Construction of harmonic diffeomorphisms and minimal graphs

Abstract

We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence of such a graph. We then apply this to construct entire minimal graphs in HxR that are conformally the complex plane C. The vertical projection of such a graph yields a harmonic diffeomorphism from C onto H, disproving a conjecture of Rick Schoen.