Proper Harmonic Maps from Hyperbolic Riemann Surfaces into the Euclidean Plane

Abstract

Let Σ be a compact Riemann surface and D 1 , . . . , D n a finite number of pairwise disjoint closed disks of Σ. We prove the existence of a proper harmonic map into the Euclidean plane from a hyperbolic domain Ω containing $${\Sigma \backslash \cup_{j=1}^n D_j}$$ and of its topological type. Here, Ω can be chosen as close as necessary to $${\Sigma \backslash \cup_{j=1}^n D_j}$$ .