1549055259-Ubiquitous_Quasidisk__The__Gehring_

9.4. UNIFORM DOMAINS ARE RIGID 131

PROOF. By hypothesis, there exists a constant c 2: 1 such that each pair of
points z 1 , z2 E D can be joined by an arc / C D such that
length(/):::; clz1 - z2I,
(9.4.17) min length(/1):::; c dist(z, aD),
J=l,2
where 11 , 1 2 are the components of/\ { z}. We will show that there exists a constant
L(c) > 1 such that each function f locally L-bilipschitz in D with 1 < L < L(c) is
injective; we may assume L > 1 since otherwise f is trivially injective. We do this
by proving that for any such function and any two points z 1 , z 2 E D , there is an
isometry g of R^2 such that

(9 .4 .18)

for j = 1, 2. From this it follows that f (z 1 ) -=f. f (z 2 ) since otherwise we would have

lz1 - z2I = lg(z1) - g(z2)I ::=; lf(zi) - g(z1)I + lf(z2) - g(z2)I < lz1 - z2I·

To establish (9 .4 .18), we join the points z 1 , z2 by the arc / which satisfies

(9.4.17) and let z 0 be the midpoint of/, i.e.,

length(/( z 1 , zo)) = length(/( z2, z 0 )) = l

where 1 (z, z 0 ) denotes the subarc with z 0 , z as endpoints. Let z = z (s ) be the
arclength representation for 1(z1, zo) with z(O) = zo and z(l) = z1. Then

by (9 .4 .17). Set

and for j = 0, 1, 2, ... let

(9.4.19) s1=(l-qi)z,

Then

l-s

dist(z(s), 8D) 2: -

c

cL^2

q = 1 + cL2 E ( ~ ' 1)

qi

dj = -l,

c

. qi+l l
(9.4.20) l(j+l - (jl :::; Sj+l - Sj = (1 - q)q^1 l = c L 2 = Tj+l

and dist( (j' aD) > dj = L^2 Tj.
By Lemmas 9.4.l and 9.4.13 there exists for each j an isometry g 1 : R^2 -*R^2
such that

(9.4.21) lf(z) - 91(z)I:::; a(L - 1) r1

in B 1 = B((j, r1) where a< 20,

(9.4.22) 91(z) = E1(z - (1) + f((1),

and E 1 (z ) = eiO; z or E1(z ) = eiO;-z. Next since

l(j+l - (j I :::; Tj+l >

B 1 n Bi+ 1 contains a closed disk B * of radius ~ r 1 in which

lg1(z ) - 91+1(z)I ::=; lf(z ) - 91(z)I + lf(z) - 9J+1(z)I

(9.4.23)

::=; a(L - l)(r 1 + r 1 +I)