Quasiconformal mapping

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.

Intuitively, let ƒ:"D" → "D"′ be an orientation preserving homeomorphism between open sets in the plane. If "f" is continuously differentiable, then it is "K"-quasiconformal if the derivative of $f$ at every point maps circles to ellipses with eccentricity bounded by "K".

Definition

Suppose ƒ:"D" → "D"′ where "D" and "D"′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of ƒ. If ƒ is assumed to have continuous partial derivatives, then ƒ is quasiconformal provided it satisfies Beltrami's equation

for some complex valued Lebesgue measurable &mu; satisfying sup |&mu;| < 1 harv|Bers|1977. This equation admits a geometrical interpretation. Equip "D" with the metric tensor

:

where &Omega;("z") > 0. Then &fnof; satisfies (EquationNote|1) precisely when it is a conformal transformation from "D" equipped with this metric to the domain "D"&prime; equipped with the standard Euclidean metric. The function &fnof; is then called &mu;-conformal. More generally, the continuous differentiability of &fnof; can be replaced by the weaker condition that &fnof; be in the Sobolev space "W"^1,2("D") of functions whose first-order distributional derivatives are in L²("D"). In this case, &fnof; is required to be a weak solution of (EquationNote|1). When &mu; is zero almost everywhere, any homeomorphism in "W"^1,2("D") that is a weak solution of (EquationNote|1) is conformal.

Without appeal to an auxiliary metric, consider the effect of the pullback under &fnof; of the usual Euclidean metric. The resulting metric is then given by

:

which, relative to the background Euclidean metric , has eigenvalues

:$(1+|mu|)^2\; extstyle\{left|frac\{partial\; f\}\{partial\; z\}\; ight|^2\},qquad\; (1-|mu|)^2\; extstyle\{left|frac\{partial\; f\}\{partial\; z\}\; ight|^2\}.$

The eigenvalues represent, respectively, the squared length of the major and minor axis of the ellipse obtained by pulling back along "f" the unit circle in the tangent plane.

Accordingly, the "dilatation" of &fnof; at a point "z" is defined by

:$K(z)\; =\; frac\{1+|mu(z)\{1-|mu(z).$

The (essential) supremum of "K"("z") is given by

:$K\; =\; sup\_\{zin\; D\}\; |K(z)|\; =\; frac\{1+|mu|\_infty\}\{1-|mu|\_infty\}$

and is called the dilatation of &fnof;.

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

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

A few facts about quasiconformal mappings

Conformal homeomophisms are $1$-quasiconformal and conversely, a 1-quasiconformal homeomorphism is conformal.

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

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

If $f:D\; o\; D\text{'}$ is $K$ quasiconformal and $g:D\text{'}\; o\; D"$ is $K\text{'}$ quasiconformal, then $gcirc\; f$ is $K,K\text{'}$ quasiconformal.

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

Measurable Riemann mapping theorem

Of central importance in the theory of quasiconformal mappings in two dimensions is the measurable Riemann mapping theorem, proved by harvtxt|Morrey|1938. The theorem generalizes the Riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. Suppose that "D" is a simply connected domain in C that is not equal to C, and suppose that $mu:D\; o\; mathbf\; C$ is Lebesgue measurable and satisfies . Then there is a conformal homeomorphism &fnof; from "D" to the unit disk which is in the Sobolev space "W"^1,2("D") and satisfies the corresponding Beltrami equation (EquationNote|1) in the distributional sense. As with Riemann's mapping theorem, this &fnof; is unique up to 3 real parameters.

"n"-dimensional generalization

References

*citation | first=Lars V.|last=Ahlfors | authorlink=Lars Ahlfors | title=Lectures on Quasiconformal mappings | publisher=van Nostrand | year=1966
*citation|title=Quasiconformal mappings, with applications to differential equations, function theory and topology |first=Lipman|last=Bers |authorlink=Lipman Bers|journal=Bull. Amer. Math. Soc.|volume=83|issue=6|year=1977|pages=1083-1100|MR|id=0463433|url=http://ams.org/bull/1977-83-06/S0002-9904-1977-14390-5/home.html
*
*citation | first1=O.|last1=Lehto | first2=K.I.|last2=Virtanen | title=Quasiconformal mappings in the plane | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd ed | year=1973
*citation|title=On the Solutions of Quasi-Linear Elliptic Partial Differential Equations|first=Charles B. Jr.|last=Morrey|journal=Transactions of the American Mathematical Society|volume=43|number=1|year=1938|pages=126-166|url=http://www.jstor.org/stable/1989904.