Harmonic quasi-isometric maps into Gromov hyperbolic $\operatorname{CAT}(0)$-spaces

Abstract

We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a proper, Gromov hyperbolic, $\operatorname{CAT}(0)$-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist–Hulin.