There are constants $\beta_{-},\beta_{+}\in\mathbb{R}$ and $k_{-},k_{+}\in\mathbb{R}$ with

(2.30)

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

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

(2.31)

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

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