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
Another equivalent definition is as follows.
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, ).
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.
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