# Quasi-conformal figure

In function theory , a **quasi-conformal mapping is** a generalization of a biholomorphic mapping. The angular accuracy is essentially dispensed with here.

## definition

Let and be two domains of the complex plane of numbers. A homeomorphism

is called **quasi-conformal** if there is a positive real number less than 1 such that

applies. It is

the **complex dilation** , also called the **Beltrami coefficient** .

The *dilatation* of *f* at 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

where is an integrable function with is called the Beltrami equation.

## main clause

On the Riemann number sphere , the solutions of the Beltrami equation are exactly the quasi-conformal mappings.

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, i.e. that is, all almost complex structures are complex structures.

## literature

- CB Morrey:
*On the solutions of quasilinear elliptic partial differential equations*. Trans. Amer. Math. Soc., Vol. 43, 1938, pages 126-166. - V. Gol'dshtein, Yu. G. Reshet'nyak:
*Quasiconformal mappings and Sobolev spaces*. Kluwer, 1990 (translated from Russian). - A. Bejancu:
*Quasi-conformal mapping*. In: Hazewinkel, Michiel:*Encyclopaedia of Mathematics*. Springer, 2001, ISBN 978-1556080104 . - Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zurich, doi : 10.4171 / 029 , ISBN 978-3-03719-029-6 , MR2284826
- Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zurich, doi : 10.4171 / 055 , ISBN 978-3-03719-055-5 , MR2524085