The studies and applications of the implicit function theorem have been well developed in various contexts. We refer the readers to the book [KP13] for seeing the comprehensive discussions and the history of the whole methodology. For our practical and specific applications, we need the following quantitative version of implicit function theorem (Lemma 6.1), which is based on Banach contraction mapping principle.

To avoid confusions, we clarify several notations as follows:

• •

  Let $\mathscr{L}:\mathfrak{A}\to\mathfrak{B}$ be a bounded linear operator between normed linear spaces $\mathfrak{A}$ and $\mathfrak{B}$, then the operator norm of $\mathscr{L}$ is defined by

  (6.1)

  $$\|\mathscr{L}\|_{op}\equiv\inf\Big\{M_{0}\in\mathbb{R}_{+}\Big|\ \|\mathscr{L}(\bm{v})\|_{\mathfrak{B}}\leq M_{0}\cdot\|\bm{v}\|_{\mathfrak{A}},\ \forall\bm{v}\in\mathfrak{A}\Big\}.$$

• •

  We use the common notation $\bm{0}$ for the zero vector in every normed linear space.

To set up the perturbation problem in our setting, we define the Banach spaces

endowed with the weighted Hölder norms

Notice an $S^{1}$ invariant function $\phi$ on $\mathcal{M}_{T}$ can be identified with a function on the quotient $Q_{T}$, and the Neumann boundary condition $\frac{\partial\phi}{\partial n}|_{\partial\mathcal{M}_{T}}=0$ amounts to the condition $\partial_{z}\phi=0$ on $\partial Q_{T}$.

In this section, the weight parameters are specified as follows:

1. (NP1)

   (Fix $\nu$) The parameters $\nu\in\mathbb{R}$ is chosen such that

   (6.10)

   $\displaystyle\nu\in(-1,0).$

   In our context, Lemma 6.5 requires $\nu\in(-1,1)$. To effectively apply Proposition 4.23, we need $\nu\in(-1,0)$.

2. (NP2)

   (Fix $\alpha$) The Hölder order $\alpha\in(0,1)$ is chosen sufficiently small such that

   (6.11)

   $$\nu+\alpha<0.$$

3. (NP3)

   (Fix $\delta$) $\delta>0$ is chosen such that

   (6.12)

   $$0<\delta<\delta_{N}\equiv\frac{1}{n\cdot(|k_{-}|+|k_{+}|)^{n}}\cdot\min\{\delta_{b},\delta_{e},\sqrt{\lambda_{D}}\},$$

   where $\epsilon_{X}>0$ is the constant in Theorem 5.2, $\sqrt{\lambda_{D}}$ is in Lemma 6.7 (Liouville theorem on $Q$), $\delta_{b}>0$ is in Proposition 5.14 (Liouville theorem on the Calabi space $\mathcal{C}^{n}$ around the boundary of the neck), $\delta_{e}>0$ is in the error estimate Proposition 4.23.

4. (NP4)

   (Fix $\mu$) The parameter $\mu$ is fixed by

   (6.13)

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

   This condition guarantees that the weight function $\rho_{\delta,\nu,\mu}^{(\alpha)}$ with parameters specified as the above is uniformly bounded from below. This will be used in proving Proposition 6.4.

We first normalize the holomorphic volume form. For $T\gg 1$, starting with the $C^{2,\alpha}$-Kähler structure $(\omega_{T},\Omega_{T})$, we will solve the Calabi-Yau equation

Notice that

and

for some computable constant $C_{1}>0$. So by (4.14) we get

Now we replace $\Omega_{T}$ by

Then we have

where

Applying Proposition 4.23 and (6.13),

Let $\mathscr{F}$ be the map sending every $\sqrt{-1}\partial\bar{\partial}\phi\in\mathfrak{S}_{1}$ to the function which satisfies

Then (6.21) immediately tells us that

Now we are ready to state the main result in this section.

To prove Theorem 6.3, we decompose the map $\mathscr{F}:\mathfrak{S}_{1}\to\mathfrak{S}_{2}$ as follows,

for any $\sqrt{-1}\partial\bar{\partial}\phi\in\mathfrak{S}_{1}$, where

By the definition of the weight function and Lemma 4.20, we have the following nonlinear error estimate.

To apply the implicit function theorem, we still need to prove the weighted linear estimate, which will be completed in the following subsections.