Existence of supercritical pasting arcs for two sheeted spheres
type: Item
instance of: scholarly article
title: Existence of supercritical pasting arcs for two sheeted spheres
zbMATH Open document ID: 1117.31001
author: Mitsuru Nakai
published in: Kodai Mathematical Journal
publication date: 2007-02-05T00:00:00Z
review text: 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 \(B\).
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 31C15
zbMATH DE Number: 5121929
zbMATH Keywords: capacity
zbMATH Keywords: pasting arcs
zbMATH Keywords: Riemann surface
reviewed by: Victor Anandam
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1151936433
OpenAlex ID: W2022290637
DOI: 10.2996/KMJ/1151936433
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description: scientific article; zbMATH DE number 5121929