Structure of critical points of multiple Green functions on flat torus and applications

Let $E_{\tau }:=\mathbb{C}\mathrm{/}(\mathbb{Z}+\mathbb{Z}\tau )$ be a flat torus and $G(z)=G(z;\tau )$ be the Green function on $E_{\tau }$ with the singularity at $0$. Consider the multiple Green function $G_n$ on $(E_{\tau }{)}^n$: $$G_n(z_1,\cdots ,z_n):=\sum _{i<j}G(z_i-z_j)-n\sum _{i=1}^{n}G(z_i)\mathrm{.}$$ A critical point is called trivial if $\{z_1,\cdots ,z_n\}=\{-z_1,\cdots ,-z_n\}$ in $E_{\tau }$. Lin and Wang (Ann. Math. 2010) proved that $G_1=-G$ has exactly three trivial critical points and at most one pair of nontrivial critical points (depends on the choice of $\tau$). For general $n\ge 2$, Chai, Lin ang Wang (Camb. J. Math. 2015) proved that $G_n$ has at most $2n+1$ trivial critical points, but how many nontrivial critical points might exist remained completely open there. In this paper, we prove that $G_n$ has at most $n$ pairs of nontrivial critical points, and this upper bound is optimal at least for $n=2,3$. The key idea is to show that nontrivial critical points (if exist) are always non-degenerate and contribute the same degree $(-1{)}^{n+1}$. Applications to the curvature equation $\mathrm{\Delta }u+e^u=8\pi n{\delta }_0$ and the integral Lamé equation $y^{\mathrm{\prime }\mathrm{\prime }}=[n(n+1)\mathrm{\wp }(z)+B]y$ will be given. In particular, we show that the kernel of the linearized operator is one-dimensional for any solution of the curvature equation.