1549055259-Ubiquitous_Quasidisk__The__Gehring_

134 9. SECOND SERIES OF IMPLICATIONS

in which case g(z ) = f(z) in U. Hence g is lo cally L-bilipschitz in D and hence
injective by (9 .5.6). On the other hand, z2 ~ G1,

and we have a contradiction. D

The desired result is now an easy consequence of Theorem 9.5 .4.

COROLLARY 9.5.7. If D is a rigid domain in R^2 , then D is linearly locally

connected with constant c where

(9.5.8)

7r

loge> L(D) -1

PROOF. Suppose that z 1 and z2 are points in D n B(z o, r) and that a is an
arc joining z 1 , z 2 in D. If a r/.. B(z 0 , er), let w 1 and w2 be the first and last points
where a meets 8B(z 0 , r). Theorem 9.5.4 then implies that w 1 and w 2 can be joined
in D n B(zo, er). Hence z 1 and z2 can be joined in D n B(zo, er) and

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

In the same way we see that

D\B(zo,r) liesinacomponentof D\B(zo,r/e)

and hence that D is linearly locally connected with constant e. D

9.6. Uniform domains have the min-max property

A domain D C R^2 is uniform if for some constant a ;:::: 1 each pair of points

z 1 , z 2 E D can be joined by a curve / C D such that

(9.6.1)

(9.6.2)

length(/) ::=;a lz1 - z2I,

min length(lj) ::=; a dist(z, 8D)

J=l,2

for each z E /, where 11 , 12 are the components of / \ { z}.
The domain D has the min-max property if for some constant b > 1 each
z1, z2 E D can be joined by a curve / C D such that

(9.6.3 ) ~ min lzj - w l ::=; lz - wl ::=; b max lzj - wl

b J=l,2 J=l,2

for each z E /and each w ~ D.
The main result of this section, namely that a uniform domain has the min-max
property, is an immediate consequence of the following observation.

REMARK 9.6.4. Suppose that/ is a curve joining z 1 , z 2 in a domain DC R^2.
If/ satisfies (9 .6. l) and (9.6.2) for each z E '"'(, then it satisfies (9.6.3 ) for each z E '"'!
and each w ~ D with b = a + l.