4.2. LOCALLY BILIPSCHITZ MAPPINGS 63EXAMPLE 4.2.2. If f is locally 1-bilipschitz in a domain D c R^2 , then f is
injective in D. On the other hand, for each L > 1,f(z) = l~I exp(iL^2 arg(z)), I arg(z)I < 7r,
is a locally L-bilipschitz mapping of D = R^2 \ (-oo, O], which is not injective.
The following result shows that f is always injective for small L provided D is
a disk or half-plane.
THEOREM 4.2.3 (John [90], [91]). If D is a disk or half-plane and if f is a
locally L-bilipschitz mapping of D with
L::::: 21/4,
then f is injective in D.
PROOF. Suppose otherwise. Because f is a local homeomorphism, we can
choose a disk U with U C D and points z 1 , z2 E aU such that f is injective in U
with f(zi) = f(z2).
Let /3 be the circular arc in U orthogonal to aU at z 1 and z 2 and let E denote
the component of U \ /3 for which f(E) is enclosed by f(/3). Then
length(! (/3)) :::; L length(/3)because f is locally L-bilipschitz. Next
m(E) :::; L^2 m(f(E))since f is injective and locally L-bilipschitz in U. Then from elementary geometry
and the isoperimetric inequality we obtain
leng(/3)2 < m(E) :::; L2 m(f(E)) :::; L2 lengt(/3))2 :::; L4 leng~~(/3)2'
whence L^4 > 2, a contradiction. D
The constant 21 /^4 in Theorem 4.2.3 is not sharp. See, for example, Gevirtz
[71] where it is shown that Theorem 4.2.3 holds with (1 + ./2)^114 in place of 2114 ;
21 /^2 is probably the right constant. The problem of determining this sharp bound
has been open for more than forty years.
DEFINITION 4.2.4. We let L(D) denote the supremum of the numbers M 2: 1
such that f is injective whenever f is locally L-bilipschitz in D with L :::; M; D is
rigid if L(D) > l.
The following analogue of Theorem 4.1.9 describes the simply connected do-
mains that are rigid.
THEOREM 4.2.5 (Gehring [50], Martio-Sarvas [123]). A simply connected do-
main DC R^2 is a K-quasidisk if and only if L(D) > 1, where K and L(D) depend
only on each other.
The following counterpart of Theorem 4.1.10 for bilipschitz functions is a con-
sequence of the above result.
THEOREM 4.2.6 (Gehring [50]). If D C R^2 is a quasidisk and if f is locally
-2
L-bilipschitz in D with L < L(D), then f has an M-bilipschitz extension to R
where M depends only on L and L(D).