Harmonic quasiconformal self-mappings and Möbius transformations of the unit ball

David Kalaj and Miodrag S. Mateljević

Vol. 247 (2010), No. 2, 389–406

DOI: 10.2140/pjm.2010.247.389

Abstract

We prove in dimension n > 2 that a K-quasiconformal harmonic mapping u of the unit ball Bⁿ onto itself is Euclidean bi-Lipschitz if u(0) = 0 and K < 2ⁿ⁻¹. This is an extension of a similar result of Tam and Wan for hyperbolic harmonic mappings with respect to a hyperbolic metric. The proof uses Möbius transformations on the related space and a recent result of the first author, which states that harmonic quasiconformal self-mappings of the unit ball are Lipschitz continuous.

Keywords

quasiconformal map, harmonic mapping, Lipschitz condition, Möbius transformation, unit ball

Mathematical Subject Classification 2000

Primary: 30C65

Secondary: 31B05

Milestones

Received: 20 June 2009

Revised: 11 January 2010

Accepted: 16 January 2010

Published: 11 August 2010