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

Abstract

Let $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ be a flat torus and $G(z)=G(z;τ)$ be the Green function on $E_τ$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_τ)^{n}$: \[ G_{n}(z_{1},\cdots,z_{n}):=\sum_{i<j}G(z_{i}-z_{j})-n\sum_{i=1}^{n}G(z_{i}). \] A critical point is called trivial if $\{z_{1},\cdots,z_{n}\}=\{-z_{1},\cdots,-z_{n}\}$ in $E_τ$. 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 $τ$). For general $n\geq 2$, 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 $Δu+e^{u}=8πn δ_{0}$ and the integral Lamé equation $y''=[n(n+1)\wp(z)+B]y$ will be given. In particular, we show that the kernel of the linearized operator is one-dimensional for any solution of the curvature equation.