The geometry of finite topology Bryant surfaces

In this paper we shall establish that properly embedded constant mean curvature one surfaces in H of finite topology are of finite total curvature and each end is regular. In particular, this implies the horosphere is the only simply connected such example, and the catenoid cousins the only annular examples of this nature. In general each annular end of such a surface is asymptotic to an end of a horosphere or an end of a catenoid cousin. Robert Bryant discovered a holomorphic parametrization of (simply connected) mean curvature one surfaces in H which can be thought of as a generalization of the Weierstrass representation of minimal surfaces in R [2]. Each (simply connected) minimal surface in R is isometric to a mean curvature one surface in H (and vice versa); R. Bryant calls this the cousin of the minimal surface. This correspondence follows easily from Bonnet’s existence theorem for surfaces in the space forms. This may have been R. Bryant’s motivation to seek a meromorphic Weierstrass type representation of mean curvature one surfaces in H. Definition. A Bryant surface is a surface in H of constant mean curvature one.