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Dimension-free quasiconformal distortion in $ n$-space


Authors: G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen
Journal: Trans. Amer. Math. Soc. 297 (1986), 687-706
MSC: Primary 30C60
DOI: https://doi.org/10.1090/S0002-9947-1986-0854093-5
MathSciNet review: 854093
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Abstract: Most distortion theorems for $ K$-quasiconformal mappings in $ {{\mathbf{R}}^n}$, $ n \geqslant 2$, depend on both $ n$ and $ K$ in an essential way, with bounds that become infinite as $ n$ tends to $ \infty $. The present authors obtain dimension-free versions of four well-known distortion theorems for quasiconformal mappings--namely, bounds for the linear dilatation, the Schwarz lemma, the $ \Theta $-distortion theorem, and the $ \eta $-quasisymmetry property of these mappings. They show that the upper estimates they have obtained in each of these four main results remain bounded as $ n$ tends to $ \infty $ with $ K$ fixed. The proofs are based on a "dimensioncancellation" property of the function $ t \mapsto {\tau ^{ - 1}}(\tau (t)/K),\,t > 0,\,K > 0$, where $ \tau (t)$ is the capacity of a Teichmüller extremal ring in $ {{\mathbf{R}}^n}$. The authors also prove a dimension-free distortion theorem for the absolute (cross) ratio under $ K$-quasiconformal mappings of $ {\overline {\mathbf{R}} ^n}$, from which several other distortion theorems follow as special cases.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0854093-5
Article copyright: © Copyright 1986 American Mathematical Society

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