Boundary Behaviour of Weil-Petersson and Fibre Metrics for Riemann Moduli Spaces
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...
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| Format: | Article |
| Language: | English |
| Published: |
Oxford University Press (OUP),
2018-05-25T18:18:35Z.
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| Online Access: | Get fulltext |
| 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 Obitsu-Wolpert for the conformal factor relative to the model plumbing metric which in turn refined the bound obtained by Masur. A similar expansion for the Ricci metric is also obtained. |
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