We begin with some explanations about our strategy. The generalised Gibbons-Hawking ansatz is convenient for producing the metric ansatz $g^{(1)}$, but very difficult for proving nonlinear existence theorems, due to the singularity issues caused by the distributional equation. So instead we will shift to the complex geometric viewpoint on $\mathbb{C}^{3}$ and attempt to solve the complex Monge-Ampère equation.

One minor problem is that there is no guarantee for $g^{(1)}$ to be smooth at the origin. So we do a surgery at the scale $A^{1/4}|\vec{\mu}|_{a}\lesssim 1$, namely the scale $|z_{1}|,|z_{2}|,|z_{0}|\lesssim A^{-1/4}$. In the annulus $\{1\leq A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 2\}$, we write $\omega^{(1)}=\sqrt{-1}\partial\bar{\partial}\phi^{(1)}$ which is $C^{\infty}$-equivalent to $\sqrt{-1}\sum_{i}dz_{i}\wedge d\bar{z}_{i}$. Using a cutoff function

$$\chi=\begin{cases}0\quad&A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 1,\\ 1&A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\geq 1.5,\end{cases}$$

we replace $\omega^{(1)}$ in the complex ball $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 2\}$ by $\sqrt{-1}\partial\bar{\partial}(\chi\phi^{(1)})$, which is now smooth but loses positive definiteness. The remedy is to add to $\sqrt{-1}\partial\bar{\partial}(\chi\phi^{(1)})$ a smooth semipositive closed $(1,1)$-form, which is compactly supported in $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 2\}$, and larger than $C\sqrt{-1}\sum_{i}dz_{i}\wedge d\bar{z}_{i}$ on $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 1.5\}$ for some sufficiently large constant $C$. Let us call the modified Kähler form $\omega^{(2)}$, which clearly agrees with $\omega^{(1)}$ outside $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 2\}$, and is $C^{\infty}$-equivalent to $\sqrt{-1}\sum_{i}dz_{i}\wedge d\bar{z}_{i}$ inside $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 3\}$. Morever, with a little care in the construction the symmetries of $\omega^{(1)}$ persist on $\omega^{(2)}$. The associated Kähler metric is $g^{(2)}$, and we shall refer to both $g^{(2)}$ and $\omega^{(2)}$ interchangably. This metric is clearly complete.

A caveat is that the functions $\mu_{1},\mu_{2}$ on $\mathbb{C}^{3}$ are no longer the moment coordinates for $\omega^{(2)}$. Furthermore in the smooth structure induced by the complex structure on $\mathbb{C}^{3}$, the functions $\mu_{1},\mu_{2}$ may not be smooth at the origin. Henceforth in this Chapter we will abuse notation to denote $\mu_{1},\mu_{2}$ as their mollified version. In other words, we perform a surgery to the fibration $\mathbb{C}^{3}\xrightarrow{(\mu_{1},\mu_{2},\eta)}\mathbb{R}^{4}$ inside the ball $\{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}\leq A^{-1/2}\}$ to make it defined by a smooth map.

We can now introduce the global weighted Hölder norms $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}$ for $T^{2}$-invariant tensors $T$ on $\mathbb{C}^{3}$.

• •

  In the region (2.12) away from the discriminant locus, the norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}$ is equivalent to $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta+\tau}}$ defined in Section 2.2.

• •

  In the region (2.14) close to $\mathfrak{D}_{1}$ but far from the origin, the norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}$ is equivalent to $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}}$ introduced in Section 2.3. Similarly with the regions close to $\mathfrak{D}_{2},\mathfrak{D}_{3}$ and far away from the origin.

• •

  In the region $A^{1/4}|\vec{\mu}|_{a}\lesssim C_{1}$ where the metric $\omega^{(2)}$ is $C^{\infty}$-equivalent to $\sqrt{-1}\sum dz_{i}\wedge d\bar{z}_{i}$ and $|z_{i}|\lesssim A^{-1/4}$, the norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}$ is equivalent to the normalised $C^{k,\alpha}$-norm on the complex ball of radius $\sim A^{-1/4}$. For example on this ball the mollified functions $\mu_{i}$ satisfy $\left\lVert\mu_{i}\right\rVert_{C^{k,\alpha}}\lesssim A^{-1/2}$ for $i=1,2$.

The point is that these regions cover the entire $\mathbb{C}^{3}$ and the norms are equivalent on overlapping regions, where the equivalence factor is independent of $A$. These norms define the corresponding Banach spaces of $T^{2}$-invariant functions/tensors on $\mathbb{C}^{3}$. The spaces which are most relevant for us are $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})$, $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\Lambda^{1})$, $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{Sym}^{2})$ and $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\Lambda^{1,1})$, corresponding to functions, 1-forms, real symmetric 2-tensors and real (1,1)-forms.

The volume form error function $E^{(2)}$ is defined by

(2.19)

$$\frac{3}{4}(1+E^{(2)})\sqrt{-1}\Omega\wedge\overline{\Omega}=(\omega^{(2)})^{3}.$$

By construction $E^{(2)}=E^{(1)}$ outside of the compact region where the surgery takes place. It follows from the discussions of Section 2.2 and 2.3 that