A remark on one-harmonic maps from a Hadamard surface of pinched negative curvature to the hyperbolic plane

Abstract

We show that every one-harmonic map, in the sense of Trapani and Valli, from a Hadamard surface of pinched negative curvature to ${\mathbb{H}}^2$ has image the interior of the convex hull of a subset of ${\partial }_{\infty }{\mathbb{H}}^2$. The proof relies on Minkowski geometry, by interpreting one-harmonic maps as the Gauss maps of convex surfaces.

7 pages, 1 figure. Published in the proceedings of the workshop "Representations of Discrete Groups and Geometric Topology on Manifolds", Josai University