Construction of harmonic diffeomorphisms and minimal graphs

2007/01/19 by Pascal Collin, Collin, Pascal, Harold Rosenberg +1
Mathematics · #53C43 #Differential Geometry (math.DG) #FOS: Mathematics #MSC: 53A10 #math.DG #msc:53A10 #msc:53C43

paper · pdf · doi:10.48550/arxiv.math/0701547

arxiv created 2007/01/19 · arxiv updated 2009/12/01

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.

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