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We show that every one-harmonic map, in the sense of Trapani and Valli, froma Hadamard surface of pinched negative curvature to $\mathbb{H}^2$ has imagethe 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 asthe Gauss maps of convex surfaces.

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