Let $E = \Bbb C/\Lambda$ be a flat torus and $G$ be its Green function with singularity at $0$. Consider the multiple Green function $G_n$ on $E^n$: $$G_{n}(z_1,\cdots,z_n) := \sum_{i < j} G(z_{i} - z_{j}) - n \sum_{i = 1} ^{n} G(z_{i}).$$ A critical point $a = (a_1, \cdots, a_n)$ of $G_n$ is called trivial if $\{a_1, \cdots, a_n\} = \{-a_1, \cdots, -a_n\}$. For such a point $a$, two geometric quantities $D(a)$ and $H(a)$ arising from bubbling analysis of mean field equations are introduced. $D(a)$ is a global quantity measuring asymptotic expansion and $H(a)$ is the Hessian of $G_n$ at $a$. By way of geometry of Lame curves developed in our previous paper (Cambridge J. Math 3, 2015), we derive precise formulas to relate these two quantities.

中文翻译：

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由平均场方程的冒泡分析产生的几何量

令 $E = \Bbb C/\Lambda$ 是一个平面圆环，$G$ 是它的格林函数，奇点在 $0$。考虑 $E^n$ 上的多重格林函数 $G_n$： $$G_{n}(z_1,\cdots,z_n) := \sum_{i < j} G(z_{i} - z_{j}) - n \sum_{i = 1} ^{n} G(z_{i}).$$ $G_n$ 的临界点 $a = (a_1, \cdots, a_n)$ 被称为平凡如果 $\{a_1 , \cdots, a_n\} = \{-a_1, \cdots, -a_n\}$。对于这样的点$a$，引入了由平均场方程的冒泡分析产生的两个几何量$D(a)$和$H(a)$。$D(a)$ 是衡量渐近扩张的全局量，$H(a)$ 是 $a$ 处 $G_n$ 的 Hessian。通过在我们之前的论文 (Cambridge J. Math 3, 2015) 中开发的 Lame 曲线的几何形状，我们推导出了精确的公式来关联这两个量。