1549055259-Ubiquitous_Quasidisk__The__Gehring_

6 1. PRELIMINARIES

If f : D-t D' and g : D'---+ D" are both sense-preserving and quasiconformal,
then μ 901 = μ! a.e. in D if and only if g is conformal.
It is possible to prescribe the complex dilatation μ1(z), and hence the linear
dilatation H 1 (z), at almost every point z of a domain D. This result, known as
the measurable Riemann mapping theorem, has turned out to be a powerful tool in
complex analysis. See Ahlfors-Bers [9], Lehto-Virtanen [117], Morrey [133], and
Bojarski [25].

THEOREM 1.1.11. Ifμ is measurable with

llμllL= = esssup lμ(z)I < 1,

D

then there exists a sense-preserving quasiconformal mapping f of D with μ f = μ

a. e. in D. Moreover f is unique up to post composition with a conformal map.

1.2. Modulus of a curve family
The conditions for quasiconformality in Definition 1.1.3 and Theorem 1.1.
involve the local behavior of a homeomorphism. We need a way to integrate the
inequality in Theorem 1.1.8 in order to derive globa l properties of the mapping.
When K = 1, f or its complex conjugate J is conformal and the Cauchy integral
formula is available. The tool most often used to replace the Cauchy formula when
K > 1 is the method of extremal length, first formulated by Ahlfors and Beurling
in [23].

Suppose that r is a family of curves in R

2

. We say that p is an admissible
density for r, or is in adm(r), if pis nonnegative and Borel measurable in R^2 and
if

!, p(z)ldzl 2: 1

for each locally rectifiable / E r. The modulus and extremal length of the family r
are then given, respectively, by

mod(r) =inf r p(z)^2 dm

p }R 2

and

where the infimum is taken over p E adm(r).

1

.A(r) = mod(r)'

THEOREM 1.2.1. If f : D ---+ D' is conformal and if r is a family of curves in
D , then

mod(f(r)) = mod(r).

PROOF. We consider the case where D , D' c R^2. For each p' E adm(f(r)) let

z _ { p'(f(z))lf'(z) I if z ED,

p( ) - 0 if z E R^2 \ D.

Then p is nonnegative and Borel measurable in R^2. If I is locally rectifiable, then
f (t) E f (r) is locally rectifiable and

1

p(z)ldzl=1 p'(f(z))lf'(z )lldzl = r p'(w)ldwl 2: l.

"I "I J f("!)