Harmonic maps between 3-dimensional hyperbolic spaces

TL;DR

This paper proves that quasiconformal maps on the 2-sphere can be extended harmonically and quasi-isometrically into 3D hyperbolic space, confirming the Schoen Conjecture in three dimensions.

Contribution

It establishes the existence of harmonic quasi-isometric extensions for quasiconformal maps in 3D hyperbolic space, confirming a longstanding conjecture.

Findings

Quasiconformal maps extend harmonically into hyperbolic space

Confirmation of the Schoen Conjecture in dimension 3

Extension preserves quasi-isometry

Abstract

We prove that a quasiconformal map of the 2-sphere admits a harmonic quasi-isometric extension to the 3-dimensional hyperbolic space, thus confirming the well known Schoen Conjecture in dimension 3.