Boundary Behaviour of Weil–Petersson and Fibre Metrics for Riemann Moduli Spaces
Author(s)Melrose, Richard B; Zhu, Xuwen
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
International Mathematics Research Notices
Oxford University Press (OUP)
Melrose, Richard and Xuwen Zhu. “Boundary Behaviour of Weil–Petersson and Fibre Metrics for Riemann Moduli Spaces.” International Mathematics Research Notices (November 2017): 1-54 © 2017 The Authors