Harmonic quasi-isometries of pinched Hadamard surfaces are injective

TL;DR

This paper proves that harmonic quasi-isometric maps between pinched Hadamard surfaces are actually quasi-conformal diffeomorphisms, establishing a strong geometric rigidity result.

Contribution

It demonstrates that harmonic quasi-isometries in this setting are necessarily injective and conformal, extending understanding of geometric mappings between negatively curved surfaces.

Findings

Harmonic quasi-isometric maps are quasi-conformal diffeomorphisms.

Injectivity of harmonic quasi-isometries is established.

The result applies to pinched Hadamard surfaces with negative curvature.

Abstract

We prove that a harmonic quasi-isometric map between pinched Hadamard surfaces is a quasi-conformal diffeomorphism.