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On quasiconformal maps with identity boundary values. (English) Zbl 1226.30024

The present paper is devoted to the study of the class \[ \text{Id}(\partial G) = \{f : \mathbb R^n \to \mathbb R^n \text{ homeomorphism}:\; f(x) = x \text{ for all } x \in \mathbb R^n \setminus G\}, \] where \(G\) is a domain in \(\mathbb R^n\) and \(n\geq 2\). For \(K\geq 1\), the set of \(K\)-quasiconformal mappings in \(\text{Id}(\partial G)\) is denoted by \(\text{Id}_K(\partial G)\). One of the main results states that if \(f \in \text{Id}_K(\partial B^n)\), then for all \(x \in B^n\),
\[ \rho_{B^n}(f(x), x)\leq \log \frac{1-a}{a}, \]
with a certain constant \(a\), where \(\rho\) denotes the hyperbolic metric and \(B^n\) the unit ball in \(\mathbb R^n\).

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30C62 Quasiconformal mappings in the complex plane
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