1549055259-Ubiquitous_Quasidisk__The__Gehring_

152 11. FOURTH SERIES OF IMPLICATIONS

-2

G (= R \ 1)

s

... ...

..... ---. z

n*

FIGURE 11.3

map h of R
2
\ f ( /) onto H. Without loss of generality we let f ( /) correspond to /1,
the positive half of the imaginary axis. Using an additional Mobius transformation
if necessary, we may also assume that h(f(z) ) = l.
Finally, there exists a K-quasiconformal map J of H fixing 0, i, oo such that
the composition g = Joh of is conformal. This is a consequence of the measurable
Riemann mapping theorem, Theorem 1.1.11, and the composition rules for the
complex dilatation. Hence by symmetry D has the harmonic bending property if

(11.1.7) w(g(z), 11; H) :::; c

where c = c(K). But this follows from the fact that the collection of K-quasicon-
formal maps of H fixing three points on the boundary is a normal family. Hence
there must exist an n EN, n = n(K), such that ](1) is in the rectangle [~, n] x
[-n,n] and
w(g(z),11;H ):::; w(~ +in,11;H) = c.
For sufficiency we assume that oo E C = 8D (which we may do since the
harmonic bending property is Mobius invariant). Then C is a K-quasicircle if
there exists a constant K such that when z 1 , z, and z 2 are three finite points on C
in that order, then
lz1 - zl < Klz1 - z2I·
Moreover, we may assume that the open segment S connecting z 1 and z 2 does
not intersect C. We normalize the situation so that z 1 = 0 and z 2 = 1 and denote

C\ C(z1, z2) by/ and R\ /by Gas in the definition of h armonic bending. Finally,

• 2
  n and n denote the the components of R \ (I U S), and we assume without loss
  of generality that z E n.
  Since D has the harmonic bending property with constant c, we know that
  w(z, an* n 8G; G):::; c.
  Here the left-hand side is the solution of the Dirichlet problem whose boundary
  data is 1 on the n• -side of/ and 0 on the n-side, evaluated at z. Likewise

w(z, an n 8G; G):::; c.

Then

w(z, an n oG; G) :::; w(z, S; n*)