CHAPTER 3Conformal invariants
We consider here eight conformally inva riant descriptions for a quasidisk D.
Four of these compare the hyperbolic and Euclidean geometries in D , two relate
the harmonic measures of arcs in 8D at points in D and D, one compares the
moduli of quadrilaterals in D and D with common vertices in 8 D , and the last
relates the extremal distance between pairs of continua in D and in R
2
.
3.1. Conformal invariants in a Jordan domain
A Jorda n domain D together with a finite set of interior points z 1 , ... , Zm ED
and boundary points w 1 , ... , Wn E 8D taken in a certain order is said to be a
configuration 2:. Two configurations are conformally equivalent if t here exists a
conformal mapping of one domain onto the other which maps sp ecified interior and
boundary points of one domain onto corresponding points of the other.
Suppose 2:' is a second configuration consisting of a Jordan domain D' wit h
points zi , ... , z~ E D' and wi, ... , w~ E 8D'. Then si nce there exist homeomor-
phisms
f : D-+B and g: D'-+B
that are conformal in D and D', respectively, in order to decide if 2: is conformally
equivalent to 2:' it is sufficient to consider the case where D = D' = B. In this case
2: and 2:' are each determined by 2m + n real numbers.
Since B has conformal self-mappings which carry any pair of interior and
boundary points z 1 , w 1 onto any other pair zi , wi or, alternatively, any ordered
triple of boundary points w 1 , w2, w3 onto any other ordered triple wi, w2, w3, the
FIGURE 3.133w2w' 3