We let

(5.238)

$$\underline{\delta}_{0}\equiv\min(\frac{\delta}{10},\delta_{b}).$$

Let $u$ be a harmonic function on the $\delta$-asymptotically Calabi space $(X^{2n},g)$, which satisfies

(5.239)

$$u=O(e^{\underline{\delta}_{0}\cdot z^{\frac{n}{2}}}).$$

By assumption, there exists some large constant $z_{1}\gg 1$, and a diffeomorphism

(5.240)

$$\Phi:[z_{1},+\infty)\times Y^{2n-1}\to X^{2n}\setminus K$$

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

(5.241)

$$\|\nabla^{k}(\Phi^{*}g-g_{\mathcal{C}^{n}})\|_{g_{\mathcal{C}^{n}}}\leq Ce^{-\delta\cdot z^{\frac{n}{2}}}.$$

By the Lemma 5.17, there is some large constant $z_{0}\gg 1$ such that

(5.242)		$\displaystyle\Delta_{g_{\mathcal{C}^{n}}}u$	$\displaystyle=\phi,$
(5.243)		$\displaystyle|\nabla^{k}_{g_{\mathcal{C}^{n}}}\phi|$	$\displaystyle=O(e^{-\delta z^{\frac{n}{2}}})$

for all $z\geq z_{0}$ and $k\in\mathbb{N}$.

Then applying Proposition 5.16 on $[z_{0},+\infty)\times Y^{2n-1}$, there exists a solution to the equation

(5.244)

$$\Delta_{g_{\mathcal{C}^{n}}}v=\phi$$

such that

(5.245)

$$|v|+|\nabla v|=O(e^{-\ell\cdot z^{\frac{n}{2}}})$$

for any $\ell\in(0,\delta/2)$. Notice that, as $z\to+\infty$, curvatures are uniformly bounded in the Calabi space. Therefore, we have

(5.246)

$\displaystyle 0=\Delta_{g}(u)=\Delta_{g_{\mathcal{C}^{n}}}(u-v),$

and $u-v=O(e^{\underline{\delta}_{0}\cdot z^{\frac{n}{2}}})$. Now we are in a position to apply Proposition 5.14 to $u-v$, which shows that there is some harmonic function $h$ on the Calabi space such that

(5.247)

$$u-v=\kappa_{0}\cdot z+c_{0}+h,$$

where $|h|+|\nabla h|=O(e^{-\underline{\delta}\cdot z^{\frac{n}{2}}})$ for all $\underline{\delta}\in(0,\delta_{b})$. Also $|dz|_{g_{\mathcal{C}^{n}}}\rightarrow 0$ as $z\rightarrow\infty$, then

(5.248)

$$|du|_{g}\leq C|du|_{g_{\mathcal{C}^{n}}}\leq C(|dv|_{g_{\mathcal{C}^{n}}}+|dz|_{g_{\mathcal{C}^{n}}}+|dw|_{g_{\mathcal{C}^{n}}})\rightarrow 0,\ \ \ \ z\rightarrow\infty.$$

Since $\Delta_{g}u=0$, so it holds that

(5.249)

$$\Delta(du)=dd^{*}(du)=-d\Delta_{g}u=0.$$

By assumption, $(X^{2n},g)$ satisfies $\Ric_{g}\geq 0$, then Bochner’s formula implies that

(5.250)

$$\frac{1}{2}\Delta_{g}|du|^{2}=|\nabla du|^{2}+\Ric_{g}(du,du)\geq 0.$$

Applying the decay property of $|du|$ in (5.248) and the maximum principle,

(5.251)

$$|du|_{g}\equiv 0\ \text{on}\ X^{2n}.$$

Therefore, $u$ is a constant.

∎