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 ${\mathcal{M}}_{\gamma }$.

arXiv comment31 pages