Construction of harmonic diffeomorphisms and minimal graphs

Abstract

We study complete minimal graphs in H{double-struck} × R{double-struck}, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H{double-struck}. We give necessary and sufficient conditions on the "lengths" 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 H{double-struck} × R{double-struck} that are conformally the complex plane C{double-struck}. The vertical projection of such a graph yields a harmonic diffeomorphism from C{double-struck} onto H{double-struck}, disproving a conjecture of Rick Schoen and S.-T. Yau. © 2010 by Princeton University (Mathematics Department).