THE ASYMPTOTIC BEHAVIOR OF EISENSTEIN SERIES AND A COMPARISON OF THE WEIL-PETERSSON AND THE ZOGRAF-TAKHTAJIAN METRICS

摘要

Of interest to us is the asymptotic behavior of Eisenstein series for degenerating hyperbolic surfaces with cusps. 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.