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A condition of quasiconformal extendability. (English) Zbl 0929.30015

The authors investigate conditions for a quasiconformal map to be extended. They define some closed set \(E\) in the complex plane \(\mathbb{C}\) which is said to be annularly coarse and prove the following theorem: Suppose that \(f\) is a quasiconformal map of the complement of an annularly coarse set \(E\) into \(\mathbb{C}\). Then \(f\) has a quasiconformal extension to \(\widehat\mathbb{C}\). Moreover, the dilation of the extension agrees with the dilatation of \(f\).

MSC:

30C62 Quasiconformal mappings in the complex plane
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
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References:

[1] L. Ahlfors: Lectures on Quasiconformal Mappings. Van-Nostland (1966). · Zbl 0138.06002
[2] L. Ahlfors and A. Beurling: Conformal invariants and function-theoretic null-sets. Acta Math., 83 , 101-129 (1950). · Zbl 0041.20301
[3] J. Heinonen and P. Koskela: Definition of quasiconformality. Invent. Math., 120 , 61-79 (1995). · Zbl 0832.30013
[4] Y. Imayoshi and M. Taniguchi: An Introduction to Teichmüller Spaces. Springer-Verlag (1991). · Zbl 0754.30001
[5] Y. Kusunoki: Theory of Abelian integrals and its applications to conformal mappings. Mem. Coll. Sci. Univ. Kyoto (Math.), 32 , 235-258 (1959); 33 , 429-433 (1961). · Zbl 0091.07201
[6] O. Lehto and K. Virtanen: Quasiconformal Mappings in the Plane. Springer-Verlag (1973). · Zbl 0267.30016
[7] L. Sario and M. Nakai: Classification Theory of Riemann Surfaces. Springer-Verlag (1970). · Zbl 0199.40603
[8] L. Sario and K. Oikawa: Capacity Functions. Springer-Verlag (1969). · Zbl 0184.10503
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