  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.
