2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/144488In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in $H^2\times R$. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary $C$ is a Jordan curve homologous to zero in the asymptotic boundary of $ H^2\times R,$ say $\partial_\infty H^2\times R$, such that $C$ is contained in a slab between two horizontal circles of $\partial_\infty H^2\times R$ with width equal to $π.$ We construct minimal vertical graphs in $H^2\times R$ over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains $\Om$ in $H^2\times \{0\}$ are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition.This paper was presented in the International Congress on Minimal and Constant Mean Curvature Surfaces, Buzios, Brazil, August 2007 (27 pages 7 figures)Differential Geometry53C42text