Existence and uniqueness of complete constant mean curvature surfaces at infinity ofH 3

Abstract

A set of conditions are given, each equivalent to the constancy of mean curvature of a surface inH3.It is shown that analogs of these equivalences exist for surfaces inS∞2,the bounding ideal sphere ofH3,leading to a notion of constant mean curvature at infinity ofH3.A parametrization of all complete constant mean curvature surfaces at infinity ofH3is given by holomorphic quadratic differentials on Ĉ,C, andD.