Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane

Details Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane

2009-06-15

arxiv.org/abs/0906.2638v1

Mathematics

Differential Geometry

18 pages, 1 figure

Scientific paper

Let $\Sigma$ be a compact Riemann surface and $D_1,...,D_n$ a finite number of pairwise disjoint closed disks of $\Sigma$. We prove the existence of a proper harmonic map into the Euclidean plane from a hyperbolic domain $\Omega$ containing $\Sigma\backslash\cup_{j=1}^n D_j$ and of its topological type. Here, $\Omega$ can be chosen as close as necessary to $\Sigma\backslash\cup_{j=1}^n D_j$. In particular, we obtain proper harmonic maps from the unit disk into the Euclidean plane, which disproves a conjecture posed by R. Schoen and S.T. Yau.

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