Singular points on compactifications of a Riemann surface
type: Item
instance of: scholarly article
title: Singular points on compactifications of a Riemann surface
zbMATH Open document ID: 0605.30045
DOI: 10.1007/BF02772693
author: Moses Glasner
author: Mitsuru Nakai
published in: Israel Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: In this well-written paper, the authors extend the notion of capacity to a general compactification \(R^*\) of a hyperbolic Riemann surface R. The new definition involves consideration of Dirichlet integrals, and is called the Dirichlet capacity. Some of the basic theory of Dirichlet capacity on \(R^*\) is subsequently developed. As it turns out, in the case of the Kuramochi compactification, the Dirichlet capacity is a constant multiple of the Kuramochi capacity. A point \(p\in R^*\) is called singular if the Dirichlet capacity of p is positive. If \(R^*_{{\mathcal R}}\) and \(R^*_{{\mathcal K}}\) represent the Royden and Kuramochi compactifications respectively, and \(\pi\) : \(R^*_{{\mathcal R}}\to R^*_{{\mathcal K}}\) is the natural projection, the main results states that: A point \(q\in R^*_{{\mathcal K}}-R\) is singular if and only if \(\pi^{-1}(q)\) contains a singular point. Moreover, for any \(q\in R^*_{{\mathcal K}}-R\), \(\pi^{-1}(q)\) contains at most one singular point.
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 3977389
zbMATH Keywords: Royden compactification
zbMATH Keywords: hyperbolic Riemann surface
zbMATH Keywords: Dirichlet capacity
zbMATH Keywords: Kuramochi compactification
MaRDI profile type: Publication
cites work: Q5719551
cites work: Singular points on the royden harmonic boundary
cites work: Q3288235
cites work: On Iversen's property and the existence of bounded analytic functions
cites work: Q5597712
full work available at URL: https://doi.org/10.1007/bf02772693
OpenAlex ID: W2085869162
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description: scientific article; zbMATH DE number 3977389