Harmonic quasi-isometric maps III :quotients of Hadamard manifolds

TL;DR

This paper extends previous results on quasi-isometric maps between Hadamard manifolds to quotient spaces, showing such maps are close to harmonic maps even in more complex geometric settings.

Contribution

It generalizes the harmonic map approximation from simply connected Hadamard manifolds to quotients by convex cocompact groups, under local quasi-isometry conditions.

Findings

Quasi-isometric maps are within bounded distance from harmonic maps in quotient spaces.

Extension of harmonic map approximation to convex cocompact quotients.

Results apply to maps that are locally quasi-isometric at infinity.

Abstract

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