Monotone maps of \(\mathbb{R}^n\) are quasiconformal
type: Item
instance of: scholarly article
title: Monotone maps of \(\mathbb{R}^n\) are quasiconformal
zbMATH Open document ID: 1178.30025
author: Kari Astala
author: Tadeusz Iwaniec
author: Gaven J. Martin
published in: Methods and Applications of Analysis
publication date: 2009-06-30T00:00:00Z
review text: L.V. Kovalev proved the interesting fact that a \(\delta\)-monotone mapping of \(\mathbb R^n\) is \(K\)-quasiconformal for some distortion constant \(K\) depending only on \(\delta\): {Theorem.} Let \(h :\Omega\subset\mathbb R^n\to\mathbb R^n\) be \(\delta\)-monotone. Then either \(h\) is constant or a quasiconformal homeomorphism with linear distortion bounded by \[ K =\frac{1+\sqrt{1-\delta^2}}{1-\sqrt{1-\delta^2}}. \] This bound on the linear distortion is sharp for every \(\delta\in (0, 1]\). In this paper the authors give a new proof of this result using methods which are rather more elementary than those employed by L.V. Kovalev, going through a compactness argument which is more or less standard in the theory of quasiconformal mappings. The authors give also the precise estimates relating the monotonicity constant \(\delta\) and the distortion constant \(K\). Let \(\Omega\subset\mathbb R^n\) be a domain containing the closed ball \({\overline B}\left(0,\frac{3}{\delta}\right)\) and let \(\alpha\in\mathbb S^{n-1}\). Define \(\mathcal F_\alpha = \{h :\Omega\to\mathbb R^n : h\) is \(\delta\)-monotone, \(h(0) = 0\), and \(|h(\alpha)| = 1\}.\) Then there is \(H = H(\delta) <\infty\) such that \(\sup_{h\in\mathcal F_\alpha} |h(z)| < H\) for all \(|z|\leq 1\). Let \(h : \Omega\to\mathbb R^n\) be a non-constant \(\delta\)-monotone mapping in a domain \(\Omega\subset\mathbb R^n\). Then \(h\) is a continuous injection whose linear distortion is bounded by \(H(\delta)\).
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 5573042
zbMATH Keywords: monotone mapping
zbMATH Keywords: linear distortion
zbMATH Keywords: weak quasisymmetry
zbMATH Keywords: compactness
zbMATH Keywords: quasiconformality
MaRDI profile type: Publication
full work available at URL: maa.2008.v15.n1.a4
OpenAlex ID: W2088559969
DOI: 10.4310/MAA.2008.V15.N1.A4
description: scientific article; zbMATH DE number 5573042