Dependence of Dirichlet integrals upon lumps of Riemann surfaces.
type: Item
instance of: scholarly article
title: Dependence of Dirichlet integrals upon lumps of Riemann surfaces.
zbMATH Open document ID: 1108.31002
author: Mitsuru Nakai
published in: Proceedings of the Japan Academy. Series A
publication date: 2006-04-06T00:00:00Z
review text: The author takes a simple arc \(\gamma\) in an open Riemann surface \(R\) carrying a non-constant harmonic function \(u\) with a finite Dirichlet integral \(D(u;R)\). Then he forms a Riemann surface \(R_\gamma\) with lump \(\widehat C\setminus\gamma\) by pasting \(R\setminus\gamma\) with \(\widehat C\setminus\gamma\) crosswise along \(\gamma\), i.e., \(R_\gamma:=(R\setminus\gamma)\cup_\gamma (\widehat C\setminus \gamma)\), and he transplants \(u_\gamma\) of \(u\) on \(R\) to \(R_\gamma\) characterizing by its being harmonic on \(R_\gamma\) with \(D(u_\gamma;R_\gamma)<+ \infty\) and \(u_\gamma =u\) at the ideal boundary of \(R_\gamma\) and hence of \(R\) in a suitable sense. Among other things, he proves that there is a non-degenerate arc \(\gamma\) (i.e., not a point arc \(\gamma)\) in \(R\) such that \(D(u_\gamma;R_\gamma)=D(u;R)\).
reviewed by: Yusuf Avci
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 30F15
zbMATH DE Number: 5017481
zbMATH Keywords: conjugate level arc
zbMATH Keywords: level arc
zbMATH Keywords: pasting arc
zbMATH Keywords: Riemann surface with lump
zbMATH Keywords: Royden decomposition
MaRDI profile type: Publication
full work available at URL: pjaa.81.131
OpenAlex ID: W1975368452
cites work: Harmonic Function Theory
cites work: Q5719551
cites work: Q4693180
cites work: Q5597712
cites work: Normal families
DOI: 10.3792/PJAA.81.131
description: scientific article; zbMATH DE number 5017481