In this paper, we prove that there are no solutions for the curvature equation \[ \Delta u+e^{u}=8\pi n\delta_{0}\text{ on }E_{\tau}, \quad n\in\mathbb{N}, \] where $E_{\tau}$ is a flat rectangular torus and $\delta_{0}$ is the Dirac measure at the lattice points. This confirms a conjecture in \cite{CLW2} and also improves a result of Eremenko and Gabrielov \cite{EG}. The nonexistence is a delicate problem because the equation always has solutions if $8\pi n$ in the RHS is replaced by $2\pi \rho$ with $0<\rho\notin 4\mathbb{N}$. Geometrically, our result implies that a rectangular torus $E_{\tau}$ admits a metric with curvature $+1$ acquiring a conic singularity at the lattice points with angle $2\pi\alpha$ if and only if $\alpha$ is not an odd integer. Unexpectedly, our proof of the nonexistence result is to apply the spectral theory of finite-gap potential, or equivalently the algebro-geometric solutions of stationary KdV hierarchy equations. Indeed, our proof can also yield a sharp nonexistence result for the curvature equation with singular sources at three half periods and the lattice points.

中文翻译：

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矩形环面上具有四个奇异源的曲率方程的尖锐不存在结果

在本文中，我们证明了曲率方程 \[ \Delta u+e^{u}=8\pi n\delta_{0}\text{ on }E_{\tau}, \quad n 无解\in\mathbb{N}, \] 其中 $E_{\tau}$ 是平面矩形环面，$\delta_{0}$ 是格点处的狄拉克测度。这证实了\cite{CLW2} 中的一个猜想，也改进了Eremenko 和Gabrielov \cite{EG} 的结果。不存在是一个微妙的问题，因为如果 RHS 中的 $8\pi n$ 被 $2\pi \rho$ 替换为 $0<\rho\notin 4\mathbb{N}$，则方程总是有解。在几何上，我们的结果意味着矩形环面 $E_{\tau}$ 承认曲率 $+1$ 的度量在角为 $2\pi\alpha$ 的格点处获得圆锥奇点当且仅当 $\alpha$ 是不是奇数。不料，我们对不存在结果的证明是应用有限间隙势的谱理论，或等效地应用平稳 KdV 层次方程的代数几何解。事实上，对于三个半周期和格点具有奇异源的曲率方程，我们的证明也可以得出一个尖锐的不存在结果。