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.