Extending the Schwarz-Christoffel formula to universal covering maps of a Riemann surface
type: Item
instance of: scholarly article
title: Extending the Schwarz-Christoffel formula to universal covering maps of a Riemann surface
zbMATH Open document ID: 1280.30005
DOI: 10.1007/BF03321850
author: Jonathan Tsai
published in: Computational Methods and Function Theory
publication date: 2013-01-11T00:00:00Z
review text: Let \(Q(z)\,dz^2\) be a quadratic differential in a domain \(D^*\) in the Riemann sphere \(\hat{\mathbb{C}}\). A \(Q\)-polygon \(P\) is a closed Jordan curve in \(D^*\) such that \(\arg [Q(z)\,dz^2]\) is piecewise constant on \(P\). \textit{R. Kühnau} [J. Reine Angew. Math. 228, 93--132 (1967; Zbl 0176.37701)] discovered an extension of the Schwarz-Christoffel formula to a domain bounded by a \(Q\)-polygon. The author of the article under review derives an extension of the Schwarz-Christoffel formula to universal covering maps of a Riemann surface. Let \(R\) be a finite Riemann surface with boundary and \(\pi :\mathbb{D}\mapsto R\) be a universal covering map of \(R\). Let \(Q(z)\,dz^2\) be a quadratic differential on \(R\) and let \(\hat{R}\) be a sub-Riemann surface of \(R\) such that the boundary of \(\hat{R}\) is a union of finite-sided \(Q\)-polygons. If the lift of \(\hat{R}\), \(D=\pi^{-1}(\hat{R})\), is a simply connected domain, then by the Riemann mapping theorem there is a conformal map \(f\) of \(\mathbb{D}\) onto \(D\). The main result of the author is that \(f\) can be computed as a solution of a certain differential equation. Then \(\pi\circ f\) is a universal covering map of \(\hat{R}\).
reviewed by: Valentin V. Andreev
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30C30
Mathematics Subject Classification ID: 65E05
zbMATH DE Number: 6124809
zbMATH Keywords: Schwarz-Christoffel formula
zbMATH Keywords: Riemann surface
zbMATH Keywords: covering space
zbMATH Keywords: quadratic differentials
describes a project that uses: SC Toolbox
describes a project that uses: Schwarz-Christoffel
MaRDI profile type: Publication
cites work: Q5179675
cites work: Q4341343
cites work: The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains
cites work: Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions
cites work: Schwarz-Christoffel mapping of multiply connected domains
cites work: Schwarz-Christoffel Mapping
cites work: Q5729426
cites work: Über die analytische Darstellung von Abbildungsfunktionen, insbesondere von Extremalfunktionen der Theorie der konformen Abbildung.
cites work: The Loewner driving function of trajectory arcs of quadratic differentials
full work available at URL: https://doi.org/10.1007/bf03321850
OpenAlex ID: W2097217681
Recommended article: Q3985299
Recommended article: A potential-theoretic construction of the Schwarz–Christoffel map for finitely connected domains
Recommended article: A quick proof of Riemann's mapping theorem
Recommended article: An integral representation formula of the Schwarzian derivative
Recommended article: Schwarz-Christoffel mapping of multiply connected domains
Recommended article: Q3035540
Recommended article: Conformal mapping onto a doubly connected circular arc polygonal domain
Recommended article: Schwarz-Christoffel formula for multiply connected domains
Recommended article: Univalence Constraints on the Schwarz–Christoffel Parameters
Recommended article: Conformal mapping onto numerable polygon with double symmetry
description: scientific article; zbMATH DE number 6124809