The cardinality of quasiconformally nonequivalent topological 3-balls with flat boundaries is 𝔠

Näkki, Raimo

doi:10.1090/S0002-9939-1979-0529214-6

The cardinality of quasiconformally nonequivalent topological $3$-balls with flat boundaries is $\mathfrak {c}$
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Proc. Amer. Math. Soc. 75 (1979), 63-68 Request permission

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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