Quasiconformal Mapping - Definition

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 the Beltrami equation

(1)

for some complex valued Lebesgue measurable μ satisfying sup |μ| < 1 (Bers 1977). This equation admits a geometrical interpretation. Equip D with the metric tensor

where Ω(z) > 0. Then ƒ 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 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 W1,2(D) of functions whose first-order distributional derivatives are in L2(D). In this case, ƒ 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 ƒ of the usual Euclidean metric. The resulting metric is then given by

which, relative to the background Euclidean metric, has eigenvalues

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

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

and is called the dilatation of ƒ.

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

If ƒ is K-quasiconformal for some finite K, then ƒ is quasiconformal.