AI 中文总结
本文作为研究两个格林函数之和系列论文的第三篇，聚焦菱形环面，通过开发不同方法，研究环面几何形状及奇点位置对格林函数临界点结构的影响，还表明环面上曲率方程恰有\(0\)、\(1\)或\(2\)个偶轴对称解且各数量均会出现。
AI 中文摘要
设\(G(z)\)是平坦环面\(E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)\)上在\(0\)处有奇点的格林函数。林和王（《数学年刊》2010年）证明\(G(z)\)至多有一对非平凡临界点。本文是研究两个格林函数之和（可简化为\(G_p(z):=\frac{1}{2}(G(z + p)+G(z - p))\)）系列论文的第三篇。我们研究环面的几何形状和奇点\(\pm p\)的位置如何影响\(G_p(z)\)的临界点结构。在第一篇中证明了\(G_p(z)\)对所有环面至多有三对非平凡临界点；第二篇研究了\(E_{\tau}\)是矩形环面的重要情形。本文通过开发不同方法研究\(E_{\tau}\)是菱形环面这一重要但更具挑战性的情形。作为应用，我们表明\(E_{\tau}\)上的曲率方程\(\Delta u + e^{u}=4\pi(\delta_p+\delta_{-p})\)恰好有\(要么0\)、\(1\)要么\(2\)个偶轴对称解，且每个数量都实际出现。
英文摘要
Let $G(z)$ be the Green function on the flat torus $E_τ=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has at most one pair of nontrivial critical points. This is the third of a series of papers to study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We study how the geometry of the torus and the location of singularities $\pm p$ affect the structure of critical points of $G_p(z)$. In Part I \cite{CFL}, we proved that $G_p(z)$ has at most three pairs of nontrivial critical points for all tori. In Part II \cite{CFL-II} (Proc. Lond. Math. Soc. 2026), we studied the important case that $E_τ$ is a rectangular torus. In this paper, first we prove that if $G_p(z)$ has three pairs of nontrivial critical points, then critical points are all non-degenerate. Secondly, we study the other important but more challenging case that $E_τ$ is a rhombus torus, by developing different approaches from \cite{CFL, CFL-II}. As applications, we show that the curvature equation $Δu+e^{u}=4π(δ_p+δ_{-p})$ on $E_τ$ has exactly either $0$, $1$ or $2$ even axisymmetric solutions and each number really occurs.
Comments52 pages
