Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane

Abstract

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.

18 pages, 1 figure