By Proposition 8.2, it suffices to show that there exists a uniform constant $C>0$ such that

for all $\omega\in\Omega^{1}(\mathcal{M})$. We argue by contradiction and suppose no such a uniform constant exists. Then we have the following:

1. (1)

   a sequence of spaces $(\mathcal{M}_{j},g_{j})$ with the gluing parameter $\beta_{j}\to\infty$ such that

   (9.4)

   $$(\mathcal{M}_{j},g_{j},p_{j})\xrightarrow{GH}(X_{\infty},d_{\infty},p_{\infty}).$$

2. (2)

   a sequence of $1$-forms $\omega_{j}\in\Omega^{1}(\mathcal{M}_{j})$ such that

   (9.5)			$\displaystyle\|\omega_{j}\|_{C_{\delta,\nu,\mu}^{0}(\mathcal{M}_{j},g_{j})}=1$
   (9.6)			$\displaystyle\|\mathscr{D}_{g_{j}}\omega_{j}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M}_{j},g_{j})}\to 0$

   as $j\rightarrow\infty$,

3. (3)

   a sequence of points $\bm{x}_{j}\in\mathcal{M}_{j}$ satisfying

   (9.7)

   $$|\rho_{j,\delta,\nu,\mu}^{(0)}(\bm{x}_{j})\cdot\omega_{j}(\bm{x}_{j})|=1,$$

   where $\rho_{j,\delta,\nu,\mu}^{(0)}$ is a sequence of weight functions in $(\mathcal{M}_{j},g_{j})$.

Now we are in a position to rescale the above contradicting sequences to produce a contradiction. To start with, let $g_{j}$ be a sequence of contradicting metrics, and we denote the rescaling factors as follows:

1. (1)

   Rescaling of the metrics:

   Let $\tilde{g}_{j}=\lambda_{j}^{2}\cdot g_{j}$, then with respect to the fixed reference point $\bm{x}_{j}\in\mathcal{M}_{j}$ picked as the above, we have the convergence,

   (9.8)

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

2. (2)

   Rescaling of the $1$-forms:

   Let $\kappa_{j}>0$ be a sequence of rescaling factors which will be determined later, such that

   (9.9)

   $$\tilde{\omega}_{j}\equiv\kappa_{j}\cdot\omega_{j}.$$

3. (3)

   Rescaling of the weight functions:

   Since we need to distinguish between the weight functions on the sequence $\mathcal{M}_{j}$ and those on the limit spaces, we denote by $\rho_{j,\delta,\nu,\mu}^{(k+\alpha)}$ the weight functions on $\mathcal{M}_{j}$ and denote by $\rho_{\infty,\delta,\nu,\mu}^{(k+\alpha)}$ the weight functions on the limit spaces. Fix $k\in\mathbb{N}$ and $\alpha\in(0,1)$, we rescale the weight function $\rho_{j,\delta,\nu,\mu}^{(k+\alpha)}$ by

   (9.10)

   $$\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)}=\tau_{j}^{(k+\alpha)}\cdot\rho_{j,\delta,\nu,\mu}^{(k+\alpha)}.$$

The above rescaling factors are chosen to satisfy the scale-invariance property of the weighted norm,

such that in this way we will obtain a sequence of $1$-forms $\tilde{\omega}_{j}\in\Omega^{1}(\mathcal{M}_{j})$ with the property

The basic strategy is to combine the compactness arguments and the Liouville theorems. That is, if $(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ is non-collapsed, we apply Proposition 8.3 and the $C^{1,\alpha}$-compactness to obtain a limiting $1$-form $\tilde{\omega}_{\infty}\in(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ such that

We will apply the Liouville theorems to show that the above limiting $1$-form $\tilde{\omega}_{\infty}$ with controlled weighted norm is in fact vanishing on $\mathcal{M}_{\infty}$, which gives a contradiction. Next, for a collapsed limit $(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$, to understand the limiting behavior of the operators $\mathscr{D}_{\tilde{g}_{j}}$ and the contradicting $1$-forms $\tilde{\omega}_{j}$, we will lift everything to an appropriately chosen non-collapsed (local) normal cover such that the $C^{1,\alpha}$-compactness still applies on such a covering space. On the other hand, by the representation lemma of the $1$-forms, see Lemma 7.11, there are coefficient functions $f_{j}^{x}$, $f_{j}^{y}$, $f_{j}^{z}$ and $f_{j}^{t}$ such that

We will show that the $4$-tuples $(f_{j}^{x},f_{j}^{y},f_{j}^{z},f_{j}^{t})$ converge to a

which can be in effect viewed as the limits of the $1$-forms $\tilde{\omega}_{j}$. In addition, we will also show that at least one of $f_{\infty}^{x}$, $f_{\infty}^{y}$, $f_{\infty}^{z}$ and $f_{\infty}^{t}$ has a positive weighted Hölder norm at $\bm{x}_{\infty}$. Therefore, the desired contradiction just arises from various versions of Liouville theorems for harmonic functions in those different collapsed regions.

In accordance with the classification of the geometries of the rescaled limits in Section 7, we will proceed to produce the desired contradiction in each of the regions discussed in Section 7.3. Precisely, we will correctly choose the rescaling factors such that the contradicting $1$-forms $\tilde{\omega}_{j}\in\Omega^{1}(\mathcal{M}_{j})$ will converge to some limit which satisfies the norm control and satisfies the assumptions in the Liouville theorems in each region.

Assume that the reference point $\bm{x}_{j}$ is Region $\I$, then the rescaled limit is the standard Ricci-flat Taub-NUT space $(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ with a limiting monopole $p_{m,\infty}$. We choose the rescaling factors as follows,

In the above way of rescaling, we have $d_{\tilde{g}_{\infty}}(p_{m,\infty},\bm{x}_{\infty})\leq C$ and the rescaled weight function in the limit space is

Then the limiting $1$-form $\tilde{\omega}_{\infty}\in\Omega^{1}(\mathcal{M}_{\infty})$ satisfies that

Notice that the above norm bound implies that for all $\bm{x}\in\mathcal{M}_{\infty}\setminus B_{2}(p_{m,\infty})$,

Since $\tilde{\omega}_{\infty}$ is in the kernel of $\mathscr{D}_{\tilde{g}_{\infty}}$, immediately $\tilde{\omega}_{\infty}$ is harmonic with respect to the Taub-NUT metric $\tilde{g}_{\infty}$. Applying Lemma 4.17, we have $\tilde{\omega}_{\infty}\equiv 0$.

Now we discuss the case that the reference points $\bm{x}_{j}$ are in Region $\II$. As what we discussed in Section (7.3), the rescaled geometries were separated in the following cases:

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

   (9.22)

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

3. (c)

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

   (9.23)

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

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

We start with our analysis in Case (a). By Lemma 7.9, the rescaled limit in Case (a) is a Ricci-flat Taub-NUT space $(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ such that the $S^{1}$-fiber at infinity has length at least $\sigma_{0}>0$. The rescaling factors in this case are

In the rescaled limit space, the limiting reference point $\bm{x}_{\infty}$ satisfies $d_{\tilde{g}_{\infty}}(\bm{x},p_{m,\infty})=1$. Moreover, the rescaled weight function in the limit space is given by

and the limiting $1$-form $\tilde{\omega}_{\infty}\in\Omega^{1}(\mathcal{M}_{\infty})$ satisfies

The above weighted norm bound implies that for every $\bm{x}\in\mathcal{M}_{\infty}\setminus B_{2}(p_{m,\infty})$, the limiting $1$-form $\tilde{\omega}_{\infty}$ satisfies the pointwise estimate

Applying Lemma 4.17, we have $\tilde{\omega}_{\infty}\equiv 0$ on the rescaled limit $\mathcal{M}_{\infty}$, which completes the proof of Case (a).

The rescaled limit in Case (b) is the punctured Euclidean space $\mathbb{R}^{3}\setminus\{0^{3}\}$. In this case, we choose the rescaling factors as follows,

In terms of the above rescaled metric, the reference point $\bm{x}_{\infty}$ satisfies $d_{g_{0}}(\bm{x},0^{3})=1$. Moreover, the rescaled weight function in the limit space is given by

Mainly we will analyze the limiting behavior of the operator $\mathscr{D}_{\tilde{g}_{j}}$ under the collapsing sequence $(\mathcal{M},g_{j},\bm{x}_{j})$. Specifically, we will construct a globally defined $1$-form

and we will also show that the coefficient functions of $\tilde{\omega}_{\infty}$ are harmonic with respect to the Euclidean metric. Our basic strategy is to apply Lemma 7.11 to reduce the convergence of the $1$-form $\tilde{\omega}_{j}$ to the convergence of the coefficient functions. Let

then Lemma 7.11 and the circle bundle structure in this case guarantee the convergence of the frames $\{\theta_{j}^{x},\theta_{j}^{y},\theta_{j}^{z},\theta_{j}^{t}\}$.

Now we are in a position to construct the limits of the above coefficient functions. We start with the Gromov-Hausdorff convergence

with $|\bm{x}_{\infty}|=1$. For any fixed $R>10$, let $A_{\frac{1}{R},R}^{g_{0}}(0^{3})$ be an annulus in $\mathbb{R}^{3}$ with respect to the Euclidean metric $g_{0}$. The first step is to obtain the limits of the coefficient functions $f_{j}^{x}$, $f_{j}^{y}$, $f_{j}^{z}$, $f_{j}^{t}$ with controlled weighted norms in the flat annulus $A_{\frac{1}{R},R}^{g_{0}}(0^{3})$ under the above Gromov-Hausdorff convergence. Next, letting $R\to\infty$, we will apply Arzelà-Ascoli to obtain global limiting functions.

First, fix any $R>0$, we consider a Euclidean annulus $A_{\frac{1}{R},R}^{g_{0}}(0^{3})\subset\mathbb{R}^{3}$ and we claim that there are limiting functions $f_{\infty,R}^{x}$, $f_{\infty,R}^{y}$, $f_{\infty,R}^{z}$ and $f_{\infty,R}^{t}$ on $A_{\frac{1}{R},R}^{g_{0}}(0^{3})\subset\mathbb{R}^{3}$. For fixed $R>0$, there are $\bar{s}_{0}(R)>0$ and $N_{0}(R)>0$ such that $\{B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})\}_{k=1}^{N}$ with $N\leq N_{0}$ is a finite collection of Euclidean balls which covers $A_{\frac{1}{R},R}^{g_{0}}(0^{3})$ which satisfies

1. (1)

   $A_{\frac{1}{R},R}^{g_{0}}(0^{3})\subset\bigcup\limits_{s=1}^{N}B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})\subset A_{\frac{1}{3R},3R}^{g_{0}}(0^{3})$

2. (2)

   $\frac{\bar{s}_{0}}{3}\leq d_{g_{0}}(\bm{y}_{\infty,k},\bm{y}_{\infty,k^{\prime}})\leq\bar{s}_{0}$ for all $1\leq k<k^{\prime}\leq N$.

We will verify that there exists a subsequence (still denoted by $j$) such that the above finite cover satisfy the following compatibility:

1. (C1)

   $f_{j}^{x}$, $f_{j}^{y}$, $f_{j}^{z}$ and $f_{j}^{t}$ converge to harmonic functions $f_{\infty,k}^{x}$, $f_{\infty,k}^{y}$, $f_{\infty,k}^{z}$ and $f_{\infty,k}^{t}$ on every ball $B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})$.

2. (C2)

   The above locally defined limiting functions can be patched together in the sense that if $B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})\cap B_{2\bar{s}_{0}}(\bm{y}_{\infty,k^{\prime}})\neq\emptyset$, then

   (9.33)

   $\displaystyle\begin{split}f_{\infty,k}^{x}(\bm{y}_{\infty})&=f_{\infty,k^{\prime}}^{x}(\bm{y}_{\infty}),\ f_{\infty,k}^{y}(\bm{y}_{\infty})=f_{\infty,k^{\prime}}^{y}(\bm{y}_{\infty}),\\ f_{\infty,k}^{z}(\bm{y}_{\infty})&=f_{\infty,k^{\prime}}^{z}(\bm{y}_{\infty}),\ f_{\infty,k}^{t}(\bm{y}_{\infty})=f_{\infty,k^{\prime}}^{t}(\bm{y}_{\infty})\end{split}$

   holds for all $\bm{y}_{\infty}\in B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})\cap B_{2\bar{s}_{0}}(\bm{y}_{\infty,k^{\prime}})$.

The above compatibility properties immediately imply that there are well-defined harmonic limiting functions $f_{\infty,R}^{x}$, $f_{\infty,R}^{y}$, $f_{\infty,R}^{z}$ and $f_{\infty,R}^{t}$ on $A_{\frac{1}{R},R}(0^{3})$.

To show property (C1), by taking some subsequence, it suffices to show that for each ball $B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})$ in the above finite cover, there is some subsequence in the original sequence $\{j\}$ such that the coefficient functions $f_{j,k}^{x}$ converge to a harmonic function $f_{\infty,k}^{x}$. For this purpose, we need to locally unwrap the collapsed fibers and discuss the convergence of the coefficient functions $f_{j,k}^{x}$ on non-collapsed universal covers.

Now we take a sequence of geodesic balls $B_{2\bar{s}_{0}}(\bm{y}_{j,k})$ with

Denote by $\ell_{j}$ ($\to 0$) the length of the collapsed $S^{1}$-fiber at $\bm{y}_{j}$ and define

where $\epsilon_{j}>0$ are chosen such that $2\ell_{j}\leq\epsilon_{j}\leq 4\ell_{j}$. Immediately in our context, $\pi_{1}(B_{2\bar{s}_{0}}(\bm{y}_{j,k}))=\Gamma_{j}$ and $\Gamma_{j}$ is isomorphic to $\mathbb{Z}$. Now let

be the universal covering map with $B_{2\bar{s}_{0}}(\bm{y}_{j,k})=\widehat{B_{2\bar{s}_{0}}(\bm{y}_{j,k})}/\Gamma_{j}$. Now on the universal covers, we have the equivariant convergence and the following diagram,

which satisfies the following properties:

1. (e1)

   the universal covers $(\widehat{B_{2\bar{s}_{0}}(\bm{y}_{j,k})},\hat{g}_{j},\hat{\bm{y}}_{j,k})$ are non-collapsed and have uniformly bounded curvatures,

2. (e2)

   the limiting Lie group $\Gamma_{\infty}$ is diffeomorphic to $\mathbb{R}$ and acts isometrically on the limit space $(\widehat{Y}_{k},\hat{g}_{\infty},\hat{\bm{y}}_{\infty,k})$,

3. (e3)

   the universal covering maps $\pr_{j}$ converge to a Riemannian submersion

   (9.38)

   $$\pr_{\infty}:(\widehat{Y}_{k},\hat{g}_{\infty},\hat{\bm{y}}_{\infty,k})\longrightarrow(B_{2\bar{s}_{0}}(\bm{y}_{\infty,k}),g_{0},\bm{y}_{\infty,k})$$

   with $B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})=\widehat{Y}_{k}/\Gamma_{\infty}$,

4. (e4)

   for every $\hat{\bm{z}}_{\infty}\in\widehat{Y}_{k}$, the orbit $\Gamma_{\infty}\cdot\hat{z}_{\infty}$ is a geodesic in $\widehat{Y}_{k}$ and isometric to $(\mathbb{R},dt^{2})$. In particular, $(\widehat{Y}_{k},\hat{g}_{\infty},\hat{\bm{y}}_{\infty})$ is isometric to $B_{2\bar{s}_{0}}(0^{3})\times\mathbb{R}$ in the Euclidean space $\mathbb{R}^{4}$.

Indeed, property (e1) follows from Lemma 7.7. Property (e2) and (e3) follow from the definition of the equivariant convergence. Property (e4) immediately follows from Lemma 7.13. Actually, by Lemma 7.13, the second fundamental form of each $\Gamma_{\infty}$-orbit is vanishing. In other words, each $\Gamma_{\infty}$-orbit is a geodesic in $\widehat{Y}_{k}$. Combining with the facts that the limiting projection $\pr_{\infty}$ is a Riemannian submersion and $B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})$ is a Euclidean ball, then $\widehat{Y}_{k}\equiv B_{2\bar{s}_{0}}(0^{3})\times\mathbb{R}$ and $\hat{g}_{\infty}$ is isometric to the Euclidean metric.

We will apply the above equivariant convergence to construct harmonic functions $f_{\infty,k}^{x}$, $f_{\infty,k}^{y}$, $f_{\infty,k}^{z}$ and $f_{\infty,k}^{t}$ in $B_{2\bar{s}_{0}}(\bm{y}_{\infty,k})$. We only show the construction for $f_{\infty,k}^{x}$. Notice that Proposition 8.3 implies that the $\Gamma_{j}$-invariant lifted functions $\hat{f}_{j}^{x}$ satisfy the uniform weighted Schauder estimate

with respect to the lifted weight function. Applying Arzelà-Ascoli, passing to a subsequence, there is a limiting function $\hat{f}_{\infty,k}^{x}$ with

with $0<\alpha^{\prime}<\alpha<1$. Combining with the above equivariant convergence, we obtain that the limit function $\hat{f}_{\infty,k}^{x}$ is $\Gamma_{\infty}$-invariant which descends to a function $f_{\infty,k}^{x}$ in $B_{2\bar{s}_{0}}(\bm{y}_{\infty})$. Now we prove that $f_{\infty,k}^{x}$ is a harmonic function on $B_{2\bar{s}_{0}}(\bm{y}_{\infty})$. In fact, the lifted $1$-forms $\hat{\omega}_{j}$ also satisfies

and hence there is a limiting $1$-form $\hat{\omega}_{\infty,k}$ satisfying

for $0<\alpha^{\prime}<\alpha<1$. The contradiction assumption implies that $\hat{\omega}_{\infty,k}$ satisfies

The standard elliptic regularity theory for $\mathscr{D}_{\hat{g}_{\infty}}$ shows that the $1$-form $\hat{\omega}_{\infty,k}$ is $C^{\infty}$. This implies that $\hat{f}_{\infty,k}^{x}\in C^{\infty}(\widehat{Y}_{k})$, then by Lemma 7.12 gives the equation

Applying Property (e4),

The construction of the harmonic limiting functions $f_{\infty,k}^{y}$, $f_{\infty,k}^{z}$ and $f_{\infty,k}^{t}$ is verbatim.

We are ready to prove the compatibility property (C2). To this end, we take the union

with

Let $B_{j}\equiv B_{2\bar{s}_{0}}(\bm{y}_{j,k})\cup B_{2\bar{s}_{0}}(\bm{y}_{j,k^{\prime}})$, then

By the same arguments as the above, $B_{j}$ has uniformly bounded curvatures and the universal covering space $(\widetilde{B}_{j},\hat{g}_{j})$ is non-collapsed. Moreover, the equivariant convergence with property (e1)-(e5) as the above still holds in this case. By passing to some subsequence, the lifted coefficient functions $\hat{f}_{j}^{x}$ are $C^{1,\alpha^{\prime}}$-converging to some invariant limiting function $\hat{f}_{\infty,k,k^{\prime}}^{x}$ on $\widehat{B}_{\infty}$ such that

Therefore, $\hat{f}_{\infty,k,k^{\prime}}^{x}$ descends to a function $f_{\infty,k,k^{\prime}}^{x}$ on $B_{\infty}$ such that

In addition, $f_{\infty,k,k^{\prime}}^{x}$ is harmonic on $B_{\infty}$, so we have managed to extend the local harmonic limiting function $f_{\infty,k}^{x}$ to the union $B_{\infty}$. Repeating the above arguments, we can extend the limiting functions to the whole annulus $A_{\frac{1}{R},R}^{g_{0}}(0^{3})$.

Consider the $4$-tuple of harmonic functions $(f_{\infty,R}^{x},f_{\infty,R}^{y},f_{\infty,R}^{z},f_{\infty,R}^{t})$ in the flat annulus $A_{\frac{1}{R},R}^{g_{0}}(0^{3})$ obtained from the above construction, and we write

Immediately, we have the weighted norm control

where $0<\alpha^{\prime}<\alpha<1$. The above construction enables us to define a global harmonic $4$-tuple on the punctured Euclidean space $\mathbb{R}^{3}\setminus\{0^{3}\}$ by applying the standard exhaustion arguments. Let $R\to+\infty$, by applying (9.52) and Arzelà-Ascoli, there is a global $4$-tuple of harmonic functions $(f_{\infty}^{x},f_{\infty}^{y},f_{\infty}^{z},f_{\infty}^{t})$ in $\mathbb{R}^{3}\setminus\{0^{3}\}$ and we denote

Then we have the weighted norm control,

where $0<\gamma<\alpha^{\prime}<\alpha<1$. The weighted norm bound implies that the limiting functions have the following controlled behavior,

By the standard removable singularity theorem, the $4$-tuple of harmonic functions $(f_{\infty}^{x},f_{\infty}^{y},f_{\infty}^{z},f_{\infty}^{t})$ extend to the entire Euclidean space $\mathbb{R}^{3}$. Applying the standard Liouville theorem for harmonic functions, we conclude that $\tilde{\omega}_{\infty}\equiv 0$ and $f_{\infty}^{t}\equiv 0$. So the contradiction arises, which completes the proof of Case (b).

Now we consider Case (c). We have shown in Section 7.3 that the rescaled limit in Case (c) is a punctured flat cylinder $(\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}}$. More precisely, we chose a sequence of punctured unbounded domains $\mathring{U}_{j}$ containing $\bm{x}_{j}$ such that

Moreover, the curvatures of the above rescaled spaces are uniformly bounded away from the singular points in $\mathcal{P}_{m_{0}}$.

Next we study the limiting weight functions. For any fixed reference point $\bm{x}_{j}$ in this case, denote $d_{j}\equiv\min\limits_{1\leq m\leq m_{0}}\{d_{m}(p_{m},\bm{x}_{j})\}$ and we choose the following rescaling factors

So the rescaled weight function in the limit space satisfies