Boundary behaviour of Weil-Petersson and fiber metrics for Riemann moduli spaces

TL;DR

This paper provides detailed asymptotic expansions of Weil-Petersson and fiber metrics near boundary divisors in the moduli space of punctured Riemann surfaces, refining previous bounds and expansions.

Contribution

It derives full asymptotic expansions of these metrics in Fenchel-Nielsen coordinates at the boundary, improving upon earlier two-term expansions and bounds.

Findings

Established complete asymptotic expansions for Weil-Petersson and Takhtajan-Zograf metrics.

Refined the expansion of hyperbolic metrics on fibers approaching nodal curves.

Extended similar expansions to the Ricci metric.

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 fibers 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.