Recursion formula for the volumes of moduli spaces of compact hyperbolic surfaces with cone points
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911508577910784 |
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| author | Jiang, Haoyang Liu, Lixin |
| author_facet | Jiang, Haoyang Liu, Lixin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11785 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Recursion formula for the volumes of moduli spaces of compact hyperbolic surfaces with cone points Jiang, Haoyang Liu, Lixin Geometric Topology 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. |
| title | Recursion formula for the volumes of moduli spaces of compact hyperbolic surfaces with cone points |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2603.11785 |