Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces

Abstract

We construct harmonic diffeomorphisms from the complex plane ${\bf C}$ onto any Hadamard surface $\mathbb{M}$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $\mathbb{M} \times \mathbb{R}$ over domains of $\mathbb{M}$ bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains converging to an entire minimal graph in $\mathbb{M} \times \mathbb{R}$ with the conformal structure of ${\bf C}$.