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A Historical Survey of Quasiconformal Mappings

In: Zum Werk Leonhard Eulers

Author

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  • Olli Lehto

    (University of Helsinki, Department of Mathematics)

Abstract

By the classical definition, a quasiconformal mapping is a sense-preserving diffeomorphism of a plane domain onto another plane domain which maps infinitesimal circles onto infinitesimal ellipses with a uniformly bounded ratio of axes. Later it was found preferable to relax a priori differentiability conditions and define a quasiconformal mapping in the plane as a sense-preserving homeomorphism which leaves some conformai invariant quasi — invariant. The most general conformai invariant suitable for this purpose is the module of a path family. The precise requirement for quasiconfrormality is the existence of a fixed constant K such that the module of every path family lying in the domain in which the homeomorphism is considered increases at most K times.

Suggested Citation

  • Olli Lehto, 1984. "A Historical Survey of Quasiconformal Mappings," Springer Books, in: Eberhard Knobloch & Ilppo Simo Louhivaara & Jörg Winkler (ed.), Zum Werk Leonhard Eulers, pages 205-217, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7121-1_12
    DOI: 10.1007/978-3-0348-7121-1_12
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