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

TL;DR

This paper proves that one-harmonic maps from a negatively curved Hadamard surface to the hyperbolic plane have images confined within the convex hull of a boundary subset, using Minkowski geometry and Gauss map interpretations.

Contribution

It establishes a geometric characterization of one-harmonic maps in terms of convex hulls and Minkowski geometry, extending understanding of harmonic map images in hyperbolic geometry.

Findings

Images are contained within the convex hull of boundary points

Utilizes Minkowski geometry and Gauss maps for proof

Connects harmonic maps with convex surface theory

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.