1549055259-Ubiquitous_Quasidisk__The__Gehring_

40 3. CONFORMAL INVARIANTS

DEFINITION 3.5.l. An arbitrary domain D C R^2 is uniform if there exists a
constant c ~ 1 such that each pair of points z 1 , z 2 E D can be joined by an arc
'Y c D so that for each z E "(,

(3.5.2)

(3.5.3)

length("() ::; c lz1 - z2I,

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

J=i,2

where 'Yi , "( 2 are the components of 'Y \ { z}.

The hypotheses in Definition 3.4.4 are more restrictive than those above since

they require that the arc 'Y also be a hyperbolic geodesic. On the other hand, the

following result implies that when D is simply connected, inequalities (3.5.2) and

(3.5.3) hold for a hyperbolic geodesic 'Y and an absolute constant times c whenever

they hold for an arbitrary arc /3 with constant c and the same endpoints as 'Y·

LEMMA 3.5.4. Suppose /3 is an arc that joins the endpoints zi, z2 of a hyperbolic

segment 'Y in a simply connected domain D C R^2.

1° If

(3.5.5)

then

(3.5.6)

where a is an absolute constant.

20 If

(3.5.7) min length(/3j)::; cdist(w, 8D)

J=i,2

for each w E /3 where /3i, /32 are the components of /3 \ { w}, then

(3.5.8). min length('Yj)::; b(c+ l)dist(z,8D)

J=i,2

for each z E 'Y where 'Yi, "( 2 are the components of 'Y \ { z} and b is an absolute

constant.

PROOF. Since 'Y is a hyperbolic segment with the same endpoints as /3,

length("() ::; a length(/3)

where a is an absolute constant by Theorem 2 of Gehring-Hayman [62]. (See also

Theorem 4.20 in Pommerenke [145].) Hence (3.5.6) follows from (3.5.5).

If z E "(, then by Lemma 2.16 of Gehring-Hag-Martio [61], there exists a

crosscut a of D which separates the components of 'Y \ { z } such that

(3.5.9) length( a)::; bi dist(z, 8D)

where bi is an absolute constant. Hence there exists a point w E f3 n a. If the

components of /3 \ { w} and 'Y \ { z} are labeled so that Zj is t he common endpoint

of 'Yi and /3j, then a U /3j contains a curve in D which joins the endpoints Zj and z

of the hyperbolic segment 'Yj. Thus as above,

length('Yj)::; a length( a U /3j)::; alength(/3j) +a length( a).

Then

min length(/3j)::; cdist(w, 8D)::; clength(a)

J=i,2