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

Abstract

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in \({\mathbb H^2 \times \mathbb R}\). As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary Γ∞ is a Jordan curve homologous to zero in \({\partial_\infty\mathbb H^2\times \mathbb R}\) such that Γ∞ is contained in a slab between two horizontal circles of \({\partial_\infty\mathbb H^2\times \mathbb R}\) with width equal to π. We construct vertical minimal graphs in \({\mathbb H^2\times \mathbb R}\) over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains Ω in \({\mathbb H^2\times \{0\}}\) 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.