- Quasiconformal mapping
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 homeomorphismis 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 at every point maps circles to ellipses with eccentricity bounded by "K".
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
for some complex valued
Lebesgue measurableμ satisfying sup |μ| < 1 harv|Bers|1977. This equation admits a geometrical interpretation. Equip "D" with the metric tensor
where Ω("z") > 0. Then ƒ satisfies (EquationNote|1) precisely when it is a conformal transformation from "D" equipped with this metric to the domain "D"′ equipped with the standard Euclidean metric. The function ƒ is then called μ-conformal. More generally, the continuous differentiability of ƒ can be replaced by the weaker condition that ƒ be in the
Sobolev space"W"1,2("D") of functions whose first-order distributional derivatives are in L2("D"). In this case, ƒ is required to be a weak solutionof (EquationNote|1). When μ 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 ƒ of the usual Euclidean metric. The resulting metric is then given by
which, relative to the background Euclidean metric , has
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 ƒ at a point "z" is defined by
supremumof "K"("z") is given by
and is called the dilatation of ƒ.
A definition based on the notion of
extremal lengthis as follows. If there is a finite such that for every collection of curves in the extremal length of is at most times the extremal length of . Then is -quasiconformal.
If is -quasiconformal for some finite , then is quasiconformal.
A few facts about quasiconformal mappings
Conformal homeomophisms are -quasiconformal and conversely, a 1-quasiconformal homeomorphism is conformal.
The map is -quasiconformal.
The map is quasiconformal if (here is a complex number). This is an example of a quasiconformal homeomorphism that is not smooth.
If is quasiconformal and is quasiconformal, then is quasiconformal.
The inverse of a -quasiconformal homeomorphism is -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 theoremfrom 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 is Lebesgue measurableand satisfies . Then there is a conformal homeomorphism ƒ 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 ƒ is unique up to 3 real parameters.
*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.
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