Mean curvature {$1$} surfaces of Costa type in hyperbolic three-space

Abstract

In this paper we prove the existence of families of complete mean curvature one surfaces in the hyperbolic three-space. We show that for each Costa-Hoffman-Meeks embedded minimal surface of positive genus in Euclidean three-space, we can produce, by cousin correspondence, a family of complete mean curvature one surfaces in the hyperbolic three-space. These surfaces have positive genus, three ends and the same group of symmetry of the original minimal surfaces. Furthermore, two of the ends approach the same point in the ideal boundary of hyperbolic three-space and the third end is asymptotic to a horosphere. The method we use to produce these results were developed in a recent paper by W. Rossman, M. Umehara and K. Yamada.