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 Ĉ disjoint from A ∪ B, which we call a pasting arc for A and B. Construct a covering Riemann surface Ĉ_γ over Ĉ by pasting two copies of $\hat{\mathbf C}\setminus\gamma$ crosswise along γ. We embed A in one sheet and B in another sheet of two sheets of Ĉ_γ which are copies of $\hat{\mathbf C}\setminus\gamma$ so that $\hat{\mathbf C}_{\gamma}\setminus A \cup B$ is understood as being obtained by pasting ($\hat{\mathbf C}\setminus A)\setminus \gamma$ with ($\hat{\mathbf C}\setminus B)\setminus \gamma$ crosswise along γ. In the comparison of the variational 2 capacity cup(A,$\hat{\mathbf C}_{\gamma}\setminus B$) of the compact set A considered in the open set $\hat{\mathbf C}_{\gamma}\setminus B$ with the corresponding cap(A,$\hat{\mathbf C}\setminus B$), we say that the pasting arc γ for A and B is subcritical, critical, or supercritical according as cap(A,$\hat{\mathbf C}_{\gamma}\setminus B$) is less than, equal to, or greater than cap(A,$\hat{\mathbf C}\setminus B$), 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 $\hat{\mathbf C}\setminus A \cup B$ for any general admissible pair of A and B without any further requirment whatsoever.