Construction of harmonic diffeomorphisms and minimal graphs

Details Construction of harmonic diffeomorphisms and minimal graphs Construction of harmonic diffeomorphisms and minimal graphs

2007-01-19

arxiv.org/abs/math/0701547v1

Mathematics

Differential Geometry

Scientific paper

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.

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