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\=r\f{z + 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]. 5
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