The proof consists of two steps.

In the first step, we will apply separation of variables to show that if a harmonic function $u$ satisfies (5.182), then $u(z,\bm{y})=k_{0}\cdot z+c_{0}+h(z,\bm{y})$ for some $k_{0},c_{0}\in\mathbb{R}$ and $h(z,\bm{y})$ has some exponential decaying rate.

Since $u$ is smooth, for any fixed $z\geq 1$, we have the fiber-wise $L^{2}$-expansion of $u$ as follows,

where $\bm{y}\in Y^{2n-1}$ and $u_{k}$ satisfies the equation

for some $j_{k}\in\mathbb{N}$ and $\lambda_{k}\geq 0$. Notice that the expansion (5.185) converges in the $C^{\infty}$-topology. This follows from Lemma 5.13, Lemma 3.32 and the Weyl law for spectrum asymptotics.

For $k=0$ we have $j_{k}=\lambda_{k}=0$, and $u_{k}$ is a linear function of the form $\kappa_{0}\cdot z+c_{0}$. For $k\geq 1$, we can write $u_{k}$ as a linear combination of the two linearly independent solutions discussed in Section 5.2 and 5.3.

where $\mathcal{G}_{k}$ is a growing and $\mathcal{D}_{k}$ is decaying.

We claim $C_{k}^{*}=0$ for all $k\in\mathbb{Z}_{+}$. To see this, we apply Lemma 5.13 to $u(z,\bm{y})$, then for all $k\in\mathbb{Z}_{+}$

So the claim follows from the asymptotics of $\mathcal{G}_{k}(z)$ in Lemma 5.4 and 5.7 which corresponds to $j_{k}=0$ and $j_{k}\in\mathbb{Z}_{+}$ respectively.

Now we define

It suffices to show $h(z,\bm{y})$ decays at the desired rate. Let $z_{0}>(2\delta_{b})^{2/n}$ be sufficiently big so that $u$ is defined on $\{z\geq z_{0}\}$. Now we fix $K_{0}\equiv 2n+1$. Applying Lemma 5.13 to $u(z_{0},\bm{y})$ we get for all $k\in\mathbb{Z}_{+}$,

We separate in several cases. First, we consider $k\in\mathbb{Z}_{+}$ with $j_{k}=0$. Applying (5.74), then for any $\epsilon>0$ with $\underline{\delta}=(1-\epsilon)\delta_{b}<\delta_{b}$, if $z\geq\frac{1}{\epsilon^{\frac{2}{n}}}\cdot z_{0}$,

This implies that

where the eigenfunction estimate

follows from Lemma 3.32.

When $j_{k}\in\mathbb{Z}_{+}$ we divide into two cases. When $Q\geq 1$ we apply Corollary 5.9.1 and Lemma 5.11 (with $\eta=2\delta_{b}$) to get

Now when $Q\leq 1$ we apply instead Corollary 5.12.1 to get

Summing up all the above we get

Since $K_{0}=2n+1$ we see the series converges. So the proof of the first step is done.

The second step is to prove the higher decaying estimate for the error function $h(z,\bm{y})$, which follows from the uniform Schauder estimate. We have proved that the error function $h(z,\bm{y})$ as a harmonic function satisfies

By explicit and straightforward computations, a Calabi space $(\mathcal{C}^{n},g_{\mathcal{C}^{n}})$ is collapsing with bounded curvatures as $z\to+\infty$. We just lift the harmonic function $h$ to the local universal cover which is non-collapsed with uniformly bounded geometry. So the following Schauder estimate holds for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ on the local universal cover,

where $r_{0}>0$ is some fixed constant of some definite size which is independent of $\bm{x}\in\mathcal{C}^{n}$. In particular, at the center $\bm{x}=(z,\bm{y})$, we have

This completes the proof of (5.184).

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