In this subsection, we will take a closer look at the Riemannian geometric behavior of the family of incomplete Kähler metrics $(\mathcal{M}_{T},\omega_{T})$ constructed in Section 4.1 as $T\rightarrow\infty$. For clarity we now re-install the parameter $T$ throughout the rest of this section.

It is easy to see that as the parameter $T\to+\infty$, the curvatures are unbounded around the singular set $\mathcal{P}\subset\mathcal{M}_{T}$ such that the standard uniform elliptic estimates just legitimately fail. Instead, we will define some appropriate weighted Hölder spaces and establish uniformly weighted a priori estimates, which will be done in Section 4.4. Geometrically, the weighted elliptic estimate that we pursue is intimately connected with the effective regularity at definite scales of the metrics $\omega_{T}$ in various pieces of $\mathcal{M}_{T}$. More rigorously, we need the following notion.

Intuitively, the $C^{k,\alpha}$-regularity scale is the maximal zooming-in scale at which the nontrivial $C^{k,\alpha}$-geometry is uniformly bounded on the local universal cover, which maximally captures the bounded covering $C^{k,\alpha}$-geometry.

The goal of this subsection is to study the $(k,\alpha)$-regularity scale at every point for appropriate $k,\alpha$. Since the Kähler metrics $\omega_{T}$ constructed in Section 4.1 are fairly explicit, so for every $\bm{x}\in\mathcal{M}_{T}$ we will explicitly determine a canonical scale $\mathfrak{s}(\bm{x})$ which is convenient for calculations and uniformly proportional to the $(k,\alpha)$-regularity scale $r_{k,\alpha}(\bm{x})$ at $\bm{x}$, i.e.

(4.183)

$$\underline{v}_{0}\cdot r_{k,\alpha}(\bm{x})\leq\mathfrak{s}(\bm{x})\leq\bar{v}_{0}\cdot r_{k,\alpha}(\bm{x}),$$

for some uniform constants $\underline{v}_{0}$ and $\bar{v}_{0}>0$ which are independent of $T\gg 1$. For convenience, $\mathfrak{s}(\bm{x})$ will be called the regularity scale.

Before the technical computations, it is helpful to present the scenario of geometric transformations on $\mathcal{M}_{T}$ from the singular set $\mathcal{P}$ to the boundary $\partial\mathcal{M}_{T}$. First, as $T\to+\infty$, curvatures blow up if the reference point $\bm{x}$ is located around $\mathcal{P}$ so that we will rescale the metric $\omega_{T}$ giving rise to a product bubble limit $\mathbb{C}_{TN,\varrho}^{2}\times\mathbb{C}^{n-2}$, where $\mathbb{C}_{TN,\varrho}^{2}$ is the Taub-NUT space (c.f. Section 2.3) for some $\varrho>0$. This is a deepest bubble (rescaling limit) in our context. When the distance from $\bm{x}$ to $\mathcal{P}$ is increasing, the length of $S^{1}$-fiber at the infinity of the Taub-NUT space $\mathbb{C}_{TN,\varrho}^{2}\times\mathbb{C}^{n-2}$ is decreasing which corresponds to $\varrho$ is increasing. The next level of bubble corresponds to $\varrho\rightarrow\infty$, or equivalently, this amounts to getting the tangent cone at infinity of the product $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$, which is $\mathbb{R}^{2n-1}\equiv\mathbb{R}^{3}\times\mathbb{C}^{n-2}$. This is of codimension-$1$ collapse, with locally uniformly bounded curvature away from $\{0^{3}\}\times\mathbb{C}^{n-2}$. When $\bm{x}$ is getting further away from $\mathcal{P}$, the size of $D$ will be shrinking such that the next level of bubble is $D\times\mathbb{R}$. This is again a codimension-$1$ collapse, with locally uniformly bounded curvature away $P=H\times\{0\}\subset D\times\mathbb{R}$. Finally, as $\bm{x}$ moves close to the boundary $\partial\mathcal{M}_{T}$, the metrics will converge to the incomplete Calabi model metrics $\mathcal{C}^{n}_{-}$ and $\mathcal{C}^{n}_{+}$, which corresponds to applying the construction in Section 2.2 to the line bundle $L^{k_{-}}$ and $L^{k_{+}}$ over $D$.

Now we are ready to make precise subdivision for $\mathcal{M}_{T}$ and analyze different rescaling geometries (see Figure 4.1). Let $H$ be a divisor of $D$ such that the singular set $P=H\times\{0\}$ is at the slice $z=0$ of the cylinder $Q\equiv D\times\mathbb{R}$. Denote by $r(\bm{x})$ the distance from $\pi(\bm{x})$ to $P$ with respect to the product metric $g_{Q}=g_{D}+dz^{2}$ on the base $Q$.

Region $\bf{I}_{1}$:

This region consists of the points $\bm{x}$ satisfying

(4.184)

$$r(\bm{x})\leq T^{-1}.$$

In other words, this region consists of points close to the divisor $P=H\times\{0\}\subset D\times\{0\}$ which is the singular locus of the $S^{1}$-fibration.

Region $\bf{I}_{2}$:

A point $\bm{x}$ in this region satisfies

(4.185)

$$\frac{T^{-1}}{2}\leq r(\bm{x})\leq 1.$$

So this region contains the points not close, but not too far from the divisor $P$.

Region $\bf{I}_{3}$:

This region consists of the points far from the divisor $P$ such that each $\bm{x}$ satisfies the condition

(4.186)

$\displaystyle r(\bm{x})\geq\frac{1}{2},\quad T_{-}\leq z(\bm{x})\leq T_{+}.$

Notice that the above regions completely cover the neck $\mathcal{M}_{T}$ such that each overlapping region has the same geometric behavior with the adjacent regions in the above subdivision. So we will just ignore these overlaps in the following discussions.

Under the above subdivision of $\bf{I}_{1}$, $\bf{I}_{2}$ and $\bf{I}_{3}$, we will rather explicitly determine the corresponding $(k,\alpha)$-regularity scales with respect to the metric

(4.187)

$$\omega_{T}\equiv T^{\frac{2-n}{n}}(\pi^{*}\tilde{\omega}+dz\wedge\Theta).$$

Region $\bf{I}_{1}$ (the deepest bubble):

For each point $\bm{x}$ in this region, we choose

(4.188)

$$\mathfrak{s}(\bm{x})=T^{\frac{1-n}{n}}.$$

As in (2.52), let us denote by

(4.189)

$\displaystyle\begin{cases}\omega_{TN,1}=(\frac{1}{2r}+1)\cdot\frac{\sqrt{-1}}{2}\cdot dy\wedge d\bar{y}+dz\wedge\Theta_{0},\\ \Omega_{TN,1}=\sqrt{-1}((\frac{1}{2r}+1)dz+\Theta_{0})\wedge dy\end{cases}$

the Kähler form and the holomorphic form of the Taub-NUT space $\mathbb{C}_{TN}^{2}$ whose $S^{1}$-fiber at infinity has length equal to $1$.

In the following, we will carry out explicit calculations to prove that under the rescaled metric

(4.190)

$$\tilde{g}_{T}=(\mathfrak{s}(\bm{x}))^{-2}g_{T}=T^{\frac{2n-2}{n}}g_{T},$$

we have the pointed convergence

(4.191)

$$(\mathcal{M}_{T},\tilde{g}_{T},\bm{x})\xrightarrow{C^{2,\alpha}}(\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2},\omega_{TN,1}\oplus g_{\mathbb{C}^{n-2}},(\bm{0}^{2},0^{n-2}))$$

in the pointed $C^{2,\alpha}$-topology, where $\bm{0}^{2}$ is the origin of the Taub-NUT space $(\mathbb{C}_{TN}^{2},\omega_{TN,1})$. Moreover, the rescaled holomorphic volume form $T^{n-1}\cdot\Omega_{T}$ converges to $\Omega_{TN,1}$ in the $C^{2,\alpha}$-topology, where $\Omega_{TN,1}$ is the holomorphic volume form of $(\mathbb{C}_{TN}^{2},{\omega}_{TN,1})$ (c.f. Section 2.3). This implies that

(4.192)

$$\underline{v}_{0}\cdot r_{2,\alpha}(\bm{x})\leq\mathfrak{s}(\bm{x})\leq\bar{v}_{0}\cdot r_{2,\alpha}(\bm{x}),$$

where $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ are uniform constants independent of $T\gg 1$.

Fix $p\in H$, we may choose local special holomorphic coordinates $\{w_{j}\}_{j=1}^{n-1}$ in some neighborhood of $p$ in $D$ such that that

(4.193)

$$\omega_{D}=\omega_{\mathbb{C}^{n-1}}+O(|w|),$$

where

(4.194)

$$\omega_{\mathbb{C}^{n-1}}\equiv\frac{\sqrt{-1}}{2}\sum_{j=1}^{n-1}dw_{j}\wedge d\bar{w}_{j}.$$

Then by the analysis in Section 4.1, one can see that

(4.195)

$$T^{\frac{2n-2}{n}}\omega_{T}=\Big(T\omega_{TN,T}\oplus T^{2}\omega_{\mathbb{C}^{n-2}}\Big)+T^{2}\pi^{*}(\omega_{D}-\omega_{\mathbb{C}^{n-1}})+TO^{\prime}(s^{3})+Td(s^{2}\Gamma)$$

where $\omega_{TN,T}$ is the Taub-NUT metric on $\mathbb{C}^{2}_{u_{1},u_{2}}$ given by (2.52), and

(4.196)

$$\omega_{\mathbb{C}^{n-2}}\equiv\frac{\sqrt{-1}}{2}\sum_{j=2}^{n-1}dw_{j}\wedge d\bar{w}_{j}.$$

Notice that, we have already used the relations

(4.197)

$$\pi^{*}(O^{\prime}(r^{p}))=O^{\prime}(s^{2p})(p\geq 1),\quad\pi^{*}dy=\widetilde{O}(s),\quad\pi^{*}dz=\widetilde{O}(s).$$

We perform a change of coordinates

(4.198)

$$z=T^{-1}\underline{z},\ y=T^{-1}\underline{y},\ w_{j}=T^{-1}\underline{w}_{j},\ u_{k}=T^{-1/2}\underline{u}_{k}$$

and denote

(4.199)

$${\bm{w}}\equiv(\underline{w}_{2},\cdots,\underline{w}_{n-1}),\ {\bm{u}}\equiv(\underline{u}_{1},\underline{u}_{2}),\ \underline{s}=|{\bm{u}}|.$$

From now on, we write the tensors $\omega_{TN,1}$ and $\Omega_{TN,1}$ with respect to those rescaled coordinates $\bm{w}$ and $\bm{u}$, we have

(4.200)

$$T\omega_{TN,T}\equiv\omega_{TN,1},\quad T^{2}\omega_{\mathbb{C}^{n-2}}\equiv\omega_{\mathbb{C}^{n-2}},$$

where “$\equiv$” means that the two metrics are isometric. Moreover,

(4.201)

$$\begin{cases}T^{2}\pi^{*}(\omega_{D}-\omega_{\mathbb{C}^{n-1}})=O((|{\bm{w}}|+|{\bm{u}}|^{2})T^{-1})\\ TO^{\prime}(s^{3})=O(T^{-3/2}\underline{s}^{3})\\ Td(s^{2}\Gamma)=O(T^{-3/2}\underline{s}).\end{cases}$$

The above computations impies

(4.202)

$$|T^{\frac{2n-2}{n}}\omega_{T}-(\omega_{TN,1}\oplus\omega_{\mathbb{C}^{n-2}})|_{C^{2,\alpha}}=O(T^{-1}),$$

where the norm is measured with respect to the limiting product metric $\omega_{TN,1}\oplus\omega_{\mathbb{C}^{n-2}}$.

In a similar vein, by the analysis in Section 4.1, we also obtain the expansion for the holomorphic form $\Omega_{T}$,

(4.203)

$$T^{n-1}\Omega_{T}=\Omega_{TN,1}\wedge d\underline{w}_{2}\wedge\cdots\wedge d\underline{w}_{n-1}+O((|{\bm{w}}|+|{\bm{u}}|^{2})T^{-1}),$$

which gives the convergence of $T^{n-1}\Omega_{T}$.

Notice that, the above convergence is smooth away from $\bm{0}^{2}\times\mathbb{C}^{n-2}$, where $\bm{0}^{2}\in\mathbb{C}_{TN}^{2}$.

Starting from the above deepest bubble, we will let the reference point $\bm{x}$ keep away from the singular set $\mathcal{P}$ and switch to the next region where we will see that the bubbles transform from the Taub-NUT geometry to the cylindrical geometry. By definition, the reference point $\bm{x}$ in this region satisfies the relation

(4.204)

$$\frac{1}{2T}\leq r(\bm{x})\leq 1.$$

Region $\bf{I}_{2}$ (bubble transformations):

In this region, the Kähler metric $\omega_{T}$ on $\mathcal{M}_{T}$ can be viewed as the lifting metric of the Riemannian submersion $\mathcal{M}_{T}\to Q\setminus P$, i.e.,

(4.205)

$$g_{T}=T^{\frac{2-n}{n}}\cdot\Big(\pi^{*}(Tg_{0}+g_{1}+hdz^{2})+h^{-1}\Theta^{2}\Big),$$

where $g_{T}$, $g_{0}$ and $g_{1}$ are the Riemannian metrics corresponding to the Kähler forms $\omega_{T}$, $\omega_{D}$ and $\psi$ respectively.

As $r(\bm{x})$ varies from $2T^{-1}$ to $1$, the Gromov-Hausdorff limit of the rescaled space $\Big(\mathcal{M}_{T},\mathfrak{s}(\bm{x})^{-2}g_{T},\bm{x}\Big)$ will correspondingly change (see Figure 4.2 and Figure 4.3). We will show that, for each $\bm{x}\in\bf{I}_{2}$, the regularity scale is given by

(4.206)

$$\mathfrak{s}(\bm{x})=T^{\frac{1}{n}}\cdot r(\bm{x}).$$

More specifically, we will prove that under the rescaled metrics $\tilde{g}_{T}=(\mathfrak{s}(\bm{x}))^{-2}g_{T}$, the Gromov-Hausdorff convergence keeps $\frac{1}{C_{0}}\leq|\Rm_{\tilde{g}_{T}}(\bm{x})|\leq C_{0}$ as $T\to+\infty$,

(4.207)

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

Let $r_{j}\equiv r(\bm{x}_{j})$, then we divide the region $\bf{I}_{2}$ into three disjoint pieces depending on the scale of $r_{j}$, which will give different bubble limits (see Figure 4.2 and and Figure 4.3):

1. (a)

   There is some $\sigma_{0}>0$ such that

   (4.208)

   $$c_{0}\cdot T_{j}^{-1}\leq r_{j}\leq\frac{1}{(\sigma_{0})^{2}}\cdot T_{j}^{-1}.$$

2. (b)

   Assume that $r_{j}$ satisfies the following condition holds,

   (4.209)

   $$\frac{r_{j}}{T_{j}^{-1}}\to\infty,\ r_{j}\to 0.$$

3. (c)

   Assume that there is some $\underline{T}_{0}>0$ such that

   (4.210)

   $$\underline{T}_{0}\leq r_{j}\leq 1.$$

Case (a) is the same as Region $\bf{I}_{1}$ such that we have the $C^{2,\alpha}$ convergence of the spaces $(\mathcal{M}_{T},\tilde{g}_{j},\bm{x}_{j})$ towards the product space $\mathbb{C}^{2}_{TN,\varrho}\times\mathbb{C}^{n-2}$, where

(4.211)

$$\varrho\equiv\lim_{j\rightarrow\infty}T_{j}\cdot r_{j}\in[c_{0},\frac{1}{\sigma_{0}^{2}}].$$

Therefore, if we choose $\mathfrak{s}(\bm{x})=T^{\frac{1}{n}}\cdot r(\bm{x})$,

(4.212)

$$\underline{v}_{0}\cdot r_{2,\alpha}(\bm{x})\leq\mathfrak{s}(\bm{x})\leq\bar{v}_{0}\cdot r_{2,\alpha}(\bm{x}),$$

where $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ are uniform constants independent of $T\gg 1$.

In the following calculations, we will rescale the coordinates as follows

(4.213)

$$z=r_{j}\cdot\underline{z},\ y=r_{j}\cdot\underline{y},\ w_{p}=r_{j}\cdot\underline{w}_{p},\ u_{q}=r_{j}^{1/2}\cdot\underline{u}_{q},$$

where $r_{j}\equiv r(\bm{x}_{j})$, $p\in\{2,\ldots,n-1\}$, $q\in\{1,2\}$. For simplicity, we denote

(4.214)

$$s_{j}\equiv\mathfrak{s}(\bm{x}_{j})\quad\text{and}\quad\lambda_{j}\equiv s_{j}^{-1}.$$

Notice that, in Case (b) and Case (c), as $T\to+\infty$, curvatures tend to infinity along the singular set $\mathcal{P}\subset\mathcal{M}_{T}$, in the mean while, the rescaled distance $d_{\tilde{g}_{T}}(\bm{x}_{j},\mathcal{P})$ is uniformly bounded. Therefore, in the following, we will analyze both the convergence of the entire neck region $\mathcal{M}_{T}$ and the limiting behavior of the geometry bounded region which is a punctured region in $\mathcal{M}_{T}$ obtained by removing some small tubular neighborhood of $\mathcal{P}$ in $\mathcal{M}_{T}$. For any $b>a>0$, we denote

(4.215)

$$\mathfrak{U}_{j}(a,b)=\{\bm{x}\in\mathcal{M}_{T_{j}}|a\leq z(\bm{x})\leq b\}.$$

We will study the convergence of the punctured region

(4.216)

$$\mathring{\mathfrak{U}}_{j}\equiv\mathfrak{U}_{j}(z_{j}-\xi_{j},z_{j}+\xi_{j})\setminus\mathcal{S}_{j},$$

as $T_{j}\to+\infty$, where $\mathcal{S}_{j}$ is a small neighborhood of $\mathcal{P}$ to be determined later.

Case (b):

First, we study Case (b) which is in fact the limiting case of Case (a) as $\sigma_{0}\to 0$. Geometrically, the rescaled limit $(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ in Case (b) is the asymptotic cone of the product space $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ which is isometric to the product Euclidean space $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$.

For an embedded submanifold $N\subset M$, let us denote by $T_{r}(N)$ the $r$-tubular neighborhood of $N$ in $M$:

(4.217)

$$T_{r}(N)\equiv\{x\in Q|d_{M}(x,N)\leq r\}.$$

In this case, we choose the tubular neighborhood of $\mathcal{P}=\pi^{-1}(P)\subset\mathcal{M}_{T}$,

(4.218)

$$\mathcal{S}_{j}\equiv T_{r_{j}}(\mathcal{P},g_{j})$$

with respect to the original metrics $g_{j}\equiv g_{T_{j}}$. Let $\xi_{j}$ satisfy $\xi_{j}\cdot T^{-\frac{1}{n}}\geq 1$, then we will show that

(4.219)

$$(\mathring{\mathfrak{U}}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big((\mathbb{R}^{3}\times\mathbb{C}^{n-2})\setminus\Sigma_{0^{3}},g_{\mathbb{R}^{2n+1}},\bm{x}_{\infty}\Big).$$

where $\Sigma_{0^{3}}\equiv(\{0^{3}\}\times\mathbb{C}^{n-2})\subset\mathbb{R}^{3}\times\mathbb{C}^{n-2}=\mathbb{R}^{2n-1}$ and $d_{\mathbb{R}^{2n-1}}(\bm{x}_{\infty},\Sigma_{0^{3}})=1$.

To start with, it is straightforward that under the rescaled metric $\tilde{g}_{j}$,

(4.220)

$$\widetilde{\mathcal{S}}_{j}=T_{\lambda_{j}r_{j}}(\mathcal{P},\tilde{g}_{j})$$

converges to a slice $\Sigma_{0^{3}}\equiv(\{0^{3}\}\times\mathbb{C}^{n-2})$ because $\lambda_{j}r_{j}=T_{j}^{-\frac{1}{n}}\to 0$. Next, the limiting behavior of the rescaled metrics $\tilde{g}_{j}$ can be computed explicitly. Now we calculate the limit of each term in $\tilde{g}_{j}=T_{j}^{\frac{2-n}{n}}\cdot(\pi^{*}(T_{j}g_{0}+g_{1}+hdz^{2})+h^{-1}\Theta^{2}),$ which is given by (4.205): First, the scale assumption in Case (b) $r_{j}\to 0$ and $r_{j}T_{j}\to+\infty$ imply that

	$\displaystyle\lambda_{j}^{2}\cdot T_{j}^{\frac{2-n}{n}}\cdot\pi^{*}(T_{j}g_{0}+g_{1})$	$\displaystyle=r_{j}^{-2}\cdot T_{j}^{-1}\cdot\pi^{*}(T_{j}g_{0}+g_{1})$
		$\displaystyle=r_{j}^{-2}\pi^{*}(g_{0})+r_{j}^{-2}\cdot T_{j}^{-1}\cdot\pi^{*}(g_{1})$
(4.221)			$\displaystyle\to g_{\mathbb{C}^{n-1}},$

where we used the rescaled coordinates (4.213) in the computations. By the same computation,

(4.222)		$\displaystyle\lambda_{j}^{2}\cdot\pi^{*}\Big(T_{j}^{\frac{2-n}{n}}\cdot(h\cdot dz^{2})\Big)=(\Tr_{\omega_{D}}\psi+q(z))\cdot T_{j}^{-1}(d\underline{z})^{2}$	$\displaystyle\to g_{\mathbb{R}},$
(4.223)		$\displaystyle\lambda_{j}^{2}\cdot T^{\frac{2-n}{n}}\cdot(h^{-1}\Theta^{2})=r_{j}^{-2}\cdot T_{j}^{-1}\cdot(h^{-1}\Theta^{2})$	$\displaystyle\to 0.$

Therefore, we obtained the desired convergence.

Now that we have proved the convergence (4.219), so we will locally lift $B_{\frac{1}{2}}(\bm{x}_{j})$ to the universal cover $(\widetilde{B_{\frac{1}{2}}(\bm{x}_{j})},\tilde{\bm{x}}_{j})$. By explicit computations, it has uniformly bounded $C^{k,\alpha}$-geometry for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$. In fact, this can be seen from the higher order convergence of $\psi$ and $h$ in the above expressions. Therefore, if we choose $\mathfrak{s}(\bm{x}_{j})=T^{\frac{1}{n}}\cdot r(\bm{x}_{j})$, then for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$,

(4.224)

$$\underline{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j})\leq\mathfrak{s}(\bm{x}_{j})\leq\bar{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j}),$$

where $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ are uniform constants independent of $T\gg 1$.

Case (c):

We will prove that, for appropriately chosen parameters $\xi_{j}$ and $\mu_{j}$, the rescaled limit of the punctured annulus

(4.225)

$$\mathring{\mathfrak{U}}_{j}\equiv\mathfrak{U}_{j}(z_{j}-\xi_{j},z_{j}+\xi_{j})\setminus\mathcal{S}_{j}$$

with $\mathcal{S}_{j}\equiv\pi^{-1}(T_{\mu_{j}}(P))$ is a punctured cylinder $Q\setminus P$. That is, let $\xi_{j}$ and $\mu_{j}$ be a sequence of numbers satisfying the condition

(4.226)		$\displaystyle\frac{T^{\frac{1}{n}}}{\xi_{j}}$	$\displaystyle\to 0,\quad\frac{\xi_{j}}{T_{j}}\to 0,$
(4.227)		$\displaystyle\frac{1}{T_{j}\mu_{j}}$	$\displaystyle\to 0,\quad\mu_{j}\to 0,$

then we will show that

(4.228)

$$(\mathring{\mathfrak{U}}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big(Q\setminus P,g_{Q},\bm{x}_{\infty}\Big),$$

where $g_{Q}$ is a product metric on $Q\equiv D\times\mathbb{R}$ and $d_{g_{Q}}(\bm{x}_{\infty},P)=1$.

To see this, we need to estimate the size of $\mathring{\mathfrak{U}}_{j}$ and the puncture $\pi^{-1}(T_{\mu_{j}}(P))$ as $T_{j}\to+\infty$. By definition, when the reference point $\bm{x}_{j}$ is in Case (c), the distance to the divisor $r_{j}\equiv r(\bm{x}_{j})$ satisfies

(4.229)

$$\underline{T}_{0}\leq r_{j}\leq 1,$$

which implies the metric rescaling factor $\lambda_{j}$ satisfies

(4.230)

$$T_{j}^{-\frac{1}{n}}\leq\lambda_{j}\leq T_{j}^{-\frac{1}{n}}\cdot\underline{T}_{0}^{-1}.$$

Let $\lambda_{0}>0$ be a positive constant such that passing to a subsequence, $T_{j}^{\frac{1}{n}}\cdot\lambda_{j}\to\lambda_{0}$. In the following, we will show that the limit of the rescaled metric

(4.231)

$$\tilde{g}_{j}=\lambda_{j}^{2}\cdot T_{j}^{\frac{2-n}{n}}\cdot\Big(\pi^{*}(Tg_{0}+g_{1}+hdz^{2})+h^{-1}\Theta^{2}\Big)$$

is the Riemann product

(4.232)

$$g_{Q}=\lambda_{0}^{2}(dz^{2}+g_{0}).$$

In fact, by the choice of $\mu_{j}$, we have for every $\bm{y}\in\mathring{\mathfrak{U}}_{j}$, $r(\bm{y})\geq\mu_{j}$. Hence there is a smooth function $\chi$ satisfying $|\chi|=O^{\prime}(r)$ and $|\chi|\leq C\cdot\xi_{j}\ll T_{j}$ such that

(4.233)

$$\Big|h(\bm{y})-\Big(\chi_{j}(\bm{y})+T_{j}\Big)\Big|\leq\frac{1}{2\mu_{j}},$$

which implies

(4.234)

$$\Big|\frac{h(\bm{y})}{T_{j}}-1\Big|\leq\Big|\frac{h(\bm{y})}{T_{j}}-\Big(\frac{\chi_{j}(\bm{y})}{T_{j}}+1\Big)\Big|+\frac{|\xi_{j}(\bm{y})|}{T_{j}}\leq\frac{1}{2\mu_{j}T_{j}}+\frac{C\cdot\xi_{j}}{T_{j}}\to 0.$$

Therefore,

(4.235)

$\displaystyle\Big|\lambda_{j}^{2}\cdot T_{j}^{\frac{2-n}{n}}\cdot h(\bm{y})-\lambda_{0}^{2}\Big|=\Big|(\lambda_{j}\cdot T^{\frac{1}{n}})^{2}\cdot\frac{h(\bm{y})}{T_{j}}-\lambda_{0}^{2}\Big|\to|\lambda_{0}^{2}-\lambda_{0}^{2}|=0.$

Similarly, one can show that

(4.236)

$$\lambda_{j}^{2}\cdot T_{j}^{\frac{2-n}{n}}\cdot(Tg_{0}+g_{1})\to\lambda_{0}^{2}\cdot g_{0}.$$

Moreover, the above computations imply that $\mathfrak{U}_{j}$ has two ends and

(4.237)

$$\diam_{\tilde{g}_{j}}(\mathfrak{U}_{j})\approx C\cdot\xi_{j}\cdot T^{-\frac{1}{n}}\to\infty$$

and

(4.238)

$$\diam_{\tilde{g}_{j}}\Big(\pi^{-1}(T_{\mu_{j}}(P)\Big)\approx C\cdot\mu_{j}\cdot T_{j}^{-\frac{1}{n}}\to 0.$$

Therefore, applying (4.237), (4.238) and (4.235), we have

(4.239)

$$(\mathring{\mathfrak{U}}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}\Big(Q\setminus P,g_{Q},\bm{x}_{\infty}\Big),$$

where $g_{Q}=\lambda_{0}^{2}(g_{0}+dz^{2})$ is the product metric on the cylinder $Q\times\mathbb{R}$. Similar to Case (b), by choosing $\mathfrak{s}(\bm{x}_{j})=T^{\frac{1}{n}}\cdot r(\bm{x}_{j})$, then for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$,

(4.240)

$$\underline{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j})\leq\mathfrak{s}(\bm{x}_{j})\leq\bar{v}_{0}\cdot r_{k,\alpha}(\bm{x}_{j}),$$

where $\bar{v}_{0}>0$ and $\underline{v}_{0}>0$ are uniform constants independent of $T\gg 1$.

Now we care about the large scale geometries on $\mathcal{M}_{T}$ and let the reference point $\bm{x}$ keep far away from the singular set $\mathcal{P}$. More precisely, we will focus on the region consisting of the points $\bm{x}$ satisfying

(4.241)

$$r(\bm{x})\geq\frac{1}{2}.$$

Region $\bf{I}_{3}$ (large scale geometries):

We will show that the regularity scale at each point $\bm{x}$ in this region is given by

(4.242)

$$\mathfrak{s}(\bm{x})=(L_{T}(\bm{x}))^{\frac{1}{2}}\cdot T^{\frac{2-n}{2n}}.$$

Moreover, we will calculate the rescaled limit with respect to each reference point in this region. Let $z_{j}\equiv z(\bm{x}_{j})$, then depending upon the distance from the $\bm{x}_{j}$ to the singular set $\mathcal{P}$, there are three cases to analyze:

1. (a)

   (Close to the singular set $\mathcal{P}$) Assume that there is some $\zeta_{0}>0$ such that

   (4.243)

   $$r(\bm{x})\geq\frac{1}{2},\ |z_{j}|\leq\zeta_{0}.$$

2. (b)

   (Far from the singular set $\mathcal{P}$ and the boundary of $\mathcal{M}_{T}$) Assume that $z_{j}$ satisfies

   (4.244)

   $$|\zeta_{j}|\to\infty,\ \frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\to 0.$$

3. (c)

   (Close to the boundary) Assume that there is some $c_{0}>0$ such that

   (4.245)

   $\displaystyle c_{0}\leq\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\leq\begin{cases}c_{-},&\text{if}\ z_{j}<0,\\ c_{+},&\text{if}\ z_{j}>0.\end{cases}$

Case (a) is identical to Case (c) of Region $\bf{I}_{1}$ such that the rescaled limit space is a cylinder $(Q,g_{Q},\bm{x}_{\infty})$ and $d_{g_{Q}}(\bm{x}_{\infty},P)\leq C_{0}$ for $P=H\times\{0\}\subset Q$. Moreover, the convergence keeps curvatures uniformly bounded away from the singular set $P$.

Case (b):

Now we switch to calculate the limiting metric in Case (b). In this case, with respect to the reference point $\bm{x}_{j}$, the metric rescaling factor $\lambda_{j}\equiv\lambda(\bm{x}_{j})$ is chosen as

(4.246)

$$\lambda_{j}=(L_{T_{j}}(\bm{x}_{j}))^{-\frac{1}{2}}\cdot T_{j}^{\frac{n-2}{2n}}.$$

Let $\mathfrak{U}_{j}\equiv\mathfrak{U}(z_{j}-\xi_{j},z_{j}+\xi_{j})$ be the annulus centered at the slice $z=z_{j}$ such that

(4.247)

$$C^{-1}\cdot T_{j}^{\frac{n-2}{n}}\leq|\xi_{j}|\leq C\cdot T_{j}^{\frac{n-2}{n}}.$$

where $C>0$ is independent of $T_{j}$. We will show that,

(4.248)

$$(\mathfrak{U}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(Q,g_{Q},\bm{x}_{\infty}).$$

In the following computations, we will also make appropriate coordinate change along the $\mathbb{R}$-direction, that is, with respect to the reference point $\bm{x}_{j}$, we pick coordinate $w$ such that

(4.249)

$$z=z_{j}+\Big(\frac{T_{j}}{L_{T_{j}}(z_{j})}\Big)^{\frac{n-2}{2}}w.$$

In the above notations, the rescaled metric $\tilde{g}_{j}$ can be represented as

	$\displaystyle\tilde{g}_{j}$	$\displaystyle=\lambda_{j}^{2}\cdot T_{j}^{\frac{n}{2-n}}\cdot\Big(\pi^{*}(Tg_{0}+g_{1}+hdz^{2})+h^{-1}\Theta^{2}\Big)$
(4.250)			$\displaystyle=L_{T_{j}}(z_{j})^{-1}\cdot\Big(\pi^{*}((Tg_{0}+g_{1})+hdz^{2})+h^{-1}\Theta^{2}\Big).$