1549055259-Ubiquitous_Quasidisk__The__Gehring_

104 8. FIRST SERIES OF IMPLICATIONS

8.2. Hyperbolic segments and uniform domains

The following is an immediate consequence of the earlier Definition 3.5.l.

REMARK 8.2.l. Suppose that Dis a simply connected domain in R^2 and that
there is a constant c ~ 1 such that for each hyperbolic segment 'Y joining z 1 and z 2
and each z E 1,

(8.2.2)

(8.2.3)

length(!) :S c lz1 - z2I,

min length(rj) :::; c dist(z, 8D),

J=l,2

where 'Yl, 12 are the components of 'Y \ { z}. Then D is a uniform domain with
constant c.

Thus a simply connected domain with the hyperbolic segment property is uni-
form.

8.3. Uniform domains and linear local connectivity
We establish next a rather general result to show that uniform domains are
linearly locally connected.

THEOREM 8.3.l. Suppose that D is a domain in R^2 and that for some constants
a, b ~ 1 each pair of points z1, z2 in D can be joined by arcs a and (3 in D such that

(8.3.2)

(8.3.3)

diam( a) :Sa lz1 - z2I,

min lz - Zj I :S b dist(z, 8D)

J=l,2

for z E (3. Then for each zo E R^2 and each r > 0

(8.3.4)

(8.3.5)

D n B(zo, r) lies in a component of D n B(z 0 , er),

D\B(zo,r) lies in a component of D\B(z 0 ,r/c),

where c = 2 max( a, b) + l.

z

FIGURE 8.6