orientable surface S can be nonproperly embedded in H^3 as a minimal surface, too.