We denote

$$\psi=du_{2}+J_{\mathcal{C}}dv_{2}=O(e^{C\delta_{h}z}),$$

Then we have the following expansion as in Section 4.1: let $\{\Lambda_{k}\}_{k=1}$ be the spectrum of $Y^{3}$ and $\{\varphi_{k}\}_{k=1}^{\infty}$ are the corresponding eigenfunctions on $Y$ with $\mathcal{L}_{\partial_{\theta}}\varphi_{k}=\sqrt{-1}j_{k}\varphi_{k}$,

(5.30)

$$u_{2}=\sum\limits_{k}f_{k}(z)\varphi_{k}(z_{\alpha},\theta),\ v_{2}=\sum\limits_{k}g_{k}(z)\varphi_{k}(z_{\alpha},\theta),$$

which implies that

(5.31)

$$du_{2}=\sum\limits_{k}(f_{k}^{\prime}(z)\cdot\varphi_{k}\cdot dz+f_{k}(z)\cdot d\varphi_{k}),\ dv_{2}=\sum\limits_{k}(g_{k}^{\prime}(z)\varphi_{k}\cdot dz+g_{k}(z)\cdot d\varphi_{k}).$$

On $\mathcal{C}$ by the definition in Section 4.1, we have $J_{\mathcal{C}}(zdz)=d\theta$, so we have

(5.32)

$\displaystyle\begin{split}\sum\limits_{k}\big(f_{k}^{\prime}(z)-\sqrt{-1}j_{k}\cdot z\cdot g_{k}(z)\big)\varphi_{k}&=\psi(\partial_{z})\\ \sum\limits_{k}\big(z^{-1}\cdot g_{k}^{\prime}(z)+\sqrt{-1}j_{k}\cdot f_{k}(z)\big)\varphi_{k}&=\psi(\partial_{\theta}).\end{split}$

This implies that for each $k$,

(5.33)

$$\begin{split}f_{k}^{\prime}(z)-\sqrt{-1}j_{k}\cdot z\cdot g_{k}(z)=O(e^{C\delta_{h}z})\\ z^{-1}\cdot g_{k}^{\prime}(z)+\sqrt{-1}j_{k}\cdot f_{k}(z)=O(e^{C\delta_{h}z}).\end{split}$$

There are three different cases.

If $j_{k}\neq 0$, then we can write $f_{k}$ and $g_{k}$ are given by a linear combination of one growing solution $\mathcal{F}_{k}$ and one decaying solution $\mathcal{U}_{k}$. Using the analysis in Section 4 we know that the asymptotic order of $\mathcal{F}_{k}$ is $e^{\frac{j_{k}z^{2}}{2}}$ (see Lemma 4.7). The control (5.28) then implies both $f_{k}$ and $g_{k}$ can only be a multiple of the decaying solution $\mathcal{U}_{k}=O(e^{-\frac{j_{k}z^{2}}{2}})$.

If $j_{k}=0$ and $\lambda_{k}\neq 0$, then $f_{k}$ and $g_{k}$ are given by linear combinations of the exponential functions of the form $e^{\sqrt{\lambda_{k}}z}$ and $e^{-\sqrt{\lambda_{k}}z}$. Let $\underline{\delta}>0$ be the positive constant given in Proposition 4.10, we use (5.33) and the fact that $\psi=O(e^{C\delta_{h}z})$ to conclude that, if $C\delta_{h}<\underline{\delta}$, then both $f_{k}$ and $g_{k}$ must be proportional to the decaying solutions.

If $j_{k}=0$ and $\varphi_{k}$ is constant, then $f_{k}$ and $g_{k}$ are linear functions in $z$. Now since $u_{2}$ and $v_{2}$ are harmonic functions on $\mathcal{C}$, by Lemma 4.11, we conclude that

(5.34)

$$|u_{2}|=O(z),\ |v_{2}|=O(z).$$

This completes the proof of Proposition 5.5.∎