CHAPTER 1Preliminaries
We collect here some definitions and properties of plane quasiconformal map-
pings. Two basic references for this material are the books by Ahlfors [7] and Lehto
and Virtanen [117], to which we refer the reader for further details. A more recent
title is the book by Astala, Iwaniec, and Martin [16].
In what follows R^2 denotes the Euclidean plane with its usual identification
with the complex plane C. The one-point compactification R
2
= R^2 U {oo} is
equipped with the chordal metric
h( ) 2jz - wj
c z, w = JJzl2 + 1 Jlwl2 + 1 '
where we employ the usual conventions regarding oo.
Let D and D' be subdomains of R
2. We will assume, unless stated otherwise,
-2 - 2 -
that card(R \ D) 2: 2. The exterior of Dis denoted by D* = R \ D. Let B(z, r)
be the open Euclidean disk with center z E R^2 and radius r, and let B be the unit
disk B(O, 1). Finally, H will denote the upper or right half-planes
{z=x+iy:y>O} or {z = x + iy: x > O}.
1.1. Quasiconformal mappings
There are several different ways to view a quasiconformal mapping. Perhaps the
most geometrically intuitive is in terms of the linear dilatation of a homeomorphism.
Suppose that f : D -7 D' is a homeomorphism. For z E D \ { oo, 1-^1 ( oo)} and
0 < r < dist(z, 8D) we let
l1(z,r) = min lf(z) - f(w)J,
lz-wl=r
L1(z, r) = max Jf(z) - f(w)J
lz-wl=r(1.1.1)and call. L1(z, r)
H 1 (z) = hmsup l ( )
r--;O f z, r
the linear dilatation off at z. See Figure 1.1.
Recall that a homeomorphism in R^2 is either sense-preserving or sense-reversing
[117]. Menchoff showed in 1937 [129] that if D, D' c R^2 , a sense-preserving
homeomorphism f: D -7 D' is analytic, and hence conformal, whenever
(1.1.2)
for all but a countable set of z E D.
The following definition for quasiconformality is a natural counterpart of Men-
choff's theorem (Gehring [47]).
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