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 $\overrightarrow θ=(θ_1,...θ_n)$. By using the generalized McShane's identities, we show that $V_{g,m,n}(\overrightarrow L,\overrightarrow θ)$ is a polynomial of $(\ell_1,...,\ell_n,iθ_1,...,iθ_m)$. And we obtain a recursion formula for $V_{g,m,n}(\overrightarrow L,\overrightarrow θ)$, which is a generalization of Mirzakhani's result.

Figures (3)

• Figure 1: The spliting of n cone points
• Figure 2: The spliting of n cone points
• Figure 3: The Spliting of 1 geodesic boundary and n-1 cone points

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 2.1: McShane MR3389436
• Theorem 2.2: McShane MR1625712
• Definition 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3