Construction of harmonic diffeomorphisms and minimal graphs

Abstract

We study complete minimal graphs in $\mathbb{H}\times\mathbb{R}$, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon $\Gamma$ in $\mathbb{H}$. We give necessary and sufficient conditions on the “lengths” of the sides of the polygon (and all inscribed polygons in $\Gamma$) that ensure the existence of such a graph. We then apply this to construct entire minimal graphs in $\mathbb{H}\times\mathbb{R}$ that are conformally the complex plane $\mathbb{C}$. The vertical projection of such a graph yields a harmonic diffeomorphism from $\mathbb{C}$ onto $\mathbb{H}$, disproving a conjecture of Rick Schoen and S.-T. Yau.