Geometric analysis on rhombus torus: Green function with two singularities

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 at most one pair of nontrivial critical points.

This is the third of a series of papers to study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$.

We study how the geometry of the torus and the location of singularities $\pm p$ affect the structure of critical points of $G_p(z)$.

In Part I \cite{CFL}, we proved that $G_p(z)$ has at most three pairs of nontrivial critical points for all tori.

In Part II \cite{CFL-II} (Proc.

Lond.

Math.

Soc.

2026), we studied the important case that $E_{\tau}$ is a rectangular torus.

In this paper, we study the other important but more challenging case that $E_{\tau}$ is a rhombus torus, by developing different approaches from \cite{CFL, CFL-II}.

As applications, we show that the curvature equation $\Delta u+e^{u}=4\pi(\delta_p+\delta_{-p})$ on $E_{\tau}$ has exactly either $0$, $1$ or $2$ even axisymmetric solutions and each number really occurs.