The asymptotic behavior of Eisenstein series for degenerating hyperbolic surfaces with cusps is of interest to us. In order to investigate it we use integral representations of eigenfunctions for the Laplacian, the collar lemma, the interior Schauder estimates, the maximum principles for subharmonic functions and the Harnack Inequalities. As an application, we will compare the Weil–Petersson and the Zograf–Takhtajan metrics near the boundary of moduli spaces.
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Kunio Obitsu, The Asymptotic Behavior of Eisenstein Series and a Comparison of the Weil–Petersson and the Zograf–Takhtajan Metrics Klein-Gordon equations is studied in the Sobolev space Hs = Hs(. Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, pp. 449–478DOI 10.2977/PRIMS/1145477232