##

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.

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*.

Pablo Ares Gastesi
2005-08-31