Bryant Surfaces

Abstract

In these lectures we will discuss the theory of surfaces in hyperbolic 3-space of mean curvature one. We call them Bryant surfaces. Robert Bryant showed how to parametrize these surfaces by meromorphic data and began the qualitative study of their geometry. Bryant surfaces have a meromorphic Gauss map and their intimate relation (they are cousins) to minimal surfaces in ℝ³ has oriented their study.

Many important properties and examples have been found by Umehara, Yamada, Rossman, Sa Earp, Toubiana and Zu-Huan Yu. We will present some of their results.

My main goal is to present a theorem of Pascal Collin, Lurent Hauswirth and myself: a properly embedded Bryant surface in ℍ³ of finite topology has finite total curvature and the Gauss map extends meromorphically to the conformal compactification.

In fact a properly embedded Bryant annular end is asymptotic to a horosphere end or to a catenoid cousin end. Moreover if the end is part of a properly embedded Bryant surface which is not a horosphere, then the end is asymptotic to a catenoid cousin end.

We will see this implies the only simply connected properly embedded Bryant surface is a horosphere, and the only such surface with exactly two annular ends is a catenoid cousin.

We begin by a discussion of the theory of H-surfaces in the three simply connected space forms S³, ℝ³, and ℍ³, and some problems are mentionned. Then Bryant’s representation is described. We revue the theory of moving frames and indicate how this applies to Bryant surfaces. Examples are discussed. We then begin to prove our finite total curvature theorem.