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Boundary behaviour of Weil-Petersson and fibre metrics for Riemann moduli spaces. (English) Zbl 1457.32038

Summary: The Weil-Petersson and Takhtajan-Zograf metrics on the Riemann moduli spaces of complex structures for an \(n\)-fold punctured oriented surface of genus \(g\) in the stable range \(g+2n >2\), are shown here to have complete asymptotic expansions in terms of Fenchel-Nielsen coordinates at the exceptional divisors of the Knudsen-Deligne-Mumford compactification. This is accomplished by finding a full expansion for the hyperbolic metrics on the fibres of the universal curve as they approach the complete metrics on the nodal curves above the exceptional divisors and then using a push-forward theorem for conormal densities. This refines a two-term expansion due to K. Obitsu and S. A. Wolpert [Math. Ann. 341, No. 3, 685–706 (2008; Zbl 1146.30028)] for the conformal factor relative to the model plumbing metric which in turn refined the bound obtained by H. Masur [Duke Math. J. 43, 623–635 (1976; Zbl 0358.32017)]. A similar expansion for the Ricci metric is also obtained.

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
58D27 Moduli problems for differential geometric structures
14H10 Families, moduli of curves (algebraic)
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References:

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