Existence of supercritical pasting arcs for two sheeted spheres (Q862150)

Let \(A\) and \(B\) be disjoint nonempty compact subsets of the complex plane \(\mathbb{C}\) such that \(\mathbb{C}\setminus A\) and \(\mathbb{C}\setminus B\) are regular subregions of \(\widehat{\mathbb{C}}= \mathbb{C}\cup\{\infty\}\). A simple arc \(\nu\) in \(\widehat C\setminus(A\cup B)\) is referred to as a pasting arc for \(A\) and \(B\). In the context of the study of type problem, a need for the classification of pasting arcs arose. Let \(\widehat C_\nu\) be a covering Riemann surface of the base surface \(\widehat C\) with the natural projection \(\pi_\nu\). Then a pasting arc \(\nu\) is termed supercritical if \(\text{cap}(A,\widehat C_\nu\setminus B)> \text{cap}(A,\widehat C\setminus B)\). Improving a result of himself in an earlier paper [``Types of pasting arcs in two sheeted spheres'', Potential theory in Matsue, Adv. Stud. Pure Math. 44, 291--304 (2006; Zbl 1121.31001)], the author proves here that there always exists a supercritical pasting arc \(\nu\) for \(A\) and \(B\).