Types of pasting arcs in two sheeted spheres

Abstract

Fix two disjoint nondegenerate continua $A$ and $B$ in the complex plane $\mathbb{C}$ with connected complements and choose a simple arc $\gamma$ in the complex sphere $\widehat{\mathbb{C}}$ disjoint from $A \cup B$, which we call a pasting arc for $A$ and $B$. Then form a covering Riemann surface $\widehat{\mathbb{C}}_{\gamma}$ over $\widehat{\mathbb{C}}$ by pasting two copies of $\widehat{\mathbb{C}} \setminus \gamma$ crosswise along the arc $\gamma$. Viewing $A$ and $B$ as embedded in the different two sheets $\widehat{\mathbb{C}} \setminus \gamma$ of $\widehat{\mathbb{C}}_{\gamma}$, consider the variational 2-capacity $\mathrm{cap} (A, \widehat{\mathbb{C}}_{\gamma} \setminus B)$ of the set $A$ in $\widehat{\mathbb{C}}_{\gamma}$ with respect to the open subset $\widehat{\mathbb{C}}_{\gamma} \setminus B$ containing $A$. We are interested in the comparison of $\mathrm{cap} (A, \widehat{\mathbb{C}}_{\gamma} \setminus B)$ with $\mathrm{cap} (A, \widehat{\mathbb{C}} \setminus B)$. We say that the pasting arc $\gamma$ for $A$ and $B$ is subcritical, critical, or supercritical according as $\mathrm{cap} (A, \widehat{\mathbb{C}}_{\gamma} \setminus B)$ is less than, equal to, or greater than $\mathrm{cap} (A, \widehat{\mathbb{C}} \setminus B)$, respectively. The purpose of this paper is to show the existence of subcritical arc $\gamma$ for any arbitrarily given general pair of admissible $A$ and $B$ and then the existences of critical and also supercritical arcs $\gamma$ under the additional condition imposed upon $A$ and $B$ that each of $A$ and $B$ is symmetric about a common straight line in $\widehat{\mathbb{C}}$, which is the case e.g. if $A$ and $B$ are disjoint closed discs.