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Construction of harmonic diffeomorphisms and minimal graphs. (English) Zbl 1209.53010
The paper constructs harmonic diffeomorphisms from $ℂ$ onto $ℍ$. The authors use entire minimal graphs to construct such examples. The constructions are used within a general study of complete minimal graphs in $ℍ×ℝ$, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon in $ℍ$. A diffeomorphism from $ℂ$ onto $ℍ$ is constructed based on a certain entire minimal graph $ℍ×ℝ$, which disproves a conjecture of Schoen and Yau.

##### MSC:
 53A10 Minimal surfaces, surfaces with prescribed mean curvature 53C43 Differential geometric aspects of harmonic maps
##### Keywords:
minimal surfaces; minimal graphs; harmonic maps