Asymptotics of the Weil-Petersson metric

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 $\mathcal M_{\gamma}$.