Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane

TL;DR

This paper proves the existence of proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane, including from the unit disk, challenging previous conjectures and expanding understanding of harmonic map behavior.

Contribution

It demonstrates the existence of proper harmonic maps from hyperbolic domains, including the unit disk, into the Euclidean plane, with precise topological control, contradicting earlier conjectures.

Findings

Proper harmonic maps exist from hyperbolic Riemann surfaces to the Euclidean plane.

Such maps can be constructed from the unit disk, disproving a longstanding conjecture.

The domain can be chosen arbitrarily close to the original surface minus disks.

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.