Harmonic extensions of quasiregular maps

TL;DR

This paper proves that non-constant quasiregular selfmaps of the n-sphere can be extended harmonically into hyperbolic space, providing a new link between quasiregular maps and harmonic analysis.

Contribution

It establishes the existence of harmonic extensions for quasiregular maps on spheres into hyperbolic space, a novel result in geometric analysis.

Findings

Every non-constant quasiregular selfmap of the n-sphere admits a harmonic extension to hyperbolic space.

The extension exists for dimensions n ≥ 2.

The result bridges quasiregular maps and harmonic functions in geometric contexts.

Abstract

We prove that every non-constant quasiregular selfmap of the $n$-sphere $\mathbb{S}^{n}$ admits a harmonic extension to the hyperbolic space $\mathbb{H}^{n+1}$ for $n\ge 2$.