2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119355We give a global version of the Bryant representation of surfaces of constant mean curvature one (cmc-1) 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.22 pages, to appear in The Asian Journal of MathematicsDifferential Geometry58E15text