1549055259-Ubiquitous_Quasidisk__The__Gehring_

9.4. UNIFORM DOMAINS ARE RIGID 127

9.4. Uniform domains are rigid

Suppose that D C R^2 is a simply connected domain. In the last three sections

we showed that D is a quasidisk if and only if each function f, analytic and locally

injective in D, is globally injective whenever its Schwarzian derivative S 1 or pre-

Schwarzian derivative Tt is small relative to the hyperbolic density p D, that is, if

O'(D) > 0 or T(D) > 0.

We shall establish next the analogue of these results for bilipschitz maps. We

begin with some preliminary results due to F. John and R. Nevanlinna.

LEMMA 9.4.l (John [90]). Suppose that f: B(z 0 , r)-+R^2 is L-bilipschitz. Then

there exists an angle () such that

(9.4.2) lf(z) - g(z)I ::; a(L - l)r

for z E B(zo, r) where a< 20,

g(z) = E(z - zo) + f(zo),

and E(z) = ei!Jz or eie-z.

PROOF. By means of a preliminary translation and scaling we may assume
that r = 1 and f(z 0 ) = zo = 0. We may also assume that

2

(9.4.3) L < Lo = J3

since otherwise
2
lf(z) - zl::; L+ 1::; (1+Lo_1)(L-1)::; a(L-1)

and (9.4.2) will follow trivially with E(z) = z.
Before defining the map E when L < L 0 , we establish an inequality that will
be used repeatedly in what follows. Since f is L-bilipschitz with f (0) = 0,

-(L2 - l)lzjl2::; lf(zj)l2 - lzjl2::; (L2 - l)lzjl2

for j = 1,2,

-(L^2 - l)lz1 - z 212 ::; lf(z1) - f(z2)1^2 - lz1 - z21^2 ::; (L^2 - l)lz1 - z21^2 ,

and by adding these inequalities we obtain

(9.4.4) 2 jRe(f(z1)f(z2) - z1z2)I::; (L^2 - l)(lz1 - z21^2 + lz11^2 + lz21^2 )

for z 1 , z 2 E B. Next let F(z) = F(x + i y) = u +iv be the linear transformation
F: R^2 -+R^2 where

u = Ref(l) x + Ref(i) y,

v =Im f(l) x +Im f(i) y,

and let Falso denote the associated matrix of F. We may assume that det(F) 2: 0
by replacing f ( z) by f ( z) if necessary.
We will estimate the semiaxes of the ellipse F(B) by looking at the eigenvalues
of pT F - I from which bounds are easily obtained. Here

pT F = ( lf(l)_l

2

Re(f(l)f(i)))

Re(f(l)j(i)) lf(i)l^2