1549055259-Ubiquitous_Quasidisk__The__Gehring_

130 9. SECOND SERIES OF IMPLICATIONS

To establish (9.4.14), we show first that there is a unique continuous function

g defined on the radial segments of B(O, L-^1 ) such that g(O) = 0 and f o g(z) = z.

For this, given a point w with lwl = 1, we define

r=r(w)=supt

where the supremum is taken over all t > 0 for which there exists a continuous
function 9t defined on the radial segment [O, tw] with 9t(O) = 0 and f o 9t(z) = z
for z E [O, tw].
The function 9t is unique for all t < r. For otherwise there would exist for some
t, 0 < t < r, continuous functions g 1 ,t and 92,t defined on [O, tw] so that both are
inverses off with g 1 ,t(O) = g 2 ,t(O) = 0. We may assume without loss of generality
that w = 1 and let
so= inf{s > 0: 91 ,t(s) =I 92 ,t(s)}.
Then 0 < s 0 <rand g 1 ,t(so) = 92 ,t(so), since 91 ,t and 92 ,t are continuous. Moreover
there is a sequence sn-+so so that

for all n and

Since

J(g1,t(Sn)) = J(g2,t(Sn)),

this contradicts the fact that f is locally injective at g 1 ,t(so). Thus the function gt

is a function g independent of the choice of parameter t.

Next the fact that f is a local homeomorphism implies that r > 0 and that r

is not attained. Hence

lim g(t w)

t-+T

either does not exist or exists and is a point in 8B by virtue of our normalization.

By Lemma 9.4.10 applied tog on [O, tw],

lg(tw) - g(sw)I::::; L(t - s) for 0 ::::; S::::; t < T.

Hence if tn-+r, g(tnw) is a Cauchy sequence and we conclude

lim g(tnw) = z rf_ B

n-+oo

and 1::::; lzl::::; rL

which implies that r ;:::: 1/ L.

Finally, we use the functions g(t) defined on the radii [O, rw) above to define

a function on B(O, L -^1 ) which we also denote by g. Then since f is a local home-

omorphism and a continuous inverse map is determined on a segment once it is

given at one point, we conclude that g coincides locally with a continuous map and

hence is continuous. Moreover (9.4.15) follows by an argument similar to the one

used to prove that the mappings 9t were unique. D

We are now in a position to prove the main result of this section. For this we

modify a global approximation argument due to John [90] and employ the simple

criterion for injectivity introduced by Martio and Sarvas in [123].

THEOREM 9.4.16. If D is a uniform domain with constant c, then D is rigid

with L(D) = L(c) > l.