Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane

2009/06/15 by Antonio Alarcón, Antonio Alarcon, Alarcon, Antonio +3
Mathematics · #53C43 #53E20 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #math.DG #msc:53C43 #msc:53E20

paper · pdf · doi:10.48550/arxiv.0906.2638

18 pages, 1 figure

arxiv created 2009/06/15 · openalex publication_date 2009/06/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Σ be a compact Riemann surface and D₁,...,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 Σ\backslash∪_j=1ⁿ D_j and of its topological type. Here, Ω can be chosen as close as necessary to Σ\backslash∪_j=1ⁿ 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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