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Quasiconformal maps and positive boundary measure. (English) Zbl 0674.30018

A proof for the following theorem is indicated: There is a homeomorphism f of the closed unit ball \(\bar B^ n\) in \(R^ n\), \(n\geq 3\), onto the closure of a domain D such that f is quasiconformal in \(B^ n\) and \(m(\partial D)>0\). For \(n=2\) this can be made conformally with aid of the Riemann mapping theorem. Actually, a weaker result is proved in detail: There is a continuous map f: \(\bar B^ n\to R^ n\) such that (i) \(m(f\partial B^ n)>0\), (ii) f is locally K-quasiconformal in \(B^ n\), and (iii) \(| f'(x)| \in L^ n(B^ n)\). The proof makes use of pulling out spikes from the unit ball in such a way that the transformation is a similarity on some parts of the spikes and then doing a similar construction in these parts. The behavior of the mapping f is quite interesting from a measure theoretical point of view [J. Heinonen and O. Martio, Indiana Univ. Math. J. 36, 659-683 (1987; Zbl 0643.35022)].
Reviewer: O.Martio

MSC:

30C62 Quasiconformal mappings in the complex plane

Citations:

Zbl 0643.35022
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