1549055259-Ubiquitous_Quasidisk__The__Gehring_

158 11. FOURTH SERIES OF IMPLICATIONS

with s > 0, so that z 1 , w, and w" are distinct points. We get a contradiction as

before. 0

11.4. Extremal distance domains

That quasidisks have the extremal distance property follows immediately from

the following remark and Example 3. 11 .1.

REMARK 11.4.1. If a domain D h as the extremal distance property with con-

stant c and if f : R

2

---+ R

2

is K-quasiconformal, then f(D) has the extremal

distance property with constant K^2 c.

PROOF. Given continua C 1 and C 2 in R

2

and in D , respectively, then f(r)

• 2
  and f(rD) are the families of curves joining f(Ci) and f(C2) in R and in f(D),
  respectively, and
  mod(f(r))::::; Kmod(r)::::; Kcmod(rD)::::; K^2 cmod(f(rD)).
  0

The converse statement that a simply connected domain D with the extremal

distance property is a quasidisk will b e proved by showing that extremal distance

domains must be linearly locally connected. The result then follows from the first

series of implications in Chapter 8.

Our proof of linear local connectivity dep ends on the following lower bound for

t he modulus of a curve family.

LEMMA 11 .4.2. If r is the family of curves which join disjoint continua C 1 and

-2

C2 in R and if z1, W1 E C1 and z2, w2 E C2, then

7f

mod(r) 2: Jr

log4 r

where

r - .,...-lz1 ~-~-,-,-~~---, z2llw1 - w2I

• lzi - w1llz2 - w2I.
  PROOF. By performing a preliminary Mobius transformation we need only con-
  sider the case where z 1 = 0, w 1 = 1, lz 21 = r 2: 1, and w 2 = oo. Then

27f

mod(r) = modRr(r)'

where Rr(r) is the r ing domain bounded by t he intervals [O, 1] and [r, oo], and we
obtain

modRr(r)::::; log16r = 2log4Jr.

See, for example, pages 173 , 55 , 61 in Lehto-Virtanen [117]. 0

Lemma 11.4.2 yields the following lower bound for the modulus of a curve
family r which joins continua C 1 and C 2 in R
2
.

• 2
  LEMMA 11.4.3. Suppose that Ci and C2 are disjoint continua in R and that
  min diam(Cj) 2: a dist( Ci, C2),
  J=l,2

where a > 0. If r is the family of curves which join Ci
1fa
mod(r) 2: 2 a +
1
.

-2

and C2 in R , then