Quasiconformal mapping

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 ${\displaystyle sup|\mu |<1}$ (Bers 1977). This equation admits a geometrical interpretation. Equip D with the metric tensor

${\displaystyle d{s}^2=\mathrm{\Omega }(z{)}^2{\left|dz+\mu (z)d\overline{z}\right|}^2,}$

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 W^1,2(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 W^1,2(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

${\displaystyle {\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\right|}^2{\left|dz+\mu (z)d\overline{z}\right|}^2}$

which, relative to the background Euclidean metric ${\displaystyle dzd\overline{z}}$, has eigenvalues

${\displaystyle (1+|\mu |{)}^2{\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\right|}^2,(1-|\mu |{)}^2{\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\right|}^2.}$

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

${\displaystyle K(z)=\frac{1+|\mu (z)|}{1-|\mu (z)|}.}$

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

${\displaystyle K=\underset{z\in D}{sup}|K(z)|=\frac{1+\Vert \mu {\Vert }_{\mathrm{\infty }}}{1-\Vert \mu {\Vert }_{\mathrm{\infty }}}}$

and is called the dilatation of f. If f is K-quasiconformal for some finite K, then f is quasiconformal.

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 o γ : γ ∈ Γ}. Then f is K-quasiconformal.