Existence of supercritical pasting arcs for two sheeted spheres

Abstract

Take e.g. two disjoint nondegenerate compact continua A and B in the complex plane C with connected complements and pick a simple arc γ in the complex sphere C disjoint from A ∪ B, which we call a pasting arc for A and B. Construct a covering Riemann surface C_γ over C by pasting two copies of C$¥backslash$γ crosswise along γ. We embed A in one sheet and B in another sheet of two sheets of C_γ which are copies of C$¥backslash$γ so that C_γ$¥backslash$A ∪ B is understood as being obtained by pasting (C$¥backslash$A)$¥backslash$γ with (C$¥backslash$ B)$¥backslash$γ crosswise along γ. In the comparison of the variational 2 capacity cap(A, C_γ$¥backslash$B) of the compact set A considered in the open set C_γ$¥backslash$B with the corresponding cap(A,C$¥backslash$B), we say that the pasting arc γ for A and B is subcritical, critical, or supercritical according as cap(A,C_γ$¥backslash$B) is less than, equal to, or greater than cap(A,C$¥backslash$B), respectively. We have shown in our former paper [4] the existence of pasting arc γ of any one of the above three types but that of supercritical and critical type was only shown under the additional requirment on A and B that A and B are symmetric about a common straight line simultaneously. The purpose of the present paper is to show that in the above mentioned result the additional symmetry assumption is redundant: we will show the existence of supercritical and hence of critical arc γ starting from an arbitrarily given point in C$¥backslash$A ∪ B for any general admissible pair of A and B without any further requirment whatsoever.