Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations

Abstract

Let $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with $\operatorname{Im}τ>0$ be a flat torus and $G(z;τ)$ 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};τ):=\sum_{i 0\}$ such that $G_n(\cdot;τ)$ has degenerate critical points for any $τ$ on the union of these curves. In this paper, we prove that there is a measure zero subset $\mathcal{O}_n\subset \mathbb H$ (containing these curves) such that for any $τ\in \mathbb H\setminus\mathcal{O}_n$, all critical points of $G_n(\cdot;τ)$ are non-degenerate. Applications to counting the exact number of solutions of the curvature equation $Δu+e^{u}=ρδ_{0}$ on $E_τ$ will be given.