We begin with an interpretation of the equation (5.1) in terms of complex geometric data. Notice in the Tian-Yau construction we have a preferred complex structure on $X$ induced from $M$, which we denote by $J$. With respect to $J$ we can write $\gamma=\gamma^{1,0}+\gamma^{0,1}$ with $\gamma^{1,0}=\overline{\gamma^{0,1}}$. Then by the Kähler identities we have

and

Thus, equation (5.1) is equivalent to

The theorem follows from Theorem 4.3 once we prove that there exists some small $\delta>0$ and a smooth function $f=O(e^{\delta z})$ such that $\bar{\partial}f=\gamma$ (note that $\Delta f=\bar{\partial}^{*}\gamma=0$).

We next give a brief outline of the proof. In Step 1, we will construct a solution $f$ to $\bar{\partial}f=\gamma$ such that $f=O(e^{\epsilon z^{2}})$ for all $\epsilon>0$. This is done using a complex geometric argument which amounts to an application of Hörmander’s weighted $L^{2}$ estimates for the $\bar{\partial}$-operator. Interestingly it does not seem to be possible to obtain the required improvement $f=O(e^{\delta z})$ using only this type of method, owing to the fact that the function $z^{a}$ is plurisubharmonic on the Calabi model space $(\mathcal{C},g_{\mathcal{C}})$ if and only if $a\geq 2$.

To overcome this problem we use the elliptic theory on $(\mathcal{C},g_{\mathcal{C}})$ developed in Section 4. Thanks to the bound $f=O(e^{\epsilon z^{2}})$ for all $\epsilon>0$ from Step 1 and the $O(e^{-(\frac{1}{2}-\epsilon)z^{2}})$ complex structure asymptotics of Proposition 3.4, it follows that $\bar{\partial}_{\mathcal{C}}f=O(e^{\delta z})$ on $(\mathcal{C},g_{\mathcal{C}})$. In particular, since $\Delta={\rm tr}(\sqrt{-1}\partial\bar{\partial})$, the Poisson equation estimates of Proposition 4.15 imply that $f$ can be decomposed into an $O(e^{\delta z})$ part $f_{1}$ and a $g_{\mathcal{C}}$-harmonic part $f_{2}$ which is $O(e^{\epsilon z^{2}})$ for all $\epsilon>0$ (see Step 2 for details). Observe that it would not be possible to compare $\Delta_{TY}f$ and $\Delta_{\mathcal{C}}f$ directly because $g_{TY}$ and $g_{\mathcal{C}}$ are only asymptotic at rate $O(e^{-\delta z})$, which is too slow to beat the $O(e^{\epsilon z^{2}})$ growth of $f$ from Step 1.

Step 3 analyzes the $g_{\mathcal{C}}$-harmonic part $f_{2}$ of $f$. It is clear from Section 4 that $f_{2}=O(e^{Cz})$ for some large constant $C$. The required improvement $f_{2}=O(e^{\delta z})$ comes from the first-order equation $\bar{\partial}_{\mathcal{C}}f_{2}=O(e^{\delta z})$ satisfied by $f_{2}$ (in addition to $\Delta_{g_{\mathcal{C}}}f_{2}=0$). Technically this is done using separation of variables for the $\bar{\partial}_{\mathcal{C}}$-operator but the underlying idea can be easily explained: being of $O(e^{Cz})$ rather than $O(e^{Cz^{2}})$ growth, the leading terms of the harmonic function $f_{2}$ must be $S^{1}$-invariant, but on $S^{1}$-invariant functions the $\bar{\partial}_{\mathcal{C}}$-operator directly controls the radial derivative $\frac{\partial}{\partial z}$.

Step 4 concludes the proof by appealing to Theorem 4.3.

The idea of the proof of Step 2 is as follows. First, we will estimate $\Delta_{\omega_{\mathcal{C}}}u$ and $\Delta_{\omega_{\mathcal{C}}}v$ and all of their derivatives. Specifically, we will prove that they have slow exponential growth rates (as shown in (5.23)). Then applying Proposition 4.15, we can construct solutions to the Poisson equations

such that $u_{1}=O(e^{\delta z})$ and $v_{1}=O(e^{\delta z})$ This completes the desired decomposition of $u$ and $v$.

To obtain the derivative estimates for $\Delta_{\omega_{\mathcal{C}}}u$ and $\Delta_{\omega_{\mathcal{C}}}v$, we will prove the derivative estimates for $du+J_{\mathcal{C}}dv$. To start with, by the assumption on $\gamma$ and the first order equation given by Step 1,

Applying the asymptotic estimate for $J_{TY}$ in Proposition 3.4, we can convert the above growth control to the corresponding estimate for $du+J_{\mathcal{C}}dv$.

In fact, applying Item (b) of Proposition 3.4, for any $\epsilon>0$ and for any $k\in\mathbb{N}$,

We also need derivative estimates for $u$ and $v$ with respect to the model metric $\omega_{\mathcal{C}}$. Notice that by Step 1, $f=u+\sqrt{-1}v=O(e^{\epsilon z^{2}})$ for any $\epsilon>0$, which implies that

where $z$ is the natural coordinate on $\mathcal{C}$. Since $u$ and $v$ satisfy $\Delta_{\omega_{TY}}u=\Delta_{\omega_{TY}}v=0,$ by applying the same $W^{k,p}$-estimate as in the proof of Theorem 4.3, we have for all $\epsilon>0$ and $k\geq 1$,

Since the asymptotic order of harmonic functions $u$ and $v$ is dominated by $e^{\epsilon z^{2}}$ and the asymptotic order of the metric $\omega_{TY}$ is $e^{-\underline{\delta}z}$, so in terms of the model metric we have

Now we apply the assumption $|\gamma|_{\omega_{TY}}=O(e^{C\delta_{h}z})$ and the above elliptic regularity to (5.14), we get for $k\in\mathbb{N}$,

Now we proceed to prove the derivative estimates for $\Delta_{\omega_{\mathcal{C}}}u$ and $\Delta_{\omega_{\mathcal{C}}}v$ by making use of the system

The advantage of the above equation is that $\Delta_{\omega_{TY}}=\Tr_{\omega_{\mathcal{C}}}(dJ_{TY}d)$ so that the behavior $\Delta_{\omega_{TY}}$ will follow from the asymptotics of $J_{TY}$. In fact, taking the differential of (5.20),

Then using Item $(a)$ of Proposition 3.4, similar to the above we have for all $k\in\mathbb{Z}_{+}$

Taking the trace, then we obtain

Applying the linear theory for $\Delta_{\omega_{\mathcal{C}}}$ in Proposition 4.15, if $\delta_{h}\ll\underline{\delta}$, then we choose two solutions $u_{1}$ and $v_{1}$ provided by Proposition 4.15

such that $u_{1}$, $v_{1}$ satisfy

So we have finished the proof of the decomposition $u=u_{1}+u_{2}$ and $v=v_{1}+v_{2}$ such that

We also obtain that

and