Abstract
We study complete minimal graphs in $\mathbb{H}\times\mathbb{R}$, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon $\Gamma$ in $\mathbb{H}$. We give necessary and sufficient conditions on the “lengths” of the sides of the polygon (and all inscribed polygons in $\Gamma$) that ensure the existence of such a graph. We then apply this to construct entire minimal graphs in $\mathbb{H}\times\mathbb{R}$ that are conformally the complex plane $\mathbb{C}$. The vertical projection of such a graph yields a harmonic diffeomorphism from $\mathbb{C}$ onto $\mathbb{H}$, disproving a conjecture of Rick Schoen and S.-T. Yau.
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[A] U. Abresch.
@misc{A,
author={Abresch, U.},
NOTE={personal communication},
} -
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AUTHOR = {Collin, P. and Krust, R.},
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AUTHOR = {Schoen, Richard M.},
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BOOKTITLE = {Complex Geometry},
VENUE={{O}saka, 1990)},
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@book {V, MRKEY = {1929066},
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ZBLNUMBER={0999.30001},
}