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In this paper, we investigate the following curvature equation: \begin{equation} \Delta u+e^{u}=8\pi (\delta _{0}+\delta _{\frac{\omega _{k}}{2}})\text{ in } E_{\tau }\text{, }\tau \in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{\tau }$ represents a flat torus and $\frac{\omega _{k}}{2}$ is one of the half periods of $E_{\tau }$. 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*} \Delta u+e^{u}=8\pi \delta _{0}\text{ in }E_{\tau }\text{, }\tau \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 $\tau $. 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: \begi

Author

Kuo, Ting-Jung

Published

2023

Language

EN