Let $(\mathbb{T}^{2}\times\mathbb{R},g_{0})$ be a flat cylinder with a flat product metric $g_{0}$. Given a finite set $\mathcal{P}_{m_{0}}\equiv\{p_{1},\ldots,p_{m_{0}}\}\subset\mathbb{T}^{2}\times\mathbb{R}$, there is a sign changing Green’s function $V_{\infty}$ with

(2.48)

$$-\Delta_{g_{0}}V_{\infty}=2\pi\sum\limits_{k=1}^{m_{0}}\delta_{p_{k}},$$

and there are linear functions $L_{\pm}(z)=k_{\pm}z+\beta_{\pm}$ with

(2.49)

$$k_{-}=-k_{+}=\frac{\pi m_{0}}{\Area_{g_{0}}(\mathbb{T}^{2})}>0$$

such that for all $k\in\mathbb{N}$,

(2.50)

$\displaystyle\begin{split}|\nabla^{k}_{g_{0}}(V_{\infty}(z)-L_{-}(z))|&=O(e^{\sqrt{\lambda_{1}}z}),\ z\to-\infty,\\ |\nabla^{k}_{g_{0}}(V_{\infty}(z)-L_{+}(z))|&=O(e^{-\sqrt{\lambda_{1}}z}),\ z\to+\infty,\\ \end{split}$

where $\lambda_{1}>0$ is the smallest eigenvalue of $-\Delta_{\mathbb{T}^{2}}$.