Boundary Behaviour of Weil–Petersson and Fibre Metrics for Riemann Moduli Spaces

Abstract

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.