Now we are in a position to set up the whole package to implement the global weighted analysis on the glued manifold.

To begin with, let $(\omega(t),\Omega(t))$ be the $C^{1,\alpha}$-Kähler structure constructed in Section 7.3. on $\widehat{X}_{t}$. So we define the linear spaces

	$\displaystyle\mathfrak{A}$	$\displaystyle\equiv\Big\{\sqrt{-1}\partial\bar{\partial}\phi\in\Omega^{1,1}(\widehat{X}_{t})\Big|\phi\in C^{2,\alpha}(\widehat{X}_{t})\Big\},$
(7.154)		$\displaystyle\mathfrak{B}$	$\displaystyle\equiv\Big\{f\in C^{0,\alpha}(\widehat{X}_{t})\Big|\int_{\widehat{X}_{t}}f\cdot\frac{\omega(t)^{n}}{n!}=0\Big\},$

which are equipped with the weighted norms

(7.155)		$\displaystyle\|\sqrt{-1}\partial\bar{\partial}\phi\|_{\mathfrak{A}}$	$\displaystyle\equiv\|\sqrt{-1}\partial\bar{\partial}\phi\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\widehat{X}_{t})},\quad\sqrt{-1}\partial\bar{\partial}\phi\in\mathfrak{A},$
(7.156)		$\displaystyle\|f\|_{\mathfrak{B}}$	$\displaystyle\equiv\|\sqrt{-1}\partial\bar{\partial}\phi\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\widehat{X}_{t})},\quad f\in\mathfrak{B},$

such that both $\mathfrak{A}$ and $\mathfrak{B}$ are Banach spaces. As in Section 7.3.2, the parameters $\delta,\nu,\mu$ are chosen as

(7.157)		$\displaystyle 0<\delta$	$\displaystyle<\delta_{G},$
(7.158)		$\displaystyle-1<\nu$	$\displaystyle<0.$

Morevoer, $\alpha\in(0,1)$ is sufficiently small such that

(7.159)

$$\nu+\alpha<0$$

and $\mu=(1-\frac{1}{n})(\nu+2+\alpha)$.

For $|t|\ll 1$, starting with the Kähler structure $(\omega(t),\Omega(t))$, we will solve the nonlinear equation

(7.160)

$$\frac{1}{n!}(\omega(t)+\sqrt{-1}\partial\bar{\partial}\phi)^{n}=(\sqrt{-1})^{n^{2}}2^{-n}\cdot\Omega(t)\wedge\bar{\Omega}(t).$$

Let $\mathscr{F}:\mathfrak{A}\rightarrow\mathfrak{B}$ be defined by

(7.161)

$$\mathscr{F}(\sqrt{-1}\partial\bar{\partial}\phi)\cdot\omega(t)^{n}\equiv(\omega(t)+\sqrt{-1}\partial\bar{\partial}\phi)^{n}-\omega(t)^{n}(1-\mathrm{Err}_{t}).$$

Then (7.160) is equivalent to

(7.162)

$$\mathscr{F}(\sqrt{-1}\partial\bar{\partial}\phi)=0.$$

Now we write

(7.163)

$$\mathscr{F}(v)-\mathscr{F}(0)=\mathscr{L}(v)+\mathscr{N}(v),$$

for any $v\in\mathfrak{A}$, where

(7.164)

$$\mathscr{L}(\sqrt{-1}\partial\bar{\partial}\phi)=\Delta\phi,$$

is the linearization of $\mathscr{F}$ and

(7.165)

$\displaystyle\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi)\cdot\omega(t)^{n}$

$\displaystyle=(\omega(t)+\sqrt{-1}\partial\bar{\partial}\phi)^{n}-\omega(t)^{n}-n\omega(t)^{n-1}\wedge\sqrt{-1}\partial\bar{\partial}\phi.$

The proof of the following is identical to Proposition 6.4.

The global version of the weighted Schauder estimate Proposition 4.22 takes the following form. Note that the weighted Schauder estimate on the neck is given by Proposition 6.9.

The proof is similar to the proof of Proposition 4.22. From the construction of the metric $\omega(t)$ in Section 7.3 the rescaled limit geometries will be the same as in the case of the neck $\mathcal{M}_{T}$ studied in Section 4.3, except two possible incomplete Calabi model space limits replaced by the two Tian-Yau metrics on the ends. We omit the details.

The proof is very similar to the proof of Proposition 6.8, by using a contradiction argument and applying various Liouville theorems. We omit the details and only mention two different points. The first point is that from our construction of $\omega(t)$ on $\widehat{X}_{t}$, if we rescale around points in Region $\bf{IV}_{\pm}$, then we will get the Tian-Yau spaces (instead of the incomplete Calabi model spaces) as limits, and we need to use Theorem 5.2. The second point is that the other rescaled limits will be exactly the same as considered in the proof of Proposition 6.8, and this follows from the fact that by construction our metric $\omega(t)$ away from the region $\bf{IV}_{\pm}$ is essentially a small perturbation of the neck region $(\mathcal{M}_{T},\omega_{T})$.

Now given Proposition 7.15 as before it is straightforward to see that for $|t|>0$ sufficiently small, there is a $\phi(t)\in\mathfrak{A}$ solving the Calabi-Yau equation (7.160). By uniqueness of Calabi-Yau metrics, we know $T^{-\frac{2}{n}}(\omega(t)+\sqrt{-1}\partial\bar{\partial}\phi(t))$ must agree with the Calabi-Yau metric $\omega_{CY,t^{n+2}}$ on $X_{t^{n+2}}\simeq\widehat{X}_{t}$ in the Introduction. The geometric statements in Theorem 1.1 then follow from similar arguments as in Section 6.4. We omit the details.