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

Let $E_{\tau }:=\mathbb{C}\mathrm{/}(\mathbb{Z}+\mathbb{Z}\tau )$ with $\mathrm{Im}\tau >0$ be a flat torus and $G(z;\tau )$ be the Green function on $E_{\tau }$ with the singularity at $0$. Consider the multiple Green function $G_n$ on $(E_{\tau }{)}^n$: $$G_n(z_1,\cdots ,z_n;\tau ):=\sum _{i<j}G(z_i-z_j;\tau )-n\sum _{i=1}$$ Recently, Lin (J. Differ. Geom. to appear) proved that there are at least countably many analytic curves in $\mathbb{H}=\{\tau :\mathrm{Im}\tau >0\}$ such that $G_n(\cdot ;\tau )$ has degenerate critical points for any $\tau$ 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 $\tau \in \mathbb{H}\setminus {\mathcal{O}}_n$, all critical points of $G_n(\cdot ;\tau )$ are non-degenerate. Applications to counting the exact number of solutions of the curvature equation $\mathrm{\Delta }u+e^u=\rho {\delta }_0$ on $E_{\tau }$ will be given.