34 3. CONFORMAL INVARIANTS
conformal type of I: is determined by
N= 2m+n-3
real numbers. The three cases where N = 1, together with a natural real number
or conformal invariant that determines the conformal equivalence class of I:, are as
follows. See Ahlfors [8].
1° Two interior points z 1 , z 2. The conformal invariant is the hyperbolic dis-
tance hD(z 1 , z 2 ) between z 1 and z2.
2° One interior point z 1 and two boundary points w 1 , w2. The conformal in-
variant is the harmonic measure w(z 1 , 1; D) where/ is one of the boundary
arcs with endpoints w 1 , w 2.
3° Four boundary points w 1 , ... , w 4. The conformal invariant is the modulus
of the quadrilateral Q = D with vertices at w 1 , ... , w 4.vVe shall describe in this chapter how each of these invariants can be used to char-
acterize the class of quasidisks.
3.2. Hyperbolic geometryWe begin by defining in each simply connected domain D C R
2
a conformally
invariant distance, the hyperbolic metric, or hyperbolic distance, hD.
The hyperbolic density in the unit disk B is defined by
2
PB(z) = 1 - lzl2
for z E B. Next for each z 1 , z 2 E B the hyperbolic distance between t hese points is
given by
hB(z1, z2) =inf r PB(z)idzl,
f3 J f3
where the infimum is taken over all rectifiable curves (3 which join z 1 and z 2 in B.
Then there exists a unique arc / such that
hB(z1, z2) = !, PB(z)idzi.
The arc / lies in the circular crosscut (3 of B which passes through z 1 , z 2 and is
orthogonal to 8B. We call / the hyperbolic segment joining z 1 and z 2 and we call
(3 the hyperbolic line which contains/. It is not difficult to show that
h ( )-l (ll-.Z 1 z2l+lz1- z2I)
B z1, z2 - og ll - z1_ z2 I - I z1 - z2 I.The hyperbolic density in a simply connected domain D is given byPD(z) = PB(g(z))ig'(z)i,
where g is any conformal mapping of D onto B. It follows from the Schwarz lemma
that PD is independent of the choice of g. Then the hyperbolic distance between
z 1 , z2 ED is given by