Complete noncompact CMC surfaces in hyperbolic 3-space

Abstract

In this thesis we study the asymptotic Plateau problem for surfaces with constant mean curvature (CMC) in hyperbolic 3-space H3. We give a new, geometrically transparent proof of the existence of a CMC surface spanning any given Jordan curve on the sphere at infinity of H3, for mean curvature lying in the range (-1,1). Our proof does not require methods from geometric measure theory, and yields an immersed disk as solution. We then study the dependence of the solution surface on the boundary data. We view the set of H-surfaces (CMC surfaces with mean curvature equal to H) as consisting of the conformal H-harmonic maps. We therefore begin by showing smooth dependence on boundary data for H-harmonic maps (with |H| < 1) which solve a Dirichlet problem at infinity. This is achieved by showing that the linearised H-harmonic map operator is invertible as a map between appropriate function spaces. Finally we show smooth dependence on boundary data for H-surfaces which lie in a neighbourhood of the totally umbilic spherical caps {H}. This is achieved by studying the mapping properties of the so-called conformality operator. We use methods from complex geometry to show that the linearisation of this operator at a cap H is an isomorphism for all H ∈ (−1, 1)