Modulus of a boundary component
Martin Jurchescu · Pacific Journal of Mathematics · 1958
1.1 Preliminary definitions.Let R be an open Riemann surface, and let {G n } (n -1, 2, •) be an infinite sequence of subregions of R such that: (a) the relative boundary of each G n is compact, (b) G n z> G n+1 , and (c) nG n = o. W-l{G n } is said to define a boundary component γ of R in the sense of Kerekjartό [6] and Stoilow [16].Here two sequences of subregions {G n } and {G' n ) are considered to be equivalent and to define the same γ if each region G n includes a region G' m .That this is a proper equivalence relation follows immediately.Let r be a boundary component of R, and let S be a subregion of R. If there exists a defining sequence {G n } of γ with G nΰ = S, for some n 0 , we call S SL neighborhood of γ.Throughout this paper we shall consider only neighborhoods S of γ such that the relative boundary of £ is a closed analytic Jordan curve γ Qm By an exhaustion of i?, we mean an infinite sequence {R n } (n -1,2, •••) of subregions of R as follows (see [16]):(1) each R n is compact relative to R and the relative boundary β n of R n consists of a finite number of closed analytic Jordan curves β nt ,(2) R n aR n+1 , (3) u R n = R, and (4) each connected component S ni of R -R n is non-compact (relative to R) and its boundary consists of a single curve β ni .Each set R -R n is said to be a boundary neighborhood of i2.It is easy to see that, for any boundary component γ of R, there exists a single connected component S nί which is a neighborhood of γ.A property is said to be a boundary property (respectively a γ-property) if the following is true.If a Riemann surface R has the property then every Riemann surface R which admits a conformal mapping from a boundary neighborhood of R (a neighborhood of γ', where f is a boundary