# Quasiconformal mapping

﻿
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 &fnof;:"D" &rarr; "D"&prime; 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 &fnof;:"D" &rarr; "D"&prime; where "D" and "D"&prime; are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of &fnof;. If &fnof; is assumed to have continuous partial derivatives, then &fnof; 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

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

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

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

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

:$K = sup_\left\{zin D\right\} |K\left(z\right)| = frac\left\{1+|mu|_infty\right\}\left\{1-|mu|_infty\right\}$

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 $\left\{fcircgamma:gammainGamma\right\}$. 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 $\left(x,y\right)mapsto\left(2x,y\right)$ is $2$-quasiconformal.

The map $zmapsto z,|z|^\left\{s\right\}$ 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 $|mu|_infty<1$. 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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Measurable Riemann mapping theorem — In the mathematical theory of quasiconformal mappings in two dimensions, the measurable Riemann mapping theorem, proved by Morrey (1936, 1938), generalizes the Riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is… …   Wikipedia

• Muckenhoupt weights — In mathematics, the class of Muckenhoupt weights Ap consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on Lp(dω). Specifically, we consider functions f on and their associated maximal functions M(f) defined as… …   Wikipedia

• Extremal length — In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves Gamma is a conformal invariant of Gamma. More specifically, suppose thatD is an open set in the complex plane and Gamma is a… …   Wikipedia

• Juha Heinonen — (* 23. Juli 1960 in Toivakka, Finnland; † 30. Oktober 2007)[1] war ein finnischer Mathematiker. Heinonen, 1989 Inhaltsverzeichnis …   Deutsch Wikipedia

• Lars Ahlfors — Infobox Scientist name = Lars Ahlfors image width = caption = Lars Ahlfors birth date = birth date|1907|04|18 birth place = Helsinki, Finland death date = death date and age|1996|10|11|1907|04|18 death place = nationality = Finland field =… …   Wikipedia

• List of mathematics articles (Q) — NOTOC Q Q analog Q analysis Q derivative Q difference polynomial Q exponential Q factor Q Pochhammer symbol Q Q plot Q statistic Q systems Q test Q theta function Q Vandermonde identity Q.E.D. QED project QR algorithm QR decomposition Quadratic… …   Wikipedia

• Geometric group theory — is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the… …   Wikipedia

• Eric L. Schwartz — (born 1947) is Professor of Cognitive and Neural Systems, [ [http://cns web.bu.edu/people/people.html people ] ] Professor of Electrical and Computer Engineering, [ [http://www.bu.edu/dbin/ece/web/people/faculty.php Boston University ECE… …   Wikipedia

• James W. Cannon — (b. January 30, 1943) is an American mathematician working in the areas of low dimensional topology and geometric group theory. He is an Orson Pratt Professor of Mathematics at the Brigham Young University.Biographical dataJames W. Cannon was… …   Wikipedia