Continuity of relative hyperbolic spectral theory through metric degeneration
type: Item
instance of: scholarly article
title: Continuity of relative hyperbolic spectral theory through metric degeneration
zbMATH Open document ID: 0872.58062
DOI: 10.1215/S0012-7094-96-08402-1
published in: Duke Mathematical Journal
publication date: 1997-10-19T00:00:00Z
review text: The authors discuss the Minakshisundaram-Pleijel pair of punctured Riemann surfaces in the following sense. Let \(M_1\) and \(M_2\) be compact Riemann surfaces. Let \(S_1\), \(S_2\) be finite subsets of \(M_1\) and \(M_2\), respectively. Let the metrics \(\mu_1\), \(\mu_2\) realise the conformal structures on \(M_1-S_1\) and \(M_2-S_2\). It is assumed that \(S_1\) and \(S_2\) have the same cardinality and that the metrics ``agree'' on certain neighbourhoods of the points \(S_1\) and \(S_2\). It is then possible to define a zeta function of the ``difference'' of these two objects. The authors use the theory of the heat kernel to investigate how the zeta function varies as the metric varies, and, especially, when the variation either introduces or erases cusps, i.e. points of \(S_1\) and \(S_2\) where the metric becomes singular in the same way as a hyperbolic metric does. They then apply these results to the study of the ``logarithm of a regularised determinant of the Laplacian'' and the closely related structure constant. The authors show that under certain circumstances the Polyakov integrals cancel and that consequently these quantites are constant under the deformation and equal those of the limiting objects.
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 922153
zbMATH Keywords: spectral zeta functions
zbMATH Keywords: Polyakov formulae
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: determinant of the Laplacian
zbMATH Keywords: hyperbolic metric
author: Jay Jorgenson
author: Rolf E. Lundelius
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
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full work available at URL: s0012-7094-96-08402-1
OpenAlex ID: W1495379271
description: scientific article; zbMATH DE number 922153