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

TL;DR

This paper proves the existence of Lipschitz harmonic maps at finite distance from quasi-isometric maps from negatively curved Hadamard manifolds to Gromov hyperbolic CAT(0)-spaces, extending recent results.

Contribution

It establishes the existence and Lipschitz regularity of harmonic maps in a new geometric setting, generalizing prior work by Benoist-Hulin.

Findings

Existence of harmonic maps at finite distance from quasi-isometric maps.

Harmonic maps are Lipschitz continuous.

Generalization of Benoist-Hulin's recent results.

Abstract

We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a locally compact, Gromov hyperbolic, ${\rm CAT}(0)$-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.