Even Cone Spherical Metrics: Blow-Up at a Cone Singularity
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| Format: | Preprint |
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2025
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| _version_ | 1866911151249424384 |
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| author | Kuo, Ting-Jung Liang, Xuanpu Wu, Ping-Hsiang |
| author_facet | Kuo, Ting-Jung Liang, Xuanpu Wu, Ping-Hsiang |
| contents | We study families of spherical metrics on the flat torus $E_τ$ $=$ $\mathbb{C}/Λ_τ$ with blow-up behavior at prescribed conical singularities at $0$ and $\pm p$, where the cone angle at $0$ is $6π$, and at $\pm p$ is $4π$. We prove that the existence of such a necessarily unique, even family of spherical metrics is completely determined by the geometry of the torus: such a family exists if and only if\textbf{ }the Green function $G(z;τ)$ admits a pair of nontrivial critical points $\pm a$. In this case, the cone point $p$ must equal $a$, and the corresponding monodromy data is $\left( 2r,2s\right) $, where $a=r+sτ.$
An explicit transformation relating this family to the one with a single conical singularity of angle $6π$ at the origin is established in Theorem 1.4. A rigidity result for rhombic tori is proved in Theorem 1.5. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Even Cone Spherical Metrics: Blow-Up at a Cone Singularity Kuo, Ting-Jung Liang, Xuanpu Wu, Ping-Hsiang Differential Geometry Analysis of PDEs Classical Analysis and ODEs 58J10, 53A10 We study families of spherical metrics on the flat torus $E_τ$ $=$ $\mathbb{C}/Λ_τ$ with blow-up behavior at prescribed conical singularities at $0$ and $\pm p$, where the cone angle at $0$ is $6π$, and at $\pm p$ is $4π$. We prove that the existence of such a necessarily unique, even family of spherical metrics is completely determined by the geometry of the torus: such a family exists if and only if\textbf{ }the Green function $G(z;τ)$ admits a pair of nontrivial critical points $\pm a$. In this case, the cone point $p$ must equal $a$, and the corresponding monodromy data is $\left( 2r,2s\right) $, where $a=r+sτ.$ An explicit transformation relating this family to the one with a single conical singularity of angle $6π$ at the origin is established in Theorem 1.4. A rigidity result for rhombic tori is proved in Theorem 1.5. |
| title | Even Cone Spherical Metrics: Blow-Up at a Cone Singularity |
| topic | Differential Geometry Analysis of PDEs Classical Analysis and ODEs 58J10, 53A10 |
| url | https://arxiv.org/abs/2509.10013 |