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

Abstract

Let $G(z)$ be the Green function on the flat torus $E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ 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 $\tau$). 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 $E_{\tau}$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(\tau, p)$) and each number really occurs. In the Part II of this series, we study the important case $\tau=ib$ with $b>0$, i.e. $E_{\tau}$ 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 Painlev\'{e} VI equation and the curvature equation are also given.