We focus on the region $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\gtrsim A^{1/2}\}$. The key idea is that $\tilde{\alpha}_{1}-\bar{\alpha}_{1}$ is $\Delta_{a}$-harmonic , bounded and has no zero Fourier mode in the $S^{1}$ direction defined by the $x$-variable, so the exponential decay follows from Fourier analysis. We remark that similar ideas have appeared in the recent paper [13].

We perform Fourier decomposition in the $S^{1}$ direction

$$\tilde{\alpha}_{1}-\bar{\alpha}_{1}=\sum_{n\neq 0}h_{n}(\mu_{1},\mu_{2},y)e^{2\pi inx},$$

Parseval identity combined with Lemma 3.4 shows

$$\sum_{n}|h_{n}|^{2}=\int_{0}^{1}|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|^{2}dx\leq CA^{-1/2}.$$

Now $\Delta_{a}$-harmonicity translates into the 3-dimensional Helmholtz equations:

$$\Delta_{a}^{\prime}h_{n}-4\pi^{2}n^{2}A^{-1}h_{n}=0.$$

The remaining task is conceptually speaking to estimate the Dirichlet Green’s function for the Helmholtz equation on the noncompact 3-dimensional domain $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\gtrsim A^{1/2}\}$. In practice, building an upper barrier for the Green’s function suffices for our purpose.

Recall $\varrho=|(\mu_{1},\mu_{2},y)|_{a}^{\prime}$ is the distance function for the Euclidean metric $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}_{y}$. By simple direct computation, for any $\kappa>0$,

$$(\Delta_{a}^{\prime}-4\pi^{2}n^{2}A^{-1})e^{-\kappa\varrho}\leq(\kappa^{2}-4\pi^{2}n^{2}A^{-1})e^{-\kappa\varrho},$$

so for $k_{n}=2\pi|n|A^{-1/2}$, the function $e^{-k_{n}\varrho}$ is a supersolution of the Helmholtz equation. Now we build a barrier function

$$h_{n}^{\prime}(\mu_{1},\mu_{2},y)=A^{-1/2}\int_{0}^{\infty}e^{-k_{n}|(\mu_{1},\mu_{2}-s,y)|_{a}^{\prime}}ds,$$

whose singularity lies on $\mathfrak{D}_{1}$. Since $h_{n}^{\prime}$ is a positive superposition of supersolutions, it must be itself a supersolution. Other basic properties are:

• •

  On $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\geq A^{1/2}\}$, using the saddle point method for Laplace type integrals

  $$0\leq h_{n}^{\prime}\leq CA^{-3/4}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\exp(-k_{n}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})).$$

• •

  On the boundary of $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\geq A^{1/2}\}$, we have $h_{n}^{\prime}\geq\frac{A^{-1/4}}{C|n|}$.

Since $|h_{n}|\leq CA^{-1/4}$ by the Parseval identity, the comparison principle implies

$$\begin{split}h_{n}\leq C|n|h_{n}^{\prime}\leq CA^{-3/4}|n|\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\exp(-k_{n}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})).\end{split}$$

Thus on $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\geq A^{1/2}\}$, the desired bound on $|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|$ follows by summing over these estimates over $n$. It is worth commenting that we expect the exponential decay rate to be sharp. ∎