Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations
Zhijie Chen, Erjuan Fu, Chang-Shou Lin
TL;DR
This work analyzes the critical points of the multiple Green function $G_n$ on a flat torus $E_{\tau}$ and proves that the set of parameters $\tau$ for which degenerate critical points occur has Lebesgue measure zero, implying generic non-degeneracy. It builds a bridge between the elliptic PDE setting and the Lamé equation, via the monodromy data $(r,s)$ and the hyperelliptic curve $Y_n$, with the pre-modular form $Z_{r,s}^{(n)}(\tau)$ encoding realizability of given monodromy. A key technical result is an explicit Hessian formula for nontrivial critical points, $\det D^2 G_n(\boldsymbol p;\tau)=\frac{(-1)^n n^2}{4(2\pi)^{2n+2} \operatorname{Im}\tau} c_{\boldsymbol p} |\tau_r|^2 \operatorname{Im}\left(\frac{\tau_s}{\tau_r}\right)$, yielding a concrete degeneracy criterion in terms of $\tau_r,\tau_s$. The measure-zero degeneracy result, together with a description of trivial critical points, enables precise counts of curvature equation solutions on $E_{\tau}$ and informs bubbling analysis in related elliptic problems.
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 $(E_τ)^{n}$: \[ G_{n}(z_{1},\cdots,z_{n};τ):=\sum_{i<j}G(z_{i}-z_{j};τ)-n\sum_{i=1}% ^{n}G(z_{i};τ). \] Recently, Lin (J. Differ. Geom. to appear) proved that there are at least countably many analytic curves in $\mathbb H=\{τ: \operatorname{Im}τ>0\}$ 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.