Harmonic dimensions of covering surfaces
type: Item
instance of: scholarly article
title: Harmonic dimensions of covering surfaces
zbMATH Open document ID: 0811.30028
author: Hiroaki Masaoka
author: Shigeo Segawa
published in: Kodai Mathematical Journal
publication date: 1994-12-19T00:00:00Z
review text: Let \(F\) be an open Riemann surface of null boundary which has a single ideal boundary component in the sense of Kerékjártó-Stoïlow. A relatively noncompact subregion \(\Omega\) of \(F\) is said to be an end of \(F\) if the relative boundary \(\partial \Omega\) consists of finitely many analytic Jordan curves. We denote by \({\mathcal P} (\Omega)\) the class of all nonnegative harmonic functions on \(\Omega\) with vanishing values on \(\partial \Omega\). The harmonic dimension of \(\Omega\), \(\dim {\mathcal P} (\Omega)\) in notation, is defined as the minimum number of elements of \({\mathcal P} (\Omega)\) generating \({\mathcal P} (\Omega)\) provided that such a finite set exists, otherwise as \(\infty\). It is well-known that \(\dim {\mathcal P} (\Omega)\) does not depend on a choice of end of \(F:\dim {\mathcal P} (\Omega) = \dim {\mathcal P} (\Omega')\) for any pair \((\Omega, \Omega')\) of ends of \(F\). In terms of the Martin compactification \(\dim {\mathcal P} (\Omega)\) coincides with the number of minimal points over the ideal boundary. In this note we especially concern with ends \(W\) which are subregions of \(p\)-sheeted unlimited covering surfaces of \(\{0 < | z | \leq \infty\}\). For these \(W\) it is known that \(1 \leq \dim {\mathcal P} (W) \leq p\). Consider two positive sequences \(\{a_ n\}\) and \(\{b_ n\}\) satisfying \(b_{n+1} < a_ n < b_ n < 1\) and \(\lim_{n \to \infty} a_ n = 0\). Set \(G = \{0 < | z | < 1\} - I\) where \(I = \cup^ \infty_{n = 1} I_ n\) and \(I_ n = [a_ n, b_ n]\). We take \(p(>1)\) copies \(G_ 1, \dots, G_ p\) of \(G\). Joining the upper edge of \(I_ n\) on \(G_ j\) and the lower edge of \(I_ n\) on \(G_{j+1} (j \bmod p)\) for every \(n\), we obtain a \(p\)-sheeted covering surface \(W=W^ I_ p\) of \(\{0 < | z | < 1\}\) which is naturally considered as an end of a \(p\)-sheeted covering surface of \(\{0 < | z | \leq \infty\}\). Then we proved the following. Theorem. Suppose that \(p = 2^ m (m \in \mathbb{N})\). Then (i) \…
Mathematics Subject Classification ID: 30F15
Mathematics Subject Classification ID: 31C35
zbMATH DE Number: 701577
zbMATH Keywords: covering surface
zbMATH Keywords: Martin boundary
zbMATH Keywords: Thinness
zbMATH Keywords: harmonic dimension
zbMATH Keywords: Martin compactification
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039973
OpenAlex ID: W4250106721
cites work: On topologies and boundaries in potential theory. Enlarged ed. of a course of lectures delivered in 1966
cites work: Q5719551
cites work: Riemann surfaces of infinite genus
cites work: Q5581926
cites work: Harmonic dimension and extremal length
DOI: 10.2996/KMJ/1138039973
description: scientific article; zbMATH DE number 701577