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Let $\Sigma$ be a compact Riemann surface and $D_1,...,D_n$ a finite numberof pairwise disjoint closed disks of $\Sigma$. We prove the existence of aproper 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 harmonicmaps from the unit disk into the Euclidean plane, which disproves a conjectureposed by R. Schoen and S.T. Yau.

Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane