Monotone maps of \(\mathbb{R}^n\) are quasiconformal
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 \(H(\delta)\).
scientific article; zbMATH DE number 5573042