Quasiconformal mappings

In this subsection all mappings are assumed to orientation-preserving, defined between domains (open, connected sets) of the complex plane. We will use for one such mapping. We can define the following differential operators where has partial derivatives:

We define the complex dilatation of at a point , where is differentiable with , by . Observe that (since is orientation-preserving) and .

For a mapping and a point in we can set and (these expressions make sense for small values of ). If is we have that

This equation motivates our definition of quasiconformal mapping.

Definition 1.19   An orientation-preserving homeomorphism between domains of the complex plane is called quasiconformal (qc) if the circular dilatation at ,

is bounded on . If a.e. on then we say that is -quasiconformal.

Another equivalent definition is as follows.

Definition 1.20   An orientation-preserving homeomorphism between domains of the complex plane is called quasiconformal (qc) if has distributional derivatives in , for some , (locally in ), and almost everywhere on , for some satisfying . The map is called -qc for any with .

One can easily compute that a holomorphic mapping is -qc, or equivalently that its complex dilatation is identically 0 (this is simply the Cauchy-Riemann equations, ).

Prop 1.21   A -qc homeomorphism is a biholomorphism.

Prop 1.22   The inverse of a -qc mapping is also -qc. The composition of a -qc mapping and a -qc mapping is a -qc mapping.

The existence of qc mappings on the Riemann sphere with a given dilatation (solutions of the Beltrami equation ) is an important result extensively used in the theory of Teichmüller spaces.

Theorem 1.23   Given any with , there exists a unique quasiconformal homeomorphism of , with dilatation , and fixing , 0 and (pointwise). We will denote this mapping by .

The following stronger result, that the solutions given above depend holomorphically on the dilatation , is known as the measurable Riemann mapping theorem.

Theorem 1.24 (Ahlfors-Bers [3])   For every fixed , the mapping is holomorphic: if depends holomorphically on variables , then is a holomorphic function of the variables .

Pablo Ares Gastesi 2005-08-31