Global properties of constant mean curvature surfaces in ℍ2 × ℝ

Abstract

We discuss some aspects of the global behavior of surfaces in H{double-struck}2 × R{double-struck} with constant mean curvature H (known as H-surfaces). We prove a maximum principle at infinity for complete properly embedded H-surfaces with H > 1/2, and show that the genus of a compact stable H-surface with H > 1/2 is at most three.