Quasiconformal mapping

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In mathematics, the concept of quasiconformal mapping, introduced as a technical tool in complex analysis, has blossomed into an independent subject with various applications. Informally, a conformal homeomorphism is a homeomorphism between plane domains which to first order takes small circles to small circles. A quasiconformal homeomorphism to first order takes small circles to small ellipses of bounded eccentricity.

[edit] Definition

Let $f:D\to D'$ be an orientation preserving homeomorphism between open sets in the plane. If $f$ is smooth, then it is $K$ -quasiconformal if the derivative of $f$ at every point maps circles to ellipses with eccentricity bounded by $K$ . However, some $K$ -quasiconformal maps are non-smooth. There are various equivalent definitions for $K$ -quasiconformality which apply to not necessarily smooth mappings, but each of them is somewhat technical.

A definition based on the notion of extremal length is as follows. If there is a finite $K$ such that for every collection $Γ$ of curves in $D$ the extremal length of $Γ$ is at most $K$ times the extremal length of $\{f\circ\gamma:\gamma\in\Gamma\}$ . Then $f$ is $K$ -quasiconformal.

If $f$ is $K$ -quasiconformal for some finite $K$ , then $f$ is quasiconformal.

[edit] A few facts about quasiconformal mappings

Conformal homeomophisms are $1$ -quasiconformal.

The map $(x,y)\mapsto(2x,y)$ is $2$ -quasiconformal.

The map $z\mapsto z\,|z|^{s}$ is quasiconformal if $s > - 1$ (here $z$ is a complex number). This is an example of a quasiconformal homeomorphism that is not smooth.

If $f:D\to D'$ is $K$ quasiconformal and $g:D'\to D''$ is $K'$ quasiconformal, then $g\circ f$ is $K\,K'$ quasiconformal.

The inverse of a $K$ -quasiconformal homeomorphism is $K$ -quasiconformal.

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[edit] References

Heinonen, Juha; What Is ... a QuasiconformalMapping?, Notices of the American Mathematical Society; vol. 53, no. 11 (December 2006)
Lehto, O. & Virtanen, K. I. (1973), Quasiconformal mappings in the plane (2nd ed.), Berlin, New York: Springer-Verlag