Surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends

Abstract

We investigate surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends, and prove that their irregular ends must self-intersect, which answers affirmatively a conjecture of Umehara and Yamada. Moreover we also obtain an explicit representation of a constant mean curvature one surface and a new minimal surface in the Euclidean three-space.