In mathematical complex analysis, a quasiconformal mapping is a (weakly differentiable) homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Quasiconformal mappings are a generalization of conformal mappings that permit the bounded distortion of angles locally. Quasiconformal mappings were introduced by Grötzsch (1928) and named by Ahlfors (1935), Intuitively, let f : D → D′ be an orientation-preserving homeomorphism between open sets in the plane. If f is continuously differentiable, it is K-quasiconformal if, at every point, its derivative maps circles to ellipses with the ratio of the major to minor axis bounded by K.
Definition Suppose f : D → D′ where D and D′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of f. If f is assumed to have continuous partial derivatives, then f is quasiconformal provided it satisfies the Beltrami equation
for some complex valued Lebesgue measurable μ satisfying sup | μ | < 1 {\displaystyle \sup |\mu |<1} (Bers 1977). This equation admits a geometrical interpretation. Equip D with the metric tensor
d s 2 = Ω ( z ) 2 | d z + μ ( z ) d z ¯ | 2 , {\displaystyle ds^{2}=\Omega (z)^{2}\left|dz+\mu (z)\,d{\bar {z}}\right|^{2},}
where Ω(z) > 0. Then f satisfies (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 f is then called μ-conformal. More generally, the continuous differentiability of f can be replaced by the weaker condition that f be in the Sobolev space W1,2(D) of functions whose first-order distributional derivatives are in L2(D). In this case, f is required to be a weak solution of (1). When μ is zero almost everywhere, any homeomorphism in W1,2(D) that is a weak solution of (1) is conformal. Without appeal to an auxiliary metric, consider the effect of the pullback under f of the usual Euclidean metric. The resulting metric is then given by
| ∂ f ∂ z | 2 | d z + μ ( z ) d z ¯ | 2 {\displaystyle \left|{\frac {\partial f}{\partial z}}\right|^{2}\left|dz+\mu (z)\,d{\bar {z}}\right|^{2}}
which, relative to the background Euclidean metric d z d z ¯ {\displaystyle dz\,d{\bar {z}}} , has eigenvalues
( 1 + | μ | ) 2 | ∂ f ∂ z | 2 , ( 1 − | μ | ) 2 | ∂ f ∂ z | 2 . {\displaystyle (1+|\mu |)^{2}\left|{\frac {\partial f}{\partial z}}\right|^{2},\qquad (1-|\mu |)^{2}\left|{\frac {\partial f}{\partial z}}\right|^{2}.}
The eigenvalues represent, respectively, the squared length of the major and minor axes of the ellipse obtained by pulling back along f the unit circle in the tangent plane. Accordingly, the dilatation of f at a point z is defined by
K ( z ) = 1 + | μ ( z ) | 1 − | μ ( z ) | . {\displaystyle K(z)={\frac {1+|\mu (z)|}{1-|\mu (z)|}}.}
The (essential) supremum of K(z) is given by
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