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
+ h)- f(z)\
(2.1) Hf(z) = hmsup ^ , 77^-
^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)\
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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