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Continuously removable sets for quasiconformal mappings. (Russian, English) Zbl 1080.30022

Zap. Nauchn. Semin. POMI 314, 213-220, 289 (2004); translation in J. Math. Sci., New York 133, No. 6, 1728-1731 (2006).
Null sets for the extremal distance, NED-sets, play an important role in the theory of quasiconformal mappings [L. Ahlfors, A. Beurling, Acta Math.83, 101–129 (1950; Zbl 0041.20301)]. The authors employ a slightly different definition: A compact set \(F\) in \({\mathbb R}^n\) is called an NCS-set if each point \(x_0 \in F\) has \(r_0 > 0\) such that \(M(\gamma_0,\gamma_1; B(x_0,r) \setminus F) = \infty\) for all \(0 < r < r_0\) whenever \(\gamma_i: (0,1) \to B(x_0,r) \setminus F,\;i = 0,1\), are disjoint curves with \(x_0 \in \overline{\gamma}_0 \cap \overline{\gamma}_1\). Here \(M\) refers to the \(n\)-modulus of all curves joining \(\gamma_0\) and \(\gamma_1\) in \(B(x_0,r) \setminus F\). The properties of NCS-sets are similar to those of NED–sets. If \(f: D \setminus F \to {\mathbb B}^n\) is a homeomorphism with a finite \(n\)–Dirichlet integral in \(D \setminus F\) and if \(F \subset D\) is an NCS-set, then \(f\) extends continuously to \(F\). This result is used to study removable sets for quasiconformal mappings. The authors also show that an NED set is an NCS-set and, conversely, that an NCS-set with finite \({\mathcal H}^{n-1}\)-measure is an NED-set.

MSC:

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

Citations:

Zbl 0041.20301
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