Green functions, Hitchin's formula and curvature equations on tori II: Rectangular torus

Zhijie Chen; Erjuan Fu; Chang-Shou Lin

paper

Green functions, Hitchin's formula and curvature equations on tori II: Rectangular torus

arXiv:2512.05360

Abstract

Let $G (z)$ be the Green function on the flat torus $E_{τ} = C / (Z + Z τ)$ 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 $G_{p} (z) := \frac{1}{2} (G (z + p) + G (z - p))$ . 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 $d_{1} < d_{2} < \dots < d_{7} < d_{8}$ such that if $℘ (p) \in (- \infty, d_{1}] \cup [d_{2}, d_{3}] \cup [d_{4}, d_{5}] \cup [d_{6}, d_{7}] \cup [d_{8}, + \infty),$ then $G_{p} (z)$ has no nontrivial critical points; if $℘ (p) \in (d_{1}, d_{2}) \cup (d_{3}, d_{4}) \cup (d_{5}, d_{6}) \cup (d_{7}, d_{8}),$ 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.