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

Abstract: Let $V_{g,m,n}(\overrightarrow L,\overrightarrow θ)$ 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 $\overrightarrow θ=(θ1,...θ_n)$. By using the generalized McShane's identities, we show that $V{g,m,n}(\overrightarrow L,\overrightarrow θ)$ is a polynomial of $(\ell_1,...,\ell_n,iθ1,...,iθ_m)$. And we obtain a recursion formula for $V{g,m,n}(\overrightarrow L,\overrightarrow θ)$, which is a generalization of Mirzakhani's result.