Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations
Year: 2025
Type: preprint
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 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.
Source: arXiv (Cornell University)
Authors: Chen, Zhijie, Fu, Erjuan, Lin, Chang-Shou
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979255454
Cites:
Cited by:
Related to: