3.8. HARMONIC QUASISYMMETRY 43For example, if D is the right half-plane and/ is the arc in fJD with endpoints
i a, i b where 0 :<:::: a < b, then(3. 7.2) w(l, /; D) = w( -1, 1; D*) = .!_ ( arctan b - arctan a).
7r
For more material on harmonic measure we refer the reader to the book by
Garnett and Marshall [45].
The formula in (3.7.1) defines the harmonic measure w(z) = w(z, C; B) for
any measurable set C c fJB. With a proper interpretation, harmonic measure is
a conformal invariant. It is somewhat surprising that harmonic measure is not a
"quasi-invariant" like the modulus of a curve family under quasiconformal mappings
(Theorem 1.2.2). Indeed, by a result of Beurling and Ahlfors [23] there is a K-
quasiconformal mapping f: R^2 -+ R^2 with K < 1 + E, for any E > 0, and a
compact set C C fJB such that f (B) = B and w(z, C ; B) > 0 for all z E B but
w(w, f(C); B) = 0 for every w EB. However, the situation changes if we consider
arcs on the boundary of a quasidisk.
We describe two ways of characterizing quasidisks in terms of harmonic mea-
sure.
1 ° Harmonic quasisymmetry. Here we compare the harmonic measures of
adjacent arcs 11 and 12 in fJD at a pair of fixed points z 0 and z(j in D and
its exterior D*.
2° Harmonic bending. This characterization measures in a conformally in-
variant way how much each boundary arc / C fJD bends towards the
complementary arc fJD \ f.3.8. Harmonic quasisymmetry
This characterization is based on the following symmetry property of a disk or
half-plane.
THEOREM 3.8.1 (Hag [77]). A Jordan domain D is a disk or half-plane if and
only if there exist points z 0 E D , z 0 E D* such that if 11 , 12 are adjacent arcs in
fJD with.(3.8.2)
then
(3.8.3)
PROOF. Suppose that D is a disk. By performing a preliminary similarity
mapping we may assume that D = B. Then g(z) = z -^1 maps B* conformally onto
Band
w(oo,1;B*) =w(O,g('Y);B) = lengt~;g(1)) = len;~('Y) =w(0,1;B)for each arc 'Y c fJB = fJB. This yields the desired result with z 0 = 0 and z(j = oo.
The case where D is a half-plane follows similarly.
For the converse choose conformal mappings f: B-+D and g: B-+D and let
f(O) = z 0 and g(oo) = z 0. Then f and g have homeomorphic extensions to Band
B and, by means of a preliminary rotation, we may assume that f(l) = g(l). The
homeomorphism
¢ = g-^1 of : 8B-+8B