An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H² × R

الأصل · EN

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in H²× 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 H²× R, say ∂∞ H²× R, such that C is contained in a slab between two horizontal circles of ∂∞ H²× R with width equal to π. We construct minimal vertical graphs in H²× R over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains in H²× {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.