Quasiconformal mapping

In the theory of functions a quasiconformal mapping is a generalization of a biholomorphic mapping. Here is dispensed primarily to the conformality.

Definition

Let and be two areas of the complex plane. A homeomorphism

Is quasi -compliant if it is a positive real number less than 1, so that

Applies. It is

The complex dilatation, also called the Beltrami coefficient.

Dilation of f at the point z is defined as

The supremum

Is the dilation of f

Beltrami equation

Let k be a positive real number less than 1, the partial differential equation

With an integrable function with, say Beltrami equation.

Law

Applies to the Riemannian ball number that the solutions of the Beltrami equation, the quasi -conformal mappings are accurate.

As an application of this theorem, one can show that all almost complex structures on the 2-sphere and on all other two-dimensional manifolds are integrable, ie, all almost complex structures are complex structures.