# 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}

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}

## 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