An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $H^2\times R$

In 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 ${\mathrm{\partial }}_{\mathrm{\infty }}{H}^2\times R$, such that $C$ is contained in a slab between two horizontal circles of ${\mathrm{\partial }}_{\mathrm{\infty }}{H}^2\times R$ with width equal to $\pi \mathrm{.}$ 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 $\text{Om}$ 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.