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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2308.07620 |
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| _version_ | 1866916103620395008 |
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| author | Kuo, Ting-Jung |
| author_facet | Kuo, Ting-Jung |
| contents | In this paper, we investigate the following curvature equation: \begin{equation} Δu+e^{u}=8π(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ}\text{, }τ\in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{τ}$ represents a flat torus and $\frac{ω_{k}}{2}$ is one of the half periods of $E_{τ}$. Our primary objective is to establish a necessary and sufficient criterion for the existence of a non-even family of solutions (see the definition in Section 1). Remarkably, this is equivalent to determining the presence of solutions for the equation with a single conical singularity: \begin{equation*} Δu+e^{u}=8πδ_{0}\text{ in }E_{τ}\text{, }τ\in \mathbb{ H}\text{.} \end{equation*} This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all $τ$. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: \begin{equation*} Δu+e^{u}=8πn(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ},\text{ }n\in \mathbb{N}\text{.} \end{equation*}
As an application, we offer explicit descriptions for solutions to equation (0.1) in the context of both rectangle tori and rhombus tori. See Corollary 1.4 as well as Corollary 1.5. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_07620 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp results for spherical metric on flat tori with conical angle 6$π$ at two symmetric points Kuo, Ting-Jung Analysis of PDEs Classical Analysis and ODEs 58J10, 53A10 In this paper, we investigate the following curvature equation: \begin{equation} Δu+e^{u}=8π(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ}\text{, }τ\in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{τ}$ represents a flat torus and $\frac{ω_{k}}{2}$ is one of the half periods of $E_{τ}$. Our primary objective is to establish a necessary and sufficient criterion for the existence of a non-even family of solutions (see the definition in Section 1). Remarkably, this is equivalent to determining the presence of solutions for the equation with a single conical singularity: \begin{equation*} Δu+e^{u}=8πδ_{0}\text{ in }E_{τ}\text{, }τ\in \mathbb{ H}\text{.} \end{equation*} This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all $τ$. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: \begin{equation*} Δu+e^{u}=8πn(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ},\text{ }n\in \mathbb{N}\text{.} \end{equation*} As an application, we offer explicit descriptions for solutions to equation (0.1) in the context of both rectangle tori and rhombus tori. See Corollary 1.4 as well as Corollary 1.5. |
| title | Sharp results for spherical metric on flat tori with conical angle 6$π$ at two symmetric points |
| topic | Analysis of PDEs Classical Analysis and ODEs 58J10, 53A10 |
| url | https://arxiv.org/abs/2308.07620 |