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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.16230 |
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| _version_ | 1866909698753560576 |
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| author | Chen, Zhijie Kuo, Ting-Jung Lin, Chang-Shou |
| author_facet | Chen, Zhijie Kuo, Ting-Jung Lin, Chang-Shou |
| contents | We study the curvature equation with multiple singular sources on a torus \[Δu+e^{u}=8π\sum_{k=0}^{3}n_{k}δ_{\frac{ω_{k}}{2}}% +4π\left( δ_{p}+δ_{-p}\right) \quad \text{ on }\;E_τ:=\mathbb{C}/(\mathbb Z+\mathbb{Z}τ),\] where $n_k\in\mathbb N$ and $δ_a$ denotes the Dirac measure at $a$. This is known as a critical case for which the apriori estimate does not hold, and the existence of solutions has been a long-standing problem. In this paper, by establishing a deep connection with Painlevé VI equations, we show that the existence of even solutions (i.e. $u(z)=u(-z)$) depends on the location of the singular point $p$, and we give a sharp criterion of $p$ in terms of Painlevé VI equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16230 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations Chen, Zhijie Kuo, Ting-Jung Lin, Chang-Shou Analysis of PDEs We study the curvature equation with multiple singular sources on a torus \[Δu+e^{u}=8π\sum_{k=0}^{3}n_{k}δ_{\frac{ω_{k}}{2}}% +4π\left( δ_{p}+δ_{-p}\right) \quad \text{ on }\;E_τ:=\mathbb{C}/(\mathbb Z+\mathbb{Z}τ),\] where $n_k\in\mathbb N$ and $δ_a$ denotes the Dirac measure at $a$. This is known as a critical case for which the apriori estimate does not hold, and the existence of solutions has been a long-standing problem. In this paper, by establishing a deep connection with Painlevé VI equations, we show that the existence of even solutions (i.e. $u(z)=u(-z)$) depends on the location of the singular point $p$, and we give a sharp criterion of $p$ in terms of Painlevé VI equations. |
| title | Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.16230 |