  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.
