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The cardinality of quasiconformally nonequivalent topological $ 3$-balls with flat boundaries is $ \mathfrak{c}$


Author: Raimo Näkki
Journal: Proc. Amer. Math. Soc. 75 (1979), 63-68
MSC: Primary 30C60
DOI: https://doi.org/10.1090/S0002-9939-1979-0529214-6
MathSciNet review: 529214
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Abstract: The theorem mentioned in the title is proved. During the course of the proof, the failure for $ n = 3$ of the following 2-dimensional result will also be established: The boundary of a Jordan domain D in n-space is a quasiconformal $ (n - 1)$-sphere if every quasiconformal self-mapping of D can be extended to a quasiconformal self-mapping of the whole space.


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  • [1] M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 0117695 (22:8470b)
  • [2] -, Locally flat imbeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331-341. MR 0133812 (24:A3637)
  • [3] F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393. MR 0139735 (25:3166)
  • [4] -, Extension theorems for quasiconformal mappings in n-space, J. Analyse Math. 19 (1967), 149-169. MR 0215987 (35:6822)
  • [5] F. W. Gehring and J. Väisälä, The coefficients of quasiconformality of domains in space, Acta Math. 114 (1965), 1-70. MR 0180674 (31:4905)
  • [6] A. P. Kopylov, On the richness of the class of quasiconformal mappings of domains in three-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 172 (1967), 527-528. (Russian) MR 0207981 (34:7793)
  • [7] R. Näkki, Extension of quasiconformal mappings between wedges, Proceedings of the Romanian-Finnish seminar on Teichmüller spaces and quasiconformal mappings, Brašov, 1971, pp. 229-233. MR 0308392 (46:7506)
  • [8] S. Rickman, Extension over quasiconformally equivalent curves, Ann. Acad. Sci. Fenn. AI 436 (1969), 1-12. MR 0245791 (39:7097)
  • [9] A. V. Syčev, Quasiconformal mappings in space, Dokl. Akad. Nauk SSSR 166 (1966), 298-300. (Russian) MR 0200444 (34:338)
  • [10] J. Väisälä, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin and New York, 1971. MR 0454009 (56:12260)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0529214-6
Keywords: Cardinality, quasiconformal mapping, modulus of a curve family, topological ball, quasiconformal sphere, flat boundary
Article copyright: © Copyright 1979 American Mathematical Society

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