The Euclidean, hyperbolic, and spherical spans of an open Riemann surface of low genus and the related area theorems
type: Item
instance of: scholarly article
title: The Euclidean, hyperbolic, and spherical spans of an open Riemann surface of low genus and the related area theorems
zbMATH Open document ID: 0782.30036
author: Masakazu Shiba
published in: Kodai Mathematical Journal
publication date: 1993-09-07T00:00:00Z
review text: In this interesting paper the author organizes and extends his earlier work on the imbedding of open Riemann surfaces in compact Riemann surfaces [Hiroshima Math. J. 14, 371-399 (1984; Zbl 0567.30033); Trans. Am. Math. Soc. 301, 299-311 (1987; Zbl 0626.30046); Holomorphic functions and moduli I, Proc. Workshop, Berkeley/Calif. 1986, Publ. Math. Sci. Res. Inst. 10, 237-246 (1988; Zbl 0653.30028); Analytic functions theory of one complex variable, Pitman Res. Notes Math. Ser. 212, 287-298 (1989; Zbl 0682.30037)]. In particular he studies the moduli of (closed) tori into which a given open torus (open Riemann surface of genus one) can be imbedded subject to a normalization obtained by specifying canonical homology bases. The fundamental result is that the set of moduli forms a closed (possibly degenerate) disc in the upper half-plane. Its hyperbolic diameter is called the hyperbolic span of the given open torus. The author gives a variety of results in this connection and relations with results on maximal complementary area in the metric determined by a normaized holomorphic differential on the closed surface. He gives some analogues for plane domains including area results in the Euclidean and spherical metrics.
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 30F30
zbMATH DE Number: 409648
reviewed by: James A. Jenkins
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039710
OpenAlex ID: W1968325809
cites work: Q5593095
cites work: Q3310101
cites work: Q5574105
cites work: Q4146966
cites work: Q3253223
cites work: On some span theorems
cites work: Q5815016
cites work: Q4052427
cites work: Q5542302
cites work: The span of multiply connected domains
cites work: The Riemann-Hurwitz relation, parallel slit covering map, and continuation of an open Riemann surface of finite genus
cites work: The Moduli of Compact Continuations of an Open Riemann Surface of Genus One
cites work: Q3798945
cites work: Q4731436
cites work: A classification problem for Riemann surfaces of genus 1.
cites work: Q5835847
DOI: 10.2996/KMJ/1138039710
description: scientific article; zbMATH DE number 409648