1549055259-Ubiquitous_Quasidisk__The__Gehring_

64 4. INJECTIVITY CRITERIA

SKETCH OF PROOF. Let D' = f(D) and suppose that g is locally L'-bilipschitz
in D' with
L' < L(D)
L.

Then go f is locally LL'-bilipschitz in D with LL'< L(D). Thus go f and fare
injective in D , g is injective in D', and

L(D') 2 L~) > l.

Theorem 4.2.5 then implies that D and D' are K-quasidisks where K depends only

on L(D) and L(D)/ L.

Next by Theorem 3.4.5 there exists a constant e, which depends only on K,

such that z 1 , z 2 E D and w 1 , w 2 E D' can be joined by hyperbolic segments / c D

and /^1 CD' where

This and the fact that f is injective and locally L-bilipschitz in D imply that

If (z 1 ) - f (z2) I :<::: length(!(/)) :<::: L length(/) :<::: Le lz1 - z2I

and

1r^1 (w1) - 1-^1 (w2)I :<::: length(r^1 (1')) :<::: Llength(I') :<:::Le lw1 - w2I·

Thus f is Le-bilipschitz in D and hence has a homeomorphic extension f* : D-+D'.

If D is unbounded, then so is D' and, as in the proof of Theorem 4.1.10,

Theorem 2.1.8 implies that f* has an M-bilipschitz extension to R

2

where M

depends only on K and L. If D is bounded, then we can choose an auxiliary

Mobius transformation¢> so that ¢>(D) and c/>(D') are unbounded and complete the

proof as above. See Gehring [50] for the details. D

Theorem 4.2.6 has the following physical interpretation. Suppose that D is an

elastic body in R^2 and let f denote the deformation of D under a force field. Then

. ( lf(z + h) - f(z)I lhl )
Li(z ) = ln;:_:~ip max lhl ' lf(z + h) - f(z)I
measures the strain in D at the point z caused by the force field, f is locally
bilipschitz if and only if L1(z) is bounded, and L(D) is the supremum of allowable
strains before D collapses. Theorem 4.2.6 says that if
sup L(z ) < L(D),
zED
then the shape of the deformed body f(D) is roughly the same as that of the
original body D.
The related problem in higher dimensions is considered in Martio-Sarvas [123].

4.3. Locally quasiconformal mappings

A mapping f is locally K -quasi conformal in a domain D if each point of D has

a neighborhood in which f is K-quasiconformal. We consider here analogues of the

preceding injectivity results for lo cally quasiconformal mappings. In order to do so

we must first find a substitute for the quantities

sup IT1(z)I PD(z)-^1 , sup

zED