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

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 $\overrightarrow{L}=({\mathrm{\ell }}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\ell }}_m)$ and $n$ cone points of angle $\overrightarrow{\theta }=({\theta }_1,\mathrm{.}\mathrm{.}\mathrm{.}{\theta }_n)$. By using the generalized McShane’s identities, we show that $V_{g,m,n}(\overrightarrow{L},\overrightarrow{\theta })$ is a polynomial of $({\mathrm{\ell }}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\ell }}_n,i{\theta }_1,\mathrm{.}\mathrm{.}\mathrm{.},i{\theta }_m)$. And we obtain a recursion formula for $V_{g,m,n}(\overrightarrow{L},\overrightarrow{\theta })$, which is a generalization of Mirzakhani’s result.