By Robert M. Miura

Lawsuits of the NSF examine Workshop on touch changes, Held in Nashville, Tennessee, 1974

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**Example text**

11. Extremal distance property We observed earlier that mod(Γ) is a conformally invariant outer measure of a family of curves which is large if the curves in Γ are short and plentiful and which is small if the curves are long or scarce. We now use this quantity to compare distances between two continua C1 , C2 ⊂ D, as measured by the moduli of the 2 families of curves which join them in D and in R , respectively. This, in turn, leads to another characterization for quasidisks. Given continua C1 , C2 ⊂ D, we denote by ΓD and Γ the families of all curves 2 which join C1 and C2 in D and R , respectively.

3◦ Four boundary points w1 , . . , w4 . The conformal invariant is the modulus of the quadrilateral Q = D with vertices at w1 , . . , w4 . We shall describe in this chapter how each of these invariants can be used to characterize the class of quasidisks. 2. Hyperbolic geometry 2 We begin by deﬁning in each simply connected domain D ⊂ R a conformally invariant distance, the hyperbolic metric, or hyperbolic distance, hD . The hyperbolic density in the unit disk B is deﬁned by ρB (z) = 2 1 − |z|2 for z ∈ B.

In this case Σ and Σ are each determined by 2m + n real numbers. 1 33 34 3. CONFORMAL INVARIANTS conformal type of Σ is determined by N = 2m + n − 3 real numbers. The three cases where N = 1, together with a natural real number or conformal invariant that determines the conformal equivalence class of Σ, are as follows. See Ahlfors [8]. 1◦ Two interior points z1 , z2 . The conformal invariant is the hyperbolic distance hD (z1 , z2 ) between z1 and z2 . 2◦ One interior point z1 and two boundary points w1 , w2 .