1549055259-Ubiquitous_Quasidisk__The__Gehring_

52 3. CONFORMAL INVARIANTS

Example 3.11.8 is a reformulation of the level-set inequality, first established

by Hayman and Wu [79] with an absolute constant c in place of 2 in (3.11.9).

0yma [142, 143] later showed that (3.11.9) holds with the constant 2, and Rohde

[152] proved that in fact a strict inequality holds in (3.11.9). See also Fernandez-

Heinonen-Martio [42], Flinn [43], Garnett-Gehring-Jones[46], Garnett-Marshall

[45].

3.12. Quadrilaterals and harmonic quasisymmetry

The two characterizations for a quasidisk D involving the conjugate quadrilat-

eral inequality and harmonic quasisymmetry compare the conformal geometry of

a Jordan domain D with that of its exterior D*. In particular they compare the

moduli of the quadrilaterals in D and D* determined by an ordered quadruple of

points on their common boundary fJD and the harmonic measures of two adjacent

arcs in fJD evaluated at interior points in D and D*.

We conclude this chapter by showing that these two characterizations are equiv-

alent. We do this by replacing the two Jordan domains D and D* by a single do-

main, the unit disk B , the pair of interior points in D and D* by the origin 0 E B ,

and the common boundary fJD = fJD* by a self-homeomorphism h of 8B. More

specifically we choose conformal mappings f : D -+ B and g : D* -+ B and let h

denote the induced sewing homeomorphism

(3.12.1) h = g o r^1 : 8B -+ fJB.

As we shall see, relations between D and D* are then encoded in properties of

the sewing homeomorphism h. For this we look at the cross ratios and quasisym-

metry of points on the unit circle 8B. In particular we assume in what follows that

¢is a self-homeomorphism of 8B and that z 1 , z 2 , z 3 , z 4 is an ordered quadruple of

points in 8B.

We establish two lemmas concerning cross ratios which appear in a more general

setting in a paper by Viiisiila [164].

LEMMA 3.12.2. Suppose that¢ fixes the points i, -1, -i and that a is a constant

with a~ 2. If

(3.12.3)

(3.12.4)

PROOF. We begin by determining a lower bound for

needed later. Since [1, i, -1, -i] = [i, - 1, -i, l] = 2,

1¢(1) ±ii that will be

1¢(1) + 11 Ii+ ii Ii+ ii 11 + ¢(1)1

1¢(1) +ii Ii+ ll ::; a, Ii - ¢(1)1 -11---il-::; a

by (3.12.3) and we obtain

(3.12.5) 1¢(1) ±ii ~ ~

a

since 1¢(1) + ll > v12.