In this subsection, we will focus on the rescaled geometry of each region defined in Section 7.1.3, which can be viewed as a geometric preparation for defining the weighted Hölder space. In this direction, a necessary technical preparation is to rescale $(\mathcal{M},g)$ by correctly choosing some rescaled metric $\tilde{g}=\lambda^{2}g$ such that the weighted Hölder space in the rescaled space is much easier to analyze.

In our context, we will discuss a sequence $(\mathcal{M}_{j},g_{j})$ with a sequence of gluing parameters $\beta_{j}\to\infty$. In the remaining part of this section, we will specify the following way of rescaling which will be consistent with the definition of the weight function. For every $\bm{x}_{j}\in\mathcal{M}_{j}$, we will choose the rescaling factors $\lambda_{j}>0$ and the corresponding rescaled metric $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$ we have the convergence

(7.46)

$$(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty}).$$

In the meanwhile, for applying the delicate tools in analysis, necessarily we need to improve the Gromov-Hausdorff convergence to some convergence with higher regularity. To this end, we will select subdomains $U_{j}\subset\mathcal{M}_{j}$ which is of almost full measure such that the above convergence keeps Riemann curvatures uniformly bounded in $U_{j}$. The main tool of proving the curvature estimates is given by Lemma 7.2 and Lemma 7.7.

Our main task is to appropriately define the rescaling factors $\lambda_{j}>0$ which depends on the different regions in the definition of the weight function. The primary scenario is the following: while $d_{g_{j}}(p_{m},\bm{x}_{j})$ is increasing and $\bm{x}_{j}$ is moving from the monopoles in the neck region to the Tian-Yau pieces, the limiting geometries of the rescaled limits vary in a natural way. First, around the monopoles, the rescaled limit is the standard Taub-NUT space such that the $S^{1}$-fiber at infinity equals $1$. The advantage of rescaling in this way is that the local geometry around the monopoles can be captured in the rescaled limit. When $d_{g_{j}}(p_{m},\bm{x}_{j})$ is increasing, the length of the $S^{1}$ of the Taub-NUT space is decreasing such that the rescaled limit will collapse to $\mathbb{R}^{3}$. When $\bm{x}_{j}$ is farther from the monopoles, the size of the $\mathbb{T}^{2}$-fiber will be shrinking such that the rescaled limit will become $\mathbb{T}^{2}\times\mathbb{R}$. Eventually when $\bm{x}_{j}$ is located in the Tian-Yau pieces, we choose the original scale so that we will obtain a complete Tian-Yau space.

In this subsection, we focus on the blowing-up geometry around each monopole in the neck region $\mathcal{N}_{m_{0}}^{4}(-T_{-},T_{+})$. We will prove that, by correctly rescaling the Gibbons-Hawking metric defined in the above section, the blowing-up limit around each monopole is the Taub-NUT space.

Let $(\mathbb{T}^{2}\times\mathbb{R},g_{0})$ be a cylinder with a flat product metric $g_{0}$. Given a constant $\beta>0$, let $V_{\beta}$ be a harmonic function such that

(7.47)

$\displaystyle\begin{split}-\Delta_{g_{0}}V_{\beta}&=2\pi\sum\limits_{m=1}^{m_{0}}\delta_{p_{m}}\\ V_{\beta}(\bm{x})&=\frac{1}{2|\bm{x}-p_{m}|}+h_{m}(p)+\beta,\ \bm{x}\in B_{r_{0}}(p_{m}),\end{split}$

where each $h_{m}$ is a bounded harmonic function, $r_{0}\equiv\frac{1}{2}\min\{d_{0},i_{0}\}$ and

(7.48)

$\displaystyle\begin{split}d_{0}&\equiv\min\limits_{1\leq m<l\leq m_{0}}d_{g_{0}}(p_{m},p_{l})\\ i_{0}&\equiv\InjRad_{g_{0}}(\mathbb{T}^{2}\times\mathbb{R}).\end{split}$

Let $(\mathcal{N}_{m_{0}}^{4},g_{\beta})$ be the Gibbons-Hawking space defined by

(7.49)

$$g_{\beta}\equiv V_{\beta}g_{\mathbb{T}^{2}\times\mathbb{R}}+V_{\beta}^{-1}\theta^{2},$$

where $\theta$ is a connection $1$-form with

(7.50)

$$d\theta=*dV_{\beta}.$$

Given any positive constant $\sigma>0$, we define the rescaled metric as follows,

(7.51)

$\displaystyle\begin{split}\lambda_{\sigma,\beta}&\equiv\sigma\cdot\beta^{\frac{1}{2}}\\ \tilde{g}_{\sigma,\beta}&\equiv(\lambda_{\sigma,\beta})^{2}g_{\beta}.\end{split}$

Then we have the following useful lemma.

Returning to the analysis of Region $\I$: In this case, we choose $\lambda_{j}\equiv(\beta_{j})^{\frac{1}{2}}$ and the corresponding metric $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$. Applying Lemma 7.9, the rescaled spaces converge to the standard Taub-NUT space, i.e.,

(7.60)

$$(\mathcal{M},\tilde{g}_{j},p_{m})\xrightarrow{GH}(\mathbb{R}^{4},\tilde{g}_{\infty},p_{m,\infty}),$$

where the length of the $S^{1}$-fiber at infinity equals $1$. By the regularity theory of non-collapsing Einstein manifolds, the above convergence can be improved to $C^{\infty}$ everywhere.

We will analyze the convergence rescaled spaces for every fixed reference point $\bm{x}_{j}$ in Region $\II$. To understand the geometries of the rescaled limits, we will break down this region in three different cases which depend on the distance of a reference point $\bm{x}_{j}$ to the monopoles:

1. (a)

   There is a uniform constant $\sigma_{0}>0$ such that $2\beta_{j}^{-\frac{1}{2}}\leq d_{m}(\bm{x}_{j})\leq\frac{1}{\sigma_{0}}\cdot\beta_{j}^{-\frac{1}{2}}$.

2. (b)

   The distance to a pole $d_{m}(\bm{x}_{j})$ satisfies

   (7.61)

   $\displaystyle\frac{d_{m}(\bm{x}_{j})}{\beta_{j}^{-\frac{1}{2}}}\to\infty,\mbox{ and }\frac{d_{m}(\bm{x}_{j})}{\beta_{j}^{\frac{1}{2}}}\to 0.$

3. (c)

   There is some uniform constant $C_{0}>0$ such that

   (7.62)

   $$0<C_{0}\cdot\beta_{j}^{\frac{1}{2}}\leq d_{m}(\bm{x}_{j})\leq\frac{\iota_{0}^{\prime}}{4}\cdot\beta_{j}^{\frac{1}{2}}.$$

In Case (a) and Case (b), we choose

(7.63)

$$\lambda_{j}\equiv\frac{1}{d_{m}(\bm{x}_{j})}$$

and define the rescaled metric $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$. Immediately, $d_{\tilde{g}_{j}}(p_{m},\bm{x}_{j})=1$. In Case (c), we denote $d_{j}\equiv\min\limits_{1\leq m\leq m_{0}}d_{m}(\bm{x}_{j})$ and define the following rescaled metric by $\tilde{g}_{j}\equiv\lambda_{j}^{2}g_{j}$ and

(7.64)

$$\lambda_{j}\equiv\frac{1}{d_{j}}.$$

Now we proceed to describe the rescaled limits in each of the above cases. Applying Lemma 7.9 to Case (a), the rescaled spaces converge to a Ricci-flat Taub-NUT space with a monopole $p_{m,\infty}$, i.e.,

(7.65)

$$(\mathcal{M},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(\mathbb{R}^{4},\tilde{g}_{\infty},\bm{x}_{\infty}),$$

where $d_{\tilde{g}_{\infty}}(\bm{x}_{\infty},p_{m,\infty})=1$ and the $S^{1}$-fiber at infinity has length at least $\sigma_{0}>0$. Moreover, the above convergence is $C^{\infty}$ everywhere.

In Case (b), we have the convergence

(7.66)

$$\Big(\mathcal{M}\setminus B^{g_{j}}_{\beta_{j}^{-\frac{1}{2}}}(p_{m}),\tilde{g}_{j},\bm{x}_{j}\Big)\xrightarrow{GH}(\mathbb{R}^{3}\setminus\{0^{3}\},g_{\mathbb{R}^{3}},\bm{x}_{\infty}),$$

where $g_{\mathbb{R}^{3}}$ is the standard Euclidean metric in $\mathbb{R}^{3}$. In terms of the rescaled metrics $\tilde{g}_{j}$, the diameters of the fibers converge in the following way,

(7.67)

$\displaystyle\diam_{\tilde{g}_{j}}(S^{1})\leq C\cdot\beta_{j}^{-\frac{1}{2}}\cdot\frac{1}{d_{m}(\bm{x}_{j})}\to 0.$

Denote $\gamma_{j}\equiv\frac{\beta_{j}^{\frac{1}{2}}}{d_{m}(\bm{x}_{j})}$ and choose the rescaled coordinates

(7.68)

$$x_{j}\equiv\gamma_{j}\cdot x,\ y_{j}\equiv\gamma_{j}\cdot y,\ z_{j}\equiv\gamma_{j}\cdot z,$$

then one can check that the metric tensor $\tilde{g}_{j}$ in terms of the rescaled coordinates converges to the Euclidean metric $dx_{\infty}^{2}+dy_{\infty}^{2}+dz_{\infty}^{2}$, where $(x_{j},y_{j},z_{j})$ converges to $(x_{\infty},y_{\infty},z_{\infty})$ with $|dx_{\infty}|=|dy_{\infty}|=|dz_{\infty}|=1$. Therefore, by (7.67), the rescaled Gromov-Hausdorff limit is the punctured Euclidean space $\mathbb{R}^{3}\setminus\{0^{3}\}$. Moreover, applying Lemma 7.2 (or Lemma 7.7), it follows that the sequence converges with uniformly bounded curvature away from the origin.

In Case (c), we will prove the rescaled limit is a punctured flat cylinder. That is, let $s_{j}>0$ be a sequence of numbers such that

(7.69)

$\displaystyle s_{j}\to 0\ ,\frac{1}{s_{j}\beta_{j}}\to 0,$

then we claim that

(7.70)

$$\Big(\mathcal{M}\setminus\pi^{-1}\Big(\bigcup\limits_{m=1}^{m_{0}}B_{s_{j}}^{g_{0}}(p_{m})\Big),\tilde{g}_{j},\bm{x}_{j}\Big)\xrightarrow{GH}\Big((\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}},g_{0},\bm{x}_{\infty}\Big),$$

where $g_{0}$ is a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$.

To see this, we will carefully look at the convergence in a sequence of punctured domains with unbounded diameter. We denote $U(a,b)\equiv\{\bm{x}\in\mathcal{M}|a\leq z(\bm{x})\leq b\}$. Let $\xi_{j}>0$ be a sequence with $\xi_{j}/\beta_{j}\to 0$ and we choose a sequence of punctured domains

(7.71)

$$\mathring{U}_{j}\equiv U(z(\bm{x}_{j})-\xi_{j},z(\bm{x}_{j})+\xi_{j})\setminus\pi^{-1}\Big(\bigcup\limits_{m=1}^{m_{0}}B_{s_{j}}^{g_{0}}(p_{m})\Big),$$

where $B_{s_{j}}^{g_{0}}(p_{m})$ are balls of radii $s_{j}$ in the flat product metric $g_{0}$ on $\mathbb{T}^{2}\times\mathbb{R}$. It is straightforward that

(7.72)

$$\diam_{\tilde{g}_{j}}(\mathring{U}_{j})\approx C\cdot\xi_{j}\to\infty$$

and

(7.73)

$$\diam_{\tilde{g}_{j}}\Big(\pi^{-1}(B_{s_{j}}^{g_{0}}(p_{m}))\Big)\approx C\cdot s_{j}\to 0.$$

The above arguments show that the $\mathring{U}_{\infty}$ is a complete space minus $m_{0}$ points.

On the other hand, we will show that the metrics $\tilde{g}_{j}$ converge to a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$. In fact, for every $\bm{y}\in\mathring{U}_{j}$, there is a bounded harmonic function $h_{j}$ such that the Green’s function $V_{\beta_{j}}$ satisfies

(7.74)

$$\Big|V_{\beta_{j}}(\bm{y})-\Big(h_{j}(\bm{y})+\frac{2\pi b_{-}}{A}\cdot z(\bm{y})+\beta_{j}\Big)\Big|\leq\frac{1}{2s_{j}}.$$

By the assumption of Case (c), for every $j$, it holds that $\gamma_{j}\equiv\frac{d_{m}(\bm{x}_{j})}{\beta_{j}^{\frac{1}{2}}}\in[C_{0},\frac{\iota_{0}^{\prime}}{4}]$. Since $\frac{1}{s_{j}\beta_{j}}\to 0$, the following holds for some uniform constant $C>0$,

(7.75)

$\displaystyle\Big|\lambda_{j}^{2}V_{\beta_{j}}(\bm{y})-\frac{1}{\gamma_{j}^{2}}\Big|$

$\displaystyle=\frac{|V_{\beta_{j}}(\bm{y})-\beta_{j}|}{\gamma_{j}^{2}\beta_{j}}\leq\frac{C+C\cdot\xi_{j}+\frac{1}{2s_{j}}}{\gamma_{j}^{2}\beta_{j}}=\frac{\frac{C}{\beta_{j}}+\frac{C\xi_{j}}{\beta_{j}}+\frac{1}{2s_{j}\beta_{j}}}{\gamma_{j}^{2}}\to 0.$

Therefore, applying (7.72), (7.73) and (7.75), we have

(7.76)

$$(\mathring{U}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big((\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}},g_{0},\bm{x}_{\infty}\Big),$$

where $g_{0}$ is a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$ and $\mathcal{P}_{m_{0}}$ has $m_{0}$ points. Similar to Case (b), Applying Lemma 7.2 (or Lemma 7.7), it follows that the sequence converges with uniformly bounded curvature away from the monopoles.

For every fixed $\bm{x}_{j}$ in Region $\III$, we define $\lambda_{j}\equiv\beta_{j}^{-\frac{1}{2}}$ and $\tilde{g}_{j}\equiv\lambda_{j}^{2}g_{j}$. Let $s_{j}>0$ be a sequence of numbers such that

(7.77)

$\displaystyle s_{j}\to 0,\ \frac{1}{s_{j}\beta_{j}}\to 0,$

then applying the arguments in Case (c) of Region $\II$, we have

(7.78)

$$\Big(\mathcal{M}\setminus\pi^{-1}\Big(\bigcup\limits_{m=1}^{m_{0}}B_{s_{j}}^{g_{0}}(p_{m})\Big),\tilde{g}_{j},\bm{x}_{j}\Big)\xrightarrow{GH}\Big((\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}},g_{0},\bm{x}_{\infty}\Big),$$

where $g_{0}$ is a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$. Applying Lemma 7.2 (or Lemma 7.7), it follows that the sequence converges with uniformly bounded curvature away from the monopoles.

For fixed $\bm{x}_{j}$ in Region $\IV_{-}$, we choose the following rescaling factor

(7.79)

$$\lambda_{j}\equiv(L_{-}(\bm{x}_{j}))^{-1}$$

and the corresponding rescaled metric $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$. To start with, let us estimate the lower bound of the rescaled distance from $\bm{x}_{j}$ to a monopole. For every $\bm{x}_{j}$ in Region $\VI_{-}$, by the definition of this region, we have that

(7.80)

$\displaystyle d_{\tilde{g}_{j}}(p_{m},\bm{x}_{j})\geq\frac{10T_{0}^{\prime}\cdot\beta_{j}^{\frac{1}{2}}}{L_{-}(\bm{x}_{j})}=\frac{10T_{0}^{\prime}\cdot\beta_{j}^{\frac{1}{2}}}{\Big(\frac{2\pi b_{-}}{A}\cdot z(\bm{x}_{j})+\beta_{-}+\beta_{j}\Big)^{\frac{1}{2}}}.$

If $\beta_{j}>0$ is sufficiently large, then immediately

(7.81)

$$d_{\tilde{g}_{j}}(p_{m},\bm{x}_{j})\geq 5T_{0}^{\prime}>0.$$

Now we consider the following cases:

1. (a)

   There is a constant $C_{0}>10T_{0}^{\prime}$ independent of $j$ such that

   (7.82)

   $$5T_{0}^{\prime}\leq d_{\tilde{g}_{j}}(p_{m},\bm{x}_{j})\equiv\lambda_{j}\cdot d_{m}(\bm{x}_{j})\leq C_{0}$$

   for each $1\leq m\leq m_{0}$.

2. (b)

   The reference points $\bm{x}_{j}$ in Region $\IV_{-}$ satisfy

   (7.83)

   $$d_{\tilde{g}_{j}}(p_{m},\bm{x}_{j})\equiv\lambda_{j}\cdot d_{m}(\bm{x}_{j})\to\infty,$$

In Case (a), we have the convergence

(7.84)

$$(\mathcal{M},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big((\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}},g_{0},\bm{x}_{\infty}\Big),$$

where $g_{0}$ is a flat product metric and the set $\mathcal{P}_{m_{0}}$ contains $m_{0}$ point. To see this, first we notice that there is some constant $C>0$ such that

(7.85)

$$|z(\bm{x}_{j})|\leq C.$$

Let $\xi_{j}>0$ be a sequence satisfying $\xi_{j}\to\infty$ and $\frac{\xi_{j}}{\beta_{j}}\to 0$, and denote

(7.86)

$$U(a,b)\equiv\{\bm{x}\in\mathcal{M}|a\leq z(\bm{x})\leq b\}.$$

For fixed $\bm{x}_{j}$ in Region $\IV_{-}$, we choose a punctured domain

(7.87)

$$\mathring{U}_{j}\equiv U(z(\bm{x}_{j})-\xi_{j},z(\bm{x}_{j})+\xi_{j})\setminus\pi^{-1}\Big(\bigcup\limits_{m=1}^{m_{0}}B_{s_{j}}^{g_{0}}(p_{m})\Big),$$

where $B_{s_{j}}^{g_{0}}(p_{m})$ are balls of radii $s_{j}$ in the flat product metric $g_{0}$ on $\mathbb{T}^{2}\times\mathbb{R}$ and $s_{j}>0$ is a sequence of numbers satisfying

(7.88)

$\displaystyle s_{j}\to 0,\ \frac{1}{s_{j}\beta_{j}}\to 0.$

It is straightforward that

(7.89)

$$\diam_{\tilde{g}_{j}}(\mathring{U}_{j})\approx C\cdot\xi_{j}\to\infty$$

and the limit space $\mathring{U}_{\infty}$ has two ends. Moreover,

(7.90)

$$\diam_{\tilde{g}_{j}}\Big(\pi^{-1}(B_{s_{j}}^{g_{0}}(p_{m}))\Big)\approx C\cdot s_{j}\to 0.$$

Therefore, the limit space $\mathring{U}_{\infty}$ is a complete space minus $m_{0}$ points.

Next, we will show $\tilde{g}_{j}$ converges to a flat product metric $g_{0}$ on $\mathbb{T}^{2}\times\mathbb{R}$. To this end, it suffices to show that

(7.91)

$$\frac{V_{\beta_{j}}(\bm{y})}{(L_{-}(\bm{x}_{j}))^{2}}\to 1.$$

In fact, for every $\bm{y}\in\mathring{U}_{j}$, there is a bounded harmonic function $h_{j}$ such that the Green’s function $V_{\beta_{j}}$ satisfies

(7.92)

$$\Big|V_{\beta_{j}}(\bm{y})-\Big(h_{j}(\bm{y})+\frac{2\pi b_{-}}{A}\cdot z(\bm{y})+\beta_{j}\Big)\Big|\leq\frac{1}{2s_{j}}.$$

Since $\frac{1}{s_{j}\beta_{j}}\to 0$, the following holds for some uniform constant $C>0$,

(7.93)

$\displaystyle\Big|\frac{V_{\beta_{j}}(\bm{y})}{(L_{-}(\bm{x}_{j}))^{2}}-1\Big|=\frac{|V_{\beta_{j}}(\bm{y})-(L_{-}(\bm{x}_{j}))^{2}|}{\Big|\frac{2\pi b_{-}}{A}\cdot z(\bm{x}_{j})+\beta_{-}+\beta_{j}\Big|}\leq\frac{C+C\xi_{j}+\frac{1}{2s_{j}}}{\beta_{j}-C}\to 0.$

Therefore, applying (7.89), (7.90) and (7.93), we have

(7.94)

$$(\mathring{U}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big((\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}},g_{0},\bm{x}_{\infty}\Big),$$

where $g_{0}$ is a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$ and $\mathcal{P}_{m_{0}}$ has $m_{0}$ points. Moreover, by Lemma 7.2 (or Lemma 7.7), it follows that the sequence converges with uniformly bounded curvature away from the monopoles.

In Case (b), it holds that

(7.95)

$$(\mathcal{M},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(\mathbb{T}^{2}\times\mathbb{R},g_{0},\bm{x}_{\infty}),$$

where $g_{0}$ is a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$. The proof of this is similar to the previous case. Here we choose the domain

(7.96)

$$U_{j}\equiv U(z(\bm{x}_{j})-\xi_{j},z(\bm{x}_{j})+\xi_{j}),$$

where the sequence of numbers $\xi_{j}>0$ satisfy $\xi_{j}\to\infty$ and $\frac{\xi_{j}}{\beta_{j}}\to 0$. Then the same arguments show that

(7.97)

$$(U_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(\mathbb{T}^{2}\times\mathbb{R},g_{0},\bm{x}_{\infty}),$$

where $g_{0}$ is a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$. By Lemma 7.2 (or Lemma 7.7), it follows that the sequence converges with uniformly bounded curvature in any compact subset containing $\bm{x}_{j}$ of bounded diameter.

For every fixed reference point $\bm{x}_{j}$ in Region $\IV_{+}$, we choose the rescaling factor

(7.98)

$$\lambda_{j}\equiv(L_{+}(\bm{x}_{j}))^{-1}$$

and the rescaled metric $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$. The rescaled limits are the same as those in Region $\IV_{-}$

For every fixed reference point $\bm{x}_{j}$ in Region $\V_{-}$, we choose the rescaling factor

(7.99)

$$\lambda_{j}\equiv(\underline{L}_{-}(\bm{x}_{j}))^{-1}$$

and the rescaled metric $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$. Let $(X_{b_{-}}^{4},g_{b_{-}},q_{-})$ be a Tian-Yau space in our context with a fixed reference point $q_{-}\in X_{b_{-}}^{4}$. We need to analyze the following cases:

1. (a)

   Assume $z_{-}(\bm{x}_{j})\to\infty$.

2. (b)

   Assume that there is some constant $C_{0}>0$ independent of the index $j$ such that $10\zeta_{0}^{-}\leq z_{-}(\bm{x}_{j})\leq C_{0}$.

In Case (a), we have the convergence

(7.100)

$\displaystyle(\mathcal{M},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(\mathbb{T}^{2}\times\mathbb{R},g_{0},\bm{x}_{\infty}),$

where $g_{0}$ is a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$. To see this. we denote $\zeta_{j}\equiv z_{-}(\bm{x}_{j})\to\infty$. Let $\xi_{j}>0$ satisfy

(7.101)

$\displaystyle\xi_{j}\to\infty,\ \frac{\xi_{j}}{\zeta_{j}}\to 0,$

and we choose a unbounded domain

(7.102)

$$U_{j}^{-}\equiv U^{-}(\zeta_{j}-\xi_{j},\zeta_{j}+\xi_{j})=\{\bm{y}\in\mathcal{M}|\zeta_{j}-\xi_{j}\leq z_{-}(\bm{y})\leq\zeta_{j}+\xi_{j}\}.$$

We will show that

(7.103)

$$(U_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(\mathbb{T}^{2}\times\mathbb{R},g_{0},\bm{x}_{\infty}),$$

where $g_{0}$ is a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$. Applying the similar arguments as before, we have

(7.104)

$$\diam_{\tilde{g}_{j}}(U_{j})\to\infty$$

and the limit space $U_{\infty}$ has two ends. It follows that $U_{\infty}$ is complete. In addition, we need to show that $\tilde{g}_{j}$ converges to a flat product metric on $\mathbb{T}^{2}\times\mathbb{R}$. In fact,

(7.105)

$\displaystyle\Big|\frac{V_{-}(\bm{y})}{(\underline{L}_{-}(\bm{x}_{j}))^{2}}-1\Big|=\frac{|V_{-}(\bm{y})-(\underline{L}_{-}(\bm{x}_{j}))^{2}|}{\frac{2\pi b_{-}}{A}\cdot\zeta_{j}}\leq\frac{C(\xi_{j}+e^{-\frac{\epsilon_{0}\zeta_{j}}{2}})}{\frac{2\pi b_{-}}{A}\cdot\zeta_{j}}\to 0.$

By Lemma 7.2 (or Lemma 7.7), it follows that the sequence converges with uniformly bounded curvature in any compact subset containing $\bm{x}_{j}$ of bounded diameter, and this finishes the analysis of Case (a).

In Case (b), since $d(q_{-},\bm{x}_{j})\leq C_{0}$, there is some constant $C_{0}^{\prime}>0$ (depending only on the constant $C_{0}>0$ and the geometric data of $g_{b_{-}}$) such that

(7.106)

$$\frac{1}{C_{0}^{\prime}}\leq\underline{L}_{-}(\bm{x}_{j})\leq C_{0}^{\prime}.$$

Therefore, the limit space $(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ is a complete Ricci-flat Tian-Yau space which is a simple rescaling of $(X_{b_{-}}^{4},g_{b-},q_{-})$. The convergence in this case is moreover smooth on compact subsets.

For every fixed reference point $\bm{x}_{j}$ in Region $\V_{+}$, we choose the rescaling factor

(7.107)

$$\lambda_{j}\equiv(\underline{L}_{+}(\bm{x}_{j}))^{-1}$$

and the rescaled metric $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$. So the rescaling geometries are the same as those in Region $\V_{-}$.

We choose $\lambda_{j}\equiv 1$ and the limit is $(X_{b_{-}}^{4},g_{b_{-}},q_{-})$ which is a complete Tian-Yau space.

We choose $\lambda_{j}\equiv 1$ and the limit is $(X_{b_{+}}^{4},g_{b_{+}},q_{+})$ which is a complete Tian-Yau space.

The above arguments completely classify all the rescaled limit spaces. We end this section by proving the following lemmas which will be used in the proof of Proposition 9.2 in Section 9. We will choose a convenient way to study the convergence of differential $1$-forms in the rescaled spaces. The lemma below shows that, in each part with a collapsing circle bundle structure, every differential $1$-form is equivalent to its $4$-tuple of coefficient functions.

The following Lemma will be used throughout the following sections, and its simple proof is left to the reader.

The following Lemma will also be very useful in the following sections.