The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics
type: Item
instance of: scholarly article
title: The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics
zbMATH Open document ID: 1012.11043
author: Kunio Obitsu
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2003-06-29T00:00:00Z
review text: The author investigates the relationship between the Weil-Petersson metric on a Teichmüller space \(T_{g,n}\) and the Zograf-Takhtajan metric. The latter is defined analogously to the Weil-Petersson metric but with an additional weighting by Eisenstein series at the point 2. It is known that the Zograf-Takhtajan metric, like the Weil-Petersson metric, is not complete. In this paper the author uses estimates for Eisenstein series considered as functions on the Teichmüller space to compare the two metrics near the boundary of Teichmüller space. The most important estimate arises by combining Selberg's technique of point-pair invariant with a universal estimate for the injectivity radius (Shimizu-Leutbecher lemma).
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 1744045
zbMATH Keywords: Shimizu-Leutbecher lemma
zbMATH Keywords: Weil-Petersson metric
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Zograf-Takhtajan metric
zbMATH Keywords: Eisenstein series
zbMATH Keywords: injectivity radius
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145477232
OpenAlex ID: W2270820884
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DOI: 10.2977/PRIMS/1145477232
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description: scientific article; zbMATH DE number 1744045