An asymptotic theorem for minimal surfaces and existence results for minimal graphs in mathbb H2 ×mathbb R{mathbb H^2 times mathbb R}

摘要

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in mathbb H2 ×mathbb R{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 ¶_¥mathbb H2×mathbb R{partial_inftymathbb H^2times mathbb R} such that Γ_∞ is contained in a slab between two horizontal circles of ¶_¥mathbb H2×mathbb R{partial_inftymathbb H^2times mathbb R} with width equal to π. We construct vertical minimal graphs in mathbb H2×mathbb R{mathbb H^2times 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 H2×{0}{mathbb H^2times {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.