Let $E_{\tau}:= \mathbb{C}/(\mathbb{Z}+ \mathbb{Z} \tau)$ be a flat torus and $G(z; \tau)$ be the Green function on $E_{\tau}$. Consider the multiple Green function $G_{n}$ on$(E_{\tau})^{n}$: \[ G_n (z_1, \cdots ,z_n ; \tau) := \sum_{i \lt j} G(z_i - z_j ; \tau) - n \sum_{i=1}^n G(z_i ; \tau). \] We prove that for $ \tau \in i \mathbb{R}_{\gt 0}$, i.e. $E_\tau$ is a rectangular torus, $G_n$ has exactly $2n + 1$ critical points modulo the permutation group $S_n$ and all critical points are non-degenerate. More precisely, there are exactly $n$ (resp. $n+1$) critical points $ \boldsymbol{a}$’s with the Hessian satisfying $(-1)^n \det D^2 G_n (\boldsymbol{a} ; \tau) \lt 0$ (resp. $\gt 0$). This confirms a conjecture in [4]. Our proof is based on the connection between $G_n$ and the classical Lame equation from [4, 19], and one key step is to establish a precise formula of the Hessian of critical points of $G_{n}$ in terms of the monodromy data of the Lame equation. As an application, we show that the mean field equation \[ \Delta_u + e^u = \rho \delta_0 \textrm{ on } E_\tau \] has exactly $n$ solutions for $8 \pi n - \rho \gt 0$ small, and exactly $n+1$ solutions for $\rho - 8 \pi n \gt 0$ small.

中文翻译：

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矩形环面上多个格林函数临界点的精确数和非退化

令 $E_{\tau}:= \mathbb{C}/(\mathbb{Z}+ \mathbb{Z} \tau)$ 为平环面，$G(z; \tau)$ 为$E_{\tau}$。考虑多重格林函数 $G_{n}$ on$(E_{\tau})^{n}$: \[ G_n (z_1, \cdots ,z_n ; \tau) := \sum_{i \lt j} G(z_i - z_j ; \tau) - n \sum_{i=1}^n G(z_i ; \tau)。\] 我们证明对于 $ \tau \in i \mathbb{R}_{\gt 0}$，即 $E_\tau$ 是一个矩形环面，$G_n$ 正好有 $2n + 1$ 个临界点模置换组 $S_n$ 并且所有临界点都是非退化的。更准确地说，恰好有 $n$ (resp. $n+1$) 个临界点 $ \boldsymbol{a}$ 满足 $(-1)^n \det D^2 G_n (\boldsymbol{ a} ; \tau) \lt 0$ (resp. $\gt 0$)。这证实了[4]中的一个猜想。我们的证明基于 $G_n$ 与 [4, 19] 中的经典 Lame 方程之间的联系，其中一个关键步骤是根据Lame方程的单调数据建立$G_{n}$临界点Hessian的精确公式。作为一个应用，我们证明了平均场方程 \[ \Delta_u + e^u = \rho \delta_0 \textrm{ on } E_\tau \] 对于 $8 \pi n - \rho \gt 有准确的 $n$ 个解0$ 小，对于 $\rho - 8 \pi n \gt 0$ 小，正好是 $n+1$ 解决方案。