Based on the above discussions, there is a circle bundle structure in each of the following rescaled regions: Case (b) and Case (c) of Region $\II$, Region $\III$, Region $\IV_{\pm}$ and Case (a) of Region $\V_{\pm}$. In all the above cases, the collapsed rescaled limit of $\mathcal{M}$ is isometric to $\mathbb{R}^{3}$ or $\mathbb{T}^{2}\times\mathbb{R}$.

First, the proof of Case (a) of Region $\V_{-}$ is the same as the proof of Case (b) of Region $\V_{+}$. We only need to discuss Region $\V_{-}$. The original sequence $g_{j}$ is a fixed Tian-Yau metric and have the asymptotic behavior

(7.110)

$$g_{j}=V_{-}(g_{\mathbb{T}^{2}}+dz_{-}^{2})+V_{-}^{-1}\theta_{b_{-}}^{2}+O(e^{-\underline{\delta}_{1}z_{-}}).$$

In this case, the reference points $\bm{x}_{j}$ satisfy $\zeta_{j}\equiv z_{-}(\bm{x}_{j})\to\infty$. In the above discusssions, we choose the rescaling factor $\lambda_{j}\equiv\frac{1}{\underline{L}_{-}(\bm{x}_{j})}$. So under the rescaled metric $\tilde{g}_{j}$, it holds that

(7.111)

$\displaystyle|dx|_{\tilde{g}_{j}}=|dy|_{\tilde{g}_{j}}=|d\tilde{z}|_{\tilde{g}_{j}}$

$\displaystyle\to 1,\ |\theta_{b_{-}}|_{\tilde{g}_{j}}\to 0,$

where we choose the $z$-coordinate translation as $\tilde{z}(\bm{x})={z}(\bm{x})-z(\bm{x}_{j})$. In the remaining cases, the proof is very similar. We can properly rescale the $1$-forms $dx$, $dy$ and $dz$ by

(7.112)

$\displaystyle\theta_{j}^{x}\equiv\gamma_{j}\cdot dx,\ \theta_{j}^{y}\equiv\gamma_{j}\cdot dy,\ \theta_{j}^{z}\equiv\gamma_{j}\cdot d\tilde{z},\ \theta_{j}^{t}\equiv\gamma_{j}\cdot dt.$

In Case (b) and Case (c) of Region $\II$, $\gamma_{j}$ is defined by

(7.113)

$$\gamma_{j}\equiv\lambda_{j}\cdot\beta_{j}^{\frac{1}{2}},$$

where $\lambda_{j}\equiv(d_{p_{m}}(\bm{x}_{j}))^{-1}$, then by straightforward computations,

(7.114)

$\displaystyle|\theta_{j}^{x}|_{\tilde{g}_{j}}=|\theta_{j}^{y}|_{\tilde{g}_{j}}=|\theta_{j}^{z}|_{\tilde{g}_{j}}\to 1,\ |\theta_{j}^{t}|_{\tilde{g}_{j}}\to 0.$

In Region $\III$ and $\IV_{\pm}$, by the definition of the rescaled metrics,

(7.115)

$\displaystyle\begin{split}|dx|_{\tilde{g}_{j}}=|dy|_{\tilde{g}_{j}}=|dz|_{\tilde{g}_{j}}&\to 1,\ |\theta|_{\tilde{g}_{j}}\to 0.\end{split}$

So the proof is done. ∎