The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics
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).
scientific article; zbMATH DE number 1744045
scientific article