The Bers's constant in genus two
type: Item
instance of: scholarly article
title: The Bers's constant in genus two
zbMATH Open document ID: 1238.30032
author: Matthieu Gendulphe
published in: Mathematische Annalen
publication date: 2011-08-30T00:00:00Z
review text: This article deals with compact Riemann surfaces of genus \(g\geq 2\) endowed with their hyperbolic (Poincaré) metric. Lipman Bers has observed that each such Riemann surface has a family of \(3g-3\) simple and closed geodesics. Their length can be bounded by a certain constant, called the Bers constant \(B(g)\), depending on the genus \(g\). In the article it is shown that \(B(2)\) is given by \(\cosh B(2)/12=x_0\), where \(x_0\) is the unique real solution of the equation \(32x^5-32x^4-24x^3+24x^2-1=0\) which is larger than 1. The value of \(x_0\) is roughly 4.67. The surface which realises \(B(2)\) is essentially unique. The author introduces the contiguity graph, which measures in some sense the repartitions of the Weierstrass points on the surface, as a tool and uses it for the proof.
reviewed by: M. Schlichenmaier
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 5944174
zbMATH Keywords: Weierstrass points
zbMATH Keywords: hyperbolic metric
zbMATH Keywords: compact Riemann surfaces
zbMATH Keywords: contiguity graphs
MaRDI profile type: Publication
full work available at URL: s00208-010-0579-2
OpenAlex ID: W2033143485
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DOI: 10.1007/S00208-010-0579-2
description: scientific article; zbMATH DE number 5944174