SOME APPLICATIONS OF EXTREMAL LENGTH TO CONFORMAL IMBEDDINGS

Abstract

Let G be a Denjoy domain and let G' a Denjoy proper subdomain of G and homeomorphic to G. We consider conformal re-imbeddings of G' into G. Let G and G' are N-connected. We know that if N = 2, there is a re-imbedding f of G' into G such that G - cl(f(G')) has an interior point. In this note, we obtain the following theorem. If N ≥ 3, G has a Denjoy proper subdomain G' such that, for any re-imbeddings f of G' into G, G - cl(f(G')) has no interior point.