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
