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Construction of harmonic diffeomorphisms and minimal graphs

Pascal Collin·Harold Rosenberg
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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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Pascal Collin, Harold Rosenberg. 2007-01-19. Construction of harmonic diffeomorphisms and minimal graphs. https://arxiv.org/abs/math/0701547

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