Structure of critical points of multiple Green functions on flat torus and applications
Abstract
Let Eτ:=ℂ/(ℤ+ℤτ) be a flat torus and G(z)=G(z;τ) be the Green function on Eτ with the singularity at $0$. Consider the multiple Green function Gn on (Eτ)n: Gn(z1,⋯,zn):=∑i<jG(zi−zj)−n∑i=1nG(zi). A critical point is called trivial if {z1,⋯,zn}={−z1,⋯,−zn} in Eτ. Lin and Wang (Ann. Math. 2010) proved that G1=−G has exactly three trivial critical points and at most one pair of nontrivial critical points (depends on the choice of τ). For general n≥ 2, Chai, Lin ang Wang (Camb. J. Math. 2015) proved that Gn has at most 2n+1 trivial critical points, but how many nontrivial critical points might exist remained completely open there. In this paper, we prove that Gn has at most n pairs of nontrivial critical points, and this upper bound is optimal at least for n=2,3. The key idea is to show that nontrivial critical points (if exist) are always non-degenerate and contribute the same degree (−1)n+1. Applications to the curvature equation Δu+eu=8πn δ0 and the integral Lamé equation y''=[n(n+1)℘(z)+B]y will be given. In particular, we show that the kernel of the linearized operator is one-dimensional for any solution of the curvature equation.
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BibTeX
@misc{chen2026structure,
title = {{Structure of critical points of multiple Green functions on flat torus and applications}},
author = {Zhijie Chen and Chang-Shou Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07830},
eprint = {2610.07830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07830},
}
RIS
TY - UNPB
TI - Structure of critical points of multiple Green functions on flat torus and applications
AU - Zhijie Chen
AU - Chang-Shou Lin
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07830
UR - https://arxiv.org/abs/2610.07830
AB - Let E_τ:=ℂ/(ℤ+ℤτ) be a flat torus and G(z)=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,⋯,z_n):=∑_i<jG(z_i−z_j)−n∑_i=1^nG(z_i). A critical point is called trivial if {z_1,⋯,z_n}={−z_1,⋯,−z_n} in E_τ. Lin and Wang (Ann. Math. 2010) proved that G_1=−G has exactly three trivial critical points and at most one pair of nontrivial critical points (depends on the choice of τ). For general n≥ 2, Chai, Lin ang Wang (Camb. J. Math. 2015) proved that G_n has at most 2n+1 trivial critical points, but how many nontrivial critical points might exist remained completely open there. In this paper, we prove that G_n has at most n pairs of nontrivial critical points, and this upper bound is optimal at least for n=2,3. The key idea is to show that nontrivial critical points (if exist) are always non-degenerate and contribute the same degree (−1)^n+1. Applications to the curvature equation Δu+e^u=8πn δ_0 and the integral Lamé equation y''=[n(n+1)℘(z)+B]y will be given. In particular, we show that the kernel of the linearized operator is one-dimensional for any solution of the curvature equation.
ER -