Asymptotics of the Weil-Petersson metric

Abstract

We consider the Riemann moduli space $\mathcal M_γ$ of conformal structures on a compact surface of genus $γ>1$ 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 $\mathcal M_γ$.

31 pages