AI 中文总结
本文针对矩形环面上带两个奇异源的曲率方程，通过处理广义Lamé方程Hill判别式的本性奇点，完全确定条件稳定集交集，给出解存在与不存在的尖锐判据。
AI 中文摘要
我们研究平坦环面上具有两个奇异源的如下曲率方程：$$ \Delta u+e^u=8\pi(\delta_p+\delta_{-p})\quad\text{在 } E_\tau:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau) \text{ 上}, $$ 其中 $\delta_p$ 表示在 $p$ 处的狄拉克测度。情形 $\wp'(2p)=0$ 已由 Kuo 解决（J. Differential Geom. 2026）。本文研究一般情形 $\wp'(2p)\neq 0$。对于 $\wp'(2p)\neq 0$，一个新现象是：相关联的广义 Lamé 方程的 Hill 判别式具有本性奇点。我们发展新思想来完全确定两个条件稳定集的交集，这相当令人惊讶，因为由于本性奇点，似乎不可能确定这两个集合的精确结构。这导致当 $E_\tau$ 为矩形环面且 $\wp(p)\in \mathbb{R}$ 时，解存在与不存在的尖锐判据。
英文摘要
We study the following curvature equation with two singular sources on a flat torus $$ Δu+e^u=8π(δ_p+δ_{-p})\quad\text{on } E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ), $$ where $δ_p$ denotes the Dirac measure at $p$. The case $\wp'(2p)=0$ was solved by Kuo (J. Differential Geom. 2026). In this paper, we study the general case $\wp'(2p)\neq 0$. A novel phenomenon for $\wp'(2p)\neq 0$ is that the Hill discriminant of the associated generalized Lamé equation has an essential singularity. We develop new ideas to determine the intersection of two conditional stability sets completely, which is quite surprising since it seems impossible to determine the exact structure of these two sets due to the essential singularity. This leads to a sharp criteria for the existence and non-existence of solutions when $E_τ$ is a rectangular torus and $\wp(p)\in \mathbb{R}$.
Comments44 pages
