1549055259-Ubiquitous_Quasidisk__The__Gehring_

22 2. GEOMETRIC PROPERTIES

Example 1.4.5 shows that the boundary fJD of a quasidisk D can be an ex-

tremely complicated Jordan curve, one which almost has positive two-dimensional

measure. It is therefore quite surprising that all such curves admit a bilipschitz

reflection if oo E fJ D.

THEOREM 2.1.8 (Ahlfors [5]). !Joo E fJD, then Dis a K-quasidisk if and only

if it admits a Euclidean L-bilipschitz reflection, where K and L depend only on each

other.

The conformal analogue of this result also characterizes t he domains D which

are half-planes. In particular, D is a half-plane if and only if it admits a bilipschitz

reflection with L = 1 (Hag [77]). To see this, let f be a 1-bilipschitz reflection in

fJD. Fix z 1 ED and let ( E fJD, z 1 , ( :f. oo. Then

lf(z1) - (I= lf(z1) - f(()I = lz1 - (I.

Thus fJD C JU{ oo} where J is the perpendicular bisector of the segment [zi, J(z 1 )].

On the other hand, let z E J and consider the broken line 'Y = [z 1 , z] U [z, f(z 1 )].

Then 'Y must meet fJD, while by what we just observed "(llfJD C "(llJ = {(}. This

implies that z E fJD, hence fJD = J and Dis a half-plane.

Finally a third situation concerns the case where the reflection f is bilipschitz

with respect to the hyperbolic metric. If D C R

2

is a simply connected domain,

then there exists a conformal mapping f: D--tB^2 such that f(z 1 ) = 0 and f(z 2 ) = r

where 0 < r < 1. The hyperbolic distance between z 1 and z 2 is then given by

l+r

hD(z1, z2) =log--.

1-r

We will discuss this metric in more detail in Chapter 3. We have given the

above definition now so that we can introduce the notion of a hyperbolic bilipschitz

reflection.

DEFINITION 2.1.9. If D and D' are simply connected domains in R

2

, t hen

f: D--tD' is a hyperbolic L-bilipschitz mapping if

1

(2.1.10) L hD(z1, z2)::::; hD^1 (J(z1), f(z2))::::; LhD(z 1, z2)

for z1, z2 ED.

THEOREM 2.1.11 (Ahlfors [7]). A domain Dis a K-quasidisk if and only if

it admits a hyperbolic L-bilipschitz reflection, where K and L depend only on each

other.

The conformal analogue of this result characterizes the domains D which are

disks or half-planes: D is a disk or half-plane if and only if it admits a hyperbolic

L-bilipschitz reflection with L = 1.

2.2. The three-point condition

If D is a disk or half-plane, then for each pair of points z 1 , z 2 E fJD \ { oo },

min diam('Yj) ::::; lz1 - z2 I

J=l,2

where diam denotes Euclidean diameter and 'Yi, "( 2 are the components of fJD \
{ z1, z2}. More generally if D is a sector of angle a where 0 < a < 7r, then

(2.2.1) min diam('YJ)::::; a lz1 - z2I

J=l,2