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 ${\displaystyle G}$ ${\displaystyle H}$

${\displaystyle f\colon G\longrightarrow H}$

is called quasi-conformal if there is a positive real number less than 1 such that ${\displaystyle k}$

${\displaystyle \|\mu \|_{\infty }<k}$

applies. It is

${\displaystyle \mu ={\frac {f_{\bar {z}}}{f_{z}}}={\frac {\partial _{\bar {z}}f}{\partial _{z}f}}}$

the complex dilation , also called the Beltrami coefficient .

The dilatation of f at point z is defined as

${\displaystyle K(z)={\frac {1+|\mu (z)|}{1-|\mu (z)|}}.}$

The supremum

${\displaystyle K=\sup _{z\in D}|K(z)|={\frac {1+\|\mu \|_{\infty }}{1-\|\mu \|_{\infty }}}}$

is the dilation of  f .

Beltrami equation

Let k be a positive real number less than 1. The partial differential equation

${\displaystyle \partial _{\bar {z}}f=\mu (z)\partial _{z}f,}$

where is an integrable function with is called the Beltrami equation. ${\displaystyle \mu (z)}$ ${\displaystyle \|\mu \|_{\infty }<k}$

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