quasiconformal mapping

Quasiconformal mappings are mappings of the complex plane to itself that are “almost” conformal. That is, they do not distort angles arbitrarily and this “distortion” is uniformly bounded throughout their domain of definition. Alternatively one can think of quasiconformal mappings as mappings which take infinitesimal circles to infinitesimal ellipses. For example invertible linear maps are quasiconformal.

More rigorously, suppose $f$ is a mapping of the complex plane to itself, and here we will only consider sense-preserving mappings, that is mappings with a positive jacobian.

Definition.

Define the dilatation of the mapping $f$ at the point $z$ as

$$D_f(z):=\frac{|f_z|+|f_{\overline{z}}|}{|f_z|-|f_{\overline{z}}|}\ge 1,$$

and definethe maximal dilatation of the mapping as

$$K_f:=\underset{z}{sup}{D}_f(z).$$

Now we are ready to define what it means for $f$ to be quasiconformal.

Definition.

For $f$ as above, we will call $f$ quasiconformal if the maximaldilatation of $f$ is finite. We will say that $f$is $K$-quasiconformal mapping if the maximal dilatation of this mapping is $K$.

Note that sometimes the $K$-quasiconformal is used to that the dilatation is $K$ or lower.

It is easy to see that a conformal sense-preserving mapping has a dilatation of $1$ since $|f_{\overline{z}}|=0$. We can further define severalother related quantities

Definition.

For $f$ as above, define the small dilatation as

$$d_f(z):=\frac{|f_{\overline{z}}|}{|f_z|}.$$

Again for sense-preserving maps this quantity is less then 1 and it is equal to 0 if the mapping is conformal. Some authors call a map $k$-quasiconformal if the small dilatation is bounded by $k$. It is however not ambiguous as the large dilatation is always greater then or equal to 1. Furthermore this is related to the large dilatation by

$$d_f:=\frac{D_f-1}{D_f+1}.$$

Definition.

For $f$ as above, define the complex dilatation as

$${\mu }_f(z):=\frac{f_{\overline{z}}}{f_z}.$$

The complex dilatation now appears in the Beltrami differential equation

$$f_{\overline{z}}(z)={\mu }_f(z)f_z(z).$$

This means that a quasiconformal mapping is a solution to the Beltrami equation where a non-negative measurable ${\mu }_f$ is uniformly bounded by some .

The above results are stated for $f:ℂ\to ℂ$, butthe statements are exactly the same if you take $f:G\subset ℂ\to ℂ$ for an open set $G$.

The theory generalizes to other dimensions as well. For example in one real dimension, the analogous mappings are called quasisymmetric. It is a well-known theorem of Beurling and Ahlfors (http://planetmath.org/BeurlingAhlforsQuasiconformalExtension) that an of a mapping of the real line to itself is quasiconformal if and only if the mapping is quasisymmetric.