1549055259-Ubiquitous_Quasidisk__The__Gehring_

D.

CHAPTER 11

Fourth series of implications

In this chapter we prove the following statements for a simply connected domain

1° A quasidisk is a quasiconformal extension domain.

2° A quasiconformal extension domain is homogeneous with respect to the

family QC(K) for a fixed K.

3° A domain homogeneous with respect to the family QC(K) for a fixed K

is a quasidisk.

4 ° A quasidisk has the extremal distance property.

5° A domain with the extremal distance property is a linearly locally con-

nected domain.

6° A Jordan domain is a quadsidisk if and only if it has the harmonic bending

property.

These implications are indicated in Figure 11.1.

D is a quasiconformal

extension domain

D has the extremal distance

property

,11.2 /

""' /11.4

11.2 D is a quasidisk 11.4

h.3

'Z:-8.6

D is a quasiconformally

homogeneous domain

11.1

D has the harmonic

bending property

FIGURE 11.1

149

D is linearly locally

connected