Asymptotics of the Weil-Petersson metric

TL;DR

This paper studies the asymptotic behavior of the Weil-Petersson metric on the moduli space of Riemann surfaces, providing a detailed expansion near the boundary of the Deligne-Mumford compactification.

Contribution

It establishes a complete polyhomogeneous expansion of the Weil-Petersson metric in powers of short geodesic lengths near the boundary.

Findings

Polyhomogeneous expansion of the Weil-Petersson metric

Behavior of the metric near singular divisors

Insights into the geometry of moduli space boundaries

Abstract

We consider the Riemann moduli space $\mathcal M_{\gamma}$ of conformal structures on a compact surface of genus $\gamma>1$ together with its Weil-Petersson metric $g_{\mathrm{WP}}$. Our main result is that $g_{\mathrm{WP}}$ admits a complete polyhomogeneous expansion in powers of the lengths of the short geodesics up to the singular divisors of the Deligne-Mumford compactification of $\mathcal M_{\gamma}$.