3.5. UNIFORM DOMAINS 393.4. Geometry of hyperbolic segments
A second useful characterization for quasidisks is an analogue of the following
easily established observation.EXAMPLE 3.4.1. If D is a disk or half-plane and if 'Y is a hyperbolic segment
joining z 1 and z2 in D, then for each z E / ,
(3.4.2)7r
length(/)::; 2 lz1 - z2I,(3.4.3) min length('Yj)::; ~ dist(z, 8D),
J=l,2 2
where 11 , 12 are the components of/\ { z} and length denotes Euclidean length. The
constant 7r /2 cannot be replaced by a smaller constant in each of these conclusions.The situation is different if 'Y is a hyperbolic geodesic joining z 1 and z 2 in a
sector D of angle a. If 0 < a ::; 7r, then (3.4.2) holds as above in Example 3.4.1
while (3.4.3) is replaced by
. 7r 7r.
mm length('Yj)::; - -
2
d1st(z , 8D).
1 =1,2 a
On the other hand, if 7r ::; a < 2 7r, then (3.4.3) holds as in Example 3.4.l while
(3.4.2) is replaced by
27r - a 7r
length(1) ::; -7r- 2 lz1 - z2I·These facts lead to the following definition that will allow us to characterize
quasidisks in terms of the length and position of their hyperbolic geodesics.
DEFINITION 3.4.4. A simply connected domain D c R^2 has the hyperbolic
segment property if there exists a constant c ~ 1 such that for each hyperbolic
segment 'Y joining z 1 , z 2 ED and each z E /,
length(/) ::; c lz1 - z2I,
min length('Yj) ::; c dist(z, 8D),
J=l,2where 11, 12 are the components of / \ { z }.
THEOREM 3.4.5 (Gehring-Osgood [67]). A simply connected domain DC R^2
is a K -quasidisk if and only if it has the hyperbolic segment property with constant
c, where K and c depend only on each other.3.5. Uniform domains
Martio and Sarvas were the first to introduce the surprisingly useful class of
uniform domains, domains in which each pair of points z 1 and z 2 can be joined
by an arc 'Y with the properties given a bove in Definition 3.4.4. In particular
they point out in Martio-Sarvas [123] the important role these domains play in
deducing that various classes of locally injective functions are actually injective.
The classes include locally bilipschitz functions with small Lipschitz constant and
analytic functions with Schwarzian or pre-Schwarzian derivatives which are small
relative to the hyperbolic density PD·