Recursion formula for the volumes of moduli spaces of compact hyperbolic surfaces with cone points

Abstract

Let $V_{g,m,n}(\overrightarrow L,\overrightarrow \theta)$ be the Weil-Petersson volume of the moduli space of hyperbolic surfaces of genus g with m geodesic boundary components of length $\overrightarrow L=(\ell_1,...,\ell_m)$ and $n$ cone points of angle $\overrightarrow \theta=(\theta_1,...\theta_n)$. By using the generalized McShane's identities, we show that $V_{g,m,n}(\overrightarrow L,\overrightarrow \theta)$ is a polynomial of $(\ell_1,...,\ell_n,i\theta_1,...,i\theta_m)$. And we obtain a recursion formula for $V_{g,m,n}(\overrightarrow L,\overrightarrow \theta)$, which is a generalization of Mirzakhani's result.