1549055259-Ubiquitous_Quasidisk__The__Gehring_

46 3. CONFORMAL INVARIANTS

3.9. Harmonic bending

• 2
  Suppose that 'Y is a closed arc with endpoints z 1 and z 2. Then D = R \ 'Y is
  a simply connected domain and there exists a conformal mapping g which maps
  D onto the right half-plane H. Next since D is locally connected at z1 and z2, g
  has a continuous injective extension in DU {z 1 , z2} (Vaisala [163]). Hence we may
  choose g so that g( z 1 ) = 0 and g(z2) = oo. For z ED let

b(z,'Y) =max w(g(z),"/j;H),

J=l,2

where 'Yl and "1 2 are the positive and negative halves of the imaginary axis. Then

b(z, "f)-+1 if and only if Z-+"f \ {z 1 , z2}.

Hence the function b(z, "!) is a conformally invariant measure of the position of the
point z E D with respect to the interior of 'Y which attains its minimum 1/2 on the
preimage of the real axis under g.

If 'Y is a closed subarc of a Jordan curve CC R
2
, then the function b(z,'Y)
measures how much the arc C \ 'Y bends towards 'Y. In particular, if D is a disk or
half-plane, then for each closed arc 'Y C 8D,

for z E 8D \ "f.

1

b(z,"f) =

2

The following observation suggests how the bending function b(z, "!) may be

used to characterize quasidisks.

EXAMPLE 3.9.l. If Dis a sector of angle a and if 'Y is an arc in 8D, then

a a

b(z,"f) _::::: max(-

2

, 1--)
7r 27r
for each z E 8D \ 'Y· The above bound is sharp.

DEFINITION 3.9.2. A Jordan domain D has the harmonic bending property if
there exists a constant c E [~, 1) such that for each closed arc 'Y C 8D,

b(z, "!) _::::: c

for z E 8D \ 'Y·

THEOREM 3.9.3 (Fernandez-Hamilton-Heinonen [41]). A Jordan domain D is
a K -quasidisk if and only if it has the harmonic bending prnpe'T'ty with constant c,
where K and c depend only on each other.
A Jordan domain D is a disk or half-plane if and only if it has the harmonic
bending property with constant c = ~.

3.10. Quadrilaterals

A quadrilateral Q = D(z 1 , z 2 , z 3 , z 4 ) consists of a Jordan domain D c R

2

together with a quadruple of points z 1 , z 2 , z 3 , z 4 E 8D, the vertices of Q, that are
positively oriented with respect to D. The vertices z 1 , z 2 , z 3 , z 4 divide 8D into four
arcs, the sides of Q.
Each such quadrilateral Q can be mapped conformally onto a rectangle R =
R(O, a, a+ i, i) so that the vertices and sides of Q and R correspond. The modulus
of Q is then given by
mod(Q) =a.