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

Abstract

Let $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with $\operatorname{Im}\tau>0$ 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 $(E_{\tau})^{n}$: \[ G_{n}(z_{1},\cdots,z_{n};\tau):=\sum_{i<j}G(z_{i}-z_{j};\tau)-n\sum_{i=1}% ^{n}G(z_{i};\tau). \] Recently, Lin (J. Differ. Geom. to appear) proved that there are at least countably many analytic curves in $\mathbb H=\{\tau : \operatorname{Im}\tau>0\}$ 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 $\Delta u+e^{u}=\rho \delta_{0}$ on $E_{\tau}$ will be given.