Monodromy of Constant Mean Curvature Surface in Hyperbolic Space

Abstract

In this paper we give a global version of the Bryant representation of surfaces of constant mean curvature one (cmc-1 surfaces) in hyperbolic space. This allows to set the associated non-abelian period problem in the framework of flat unitary vector bundles on Riemann surfaces. We use this machinery to prove the existence of certain cmc-1 surfaces having prescribed global monodromy.