1549055259-Ubiquitous_Quasidisk__The__Gehring_

7.6. QUASICONFORMAL EQUIVALENCE OF R^3 \ 75 AND B^3 95

To see this, we may assume without loss of generality that D = S(a) where

0 <a:::; 1r. Let h: R

3

-+R

3

be given in cylindrical coordinates (r, B, x 3 ) E R^3 by

h(r,B,x3) = (r,¢(B),x3)

and h( oo) = oo, where

if 0 < - () < - ~) 2

if~<B<7r 2 - -

and ¢(-B) = -¢(B). Then h is K/2-quasiconformal, where K is as above, and h

maps D onto the real half-plane

H^2 = {(x1,X2,x3): 0 < X1 < oo, lx2I < oo, X3 = O}

-3 - -3 -2

and hence R \ D onto R \ H.

Next there exists a 2-quasiconformal mapping g which unfolds R

3

\ H

2

around

the x2-axis onto the upper half-space

H^3 = {(x1,x2,x3) E R^3 : lx1I < oo, lx2I < oo, 0 < X3 < oo}.

Finally, let f be a Mi:ibius transformation which carries H^3 onto B^3. Then fog oh

is K-quasiconformal and maps R

3
\ D onto B^3.
The problem of determining whether or not a domain DC Rn can be mapped
onto the unit ball Bn is quite difficult when n 2: 3. However the property of linear
local connectivity yields the following necessary condition.

EXAMPLE 7.6.2 (Gehring-Vaisala [70]). If D c Rn can be mapped K-quasi-
conformally onto Bn, then there exists a constant c = c( K) such that for each
xo E Rn and each r > 0
En Bn(x 0 , r) lies in a component of En Bn(xo, er),
E \ if'(x 0 , r) lies in a component of E \ if'(x 0 , r/c),

where E = Il \ D.
These examples illustrate the following interesting relation between quasicon-
formal mappings in R^2 and R^3.

THEOREM 7.6.3 (Gehring [48]). A domain D c R^2 is a quasidisk if and only
if R
3
\ D can be mapped quasiconf ormally onto B^3.
SKETCH OF PROOF. If D is a quasidisk, a construction similar to that in Ex-
ample 7.6.1 plus an important lifting theorem for quasiconformal mappings (Ahlfors
[6]) shows that R
3
\ D can be mapped quasiconformally onto B^3.
If R
3
\ D can be mapped quasiconformally onto B^3 , then Example 7.6.2 with
n = 3 implies that D is linearly locally connected and hence a quasidisk by Theo-
rem 2.4.4. 0