Green functions, Hitchin's formula and curvature equations on tori II: Rectangular torus
Zhijie Chen, Erjuan Fu, Chang-Shou Lin
TL;DR
The paper analyzes the sum of two Green functions on rectangular tori, G_p(z), and provides a complete picture of how many and what type of critical points G_p(z) can have when the torus is rectangular (τ=ib). By linking G_p to a generalized Lamé equation and introducing conditional stability sets, the authors derive an explicit 8-threshold real-line partition (d1,…,d8) for wp(p) that dictates the absence or presence of nontrivial critical points, with nondegenerate saddles in the presence of wp(p) in certain intervals. They further show that, near p=(1+τ)/4, one can get up to three pairs of nontrivial critical points, corresponding to ten total critical points including the four trivial ones, and provide sharp, interval-based counts for generic p. The work connects these spectral observations to Hitchin’s elliptic Painlevé VI equation and to curvature-type PDEs on E_τ, yielding precise existence/nonexistence and symmetry results via monodromy considerations and conditional stability sets. Overall, the results give a concrete, structured understanding of how Green-function-induced critical landscapes on rectangular tori depend on Weierstrass invariants and Weierstrass-ζ/σ data, with broad applications to isomonodromic deformations and nonlinear elliptic PDEs.
Abstract
Let $G(z)$ be the Green function on the flat torus $E_τ=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has either $3$ or $5$ critical points (depending on the choice of $τ$). Here we study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. In Part I \cite{CFL}, we proved that for any $p$ satisfying $p\neq -p$ in $E_τ$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(τ, p)$) and each number really occurs. In the Part II of this series, we study the important case $τ=ib$ with $b>0$, i.e. $E_τ$ is a rectangular torus. By developing a completely different approach from Part I, we show the existence of $8$ real values $d_1<d_2<\cdots<d_7<d_8$ such that if $$\wp(p)\in (-\infty, d_1]\cup [d_2, d_3]\cup [d_4, d_5]\cup [d_6, d_7]\cup [d_8,+\infty),$$ then $G_p(z)$ has no nontrivial critical points; if $$\wp(p)\in (d_1, d_2)\cup (d_3, d_4)\cup (d_5, d_6)\cup (d_7, d_8),$$ then $G_p(z)$ has a unique pair of nontrivial critical points that are always non-degenerate saddle points. This allows us to study the possible distribution of the numbers of critical points of $G_p(z)$ for generic $p$. Applications to the Painlevé VI equation and the curvature equation are also given.