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.
"Surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends." Tohoku Math. J. (2) 53 (2) 305 - 318, 2001. https://doi.org/10.2748/tmj/1178207483