# Harmonic quasi-isometric maps II : negatively curved manifolds

## TL;DR

This paper proves that quasi-isometric maps between negatively curved manifolds are close to unique harmonic maps, establishing a strong link between coarse geometric embeddings and harmonic analysis.

## Contribution

It demonstrates that quasi-isometric and coarse embeddings between pinched Hadamard manifolds are approximated by unique harmonic maps, extending understanding of geometric and harmonic map relationships.

## Key findings

- Quasi-isometric maps are within bounded distance from harmonic maps.
- Uniqueness of harmonic maps approximating coarse embeddings.
- Extension of harmonic map theory to negatively curved manifolds.

## Abstract

We prove that a quasi-isometric map, and more generally a coarse embedding, between pinched Hadamard manifolds is within bounded distance from a unique harmonic map.