Heights on the moduli space of Riemann surfaces with circle boundaries
type: Item
instance of: scholarly article
title: Heights on the moduli space of Riemann surfaces with circle boundaries
zbMATH Open document ID: 0755.30037
DOI: 10.1215/S0012-7094-91-06427-6
author: Hala Halim Khuri
published in: Duke Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: Let \(\Sigma=\Sigma_{p,n}\) be a compact Riemann surface of genus \(p\) with \(n\) distinct open disks removed. Associated with a smooth metric \(\sigma\) on \(\Sigma\) is the Beltrami-Laplace operator \(\Delta=\Delta_ \sigma\) with Dirichlet boundary conditions. The height function \(h(\sigma)\) is a generalization of the classical Dirichlet integral. This function is defined as \(h(\sigma)=-\log(\text{det} \Delta)\) where \(\text{det} \Delta=\exp(-Z'(0))\) and \(Z(s)=\sum_{j=1}^ \infty \lambda_ j^{-s}\) is a zeta function built up from the eigenvalues \(\lambda_ j\) of \(\Delta\). Let \(M_ u(\Sigma)\) be the set of flat metrics on \(\Sigma\) with geodesic curvature \(-1\) on \(\partial\Sigma\). The author proves --- if \(pn\neq 0\) --- that \(h: M_ u(\Sigma)\to\mathbb{R}\) is not a proper map and discusses the boundary behavior of this function.
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 58539
zbMATH Keywords: compact Riemann surface
zbMATH Keywords: Beltrami-Laplace operator
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: height function
zbMATH Keywords: Dirichlet integral
zbMATH Keywords: zeta function
zbMATH Keywords: flat metrics
zbMATH Keywords: geodesic curvature
zbMATH Keywords: proper map
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 58539