CHAPTER 2

Quasiconformal Mappings

Let fl and Q; be planar domains, / : Q — flf a homeomorphism and let

z = x + iy £ Q and r dist(z, dfi). Quasiconformal mappings are principally

mappings of "bounded distortion". If we wish to measure the distortion of / at z

it is natural to introduce the quantity

/oi \ u

(

\ v m&xlh\—\f{z

=r

+ h)- f(z)\

(2.1) Hf(z) = hmsup — ^ , 77^-

r

^o mmw=r\f(z + h)-f{z)\

This is the linear distortion function. If / is conformal, then Hf(z) = 1. Indeed the

converse is also true. This reflects the fact that infinitesimally conformal mappings

preserve angles and "roundness". If / is a diffeomorphism, then it is easy to see

that Hf(z) is finite, but not necessarily uniformly bounded as z approaches dft. A

diffeomorphism f : Q — Q' has bounded distortion if Hf (z) K for all z G ft and

some constant 1 K 00.

2.1. Analytic Definition of Quasiconformality

Unfortunately this geometric definition, while aesthetically pleasing, is difficult

to work with. These days the following analytic definition of quasiconformality is

more common.

Definition. A homeomorphism / : ft — ft' is called K-quasiconformal if / lies in

the Sobolev class W^(fi , C), of complex valued functions whose first order partial

derivatives are locally square integrable, and if its directional derivatives satisfy

(2.2) msx\daf(z)\ K mm\da f(z)\

a

oc

for almost every z G ft.

We point out that there is no gain in generality if we simply assume that / G

Wl£ (f2, C). This is because a homeomorphism of this Sobolev class has locally inte-

grable Jacobian and so the distortion inequality (2.2) implies that / G WZo'c (fi, C).

It is not difficult to see the relationship between quasiconformality and bounded

distortion for diffeomorphisms. It was in this setting that planar quasiconformal

mappings were first studied around 1928 by Grotzsch [45]. The term "quasicon-

formal" was coined by Ahlfors in 1935 [1, 2] when this class of mappings proved

an integral tool in his geometric development of Nevanlinna theory, based on the

"length-area" method. Teichmuller found a fundamental connection between quasi-

conformal mappings and quadratic differentials in his studies of extremal mappings

between Riemann surfaces [105] around 1939. Developments of the length area

method lead to the definition of quasiconformal mappings in terms of the distor-

tion of the modulus of curve families by Pfliiger [93] and then were systematically

studied in their own right by Ahlfors in 1953 [4].

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