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Martin compactifications and quasiconformal mappings. (English) Zbl 0575.30017

Given a quasiconformal mapping T of a Riemann surface \(R_ 1\) onto another \(R_ 2\), the question arises whether or not T can be extended to a homeomorphism between compactifications of the given surfaces. If one takes the Royden compactification of each Riemann surface then it is well-known that T has such an extension. In 1958 H. L. Royden [Ann. Acad. Sci. Fenn., Ser. AI 249/5, 13 p. (1958; Zbl 0081.300)] posed the question of whether or not a quasiconformal mapping T of a Riemann surface \(R_ 1\) onto another \(R_ 2\) can be extended to a homeomorphism of the Martin compactification of \(R_ 1\) onto that of \(R_ 2\). The authors deftly answer this question in the negative by taking plane regions for \(R_ 1\) and \(R_ 2\) which were first considered by A. Ancona in another context. As the authors point out, the analogous question for other compactifications such as the Kuramochi and Wiener compactifications still seems to be open.
Reviewer: J.L.Schiff

MSC:

30C62 Quasiconformal mappings in the complex plane
30F25 Ideal boundary theory for Riemann surfaces

Citations:

Zbl 0081.300
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References:

[1] Alano Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 169 – 213, x (French, with English summary). · Zbl 0377.31001
[2] Alano Ancona, Une propriété de la compactification de Martin d’un domaine euclidien, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, ix, 71 – 90 (French, with English summary). · Zbl 0589.31002
[3] Corneliu Constantinescu and Aurel Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 32, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963 (German). · Zbl 0112.30801
[4] O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band, Springer-Verlag, Berlin-New York, 1965 (German). · Zbl 0138.30301
[5] H. L. Royden, Open Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A. I, no. 249/5 (1958), 13. · Zbl 0081.30003
[6] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.40603
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