Harmonic quasi-isometric maps III :quotients of Hadamard manifolds

Abstract

In a previous paper, we proved that a quasi-isometric map $f:X⟶Y$ between two pinched Hadamard manifolds $X$ and $Y$ is within bounded distance from a unique harmonic map. We extend this result to maps $f:\Gamma \backslash X⟶Y$, where $\Gamma$ is a convex cocompact discrete group of isometries of $X$ and $f$ is locally quasi-isometric at infinity.