The proof of the uniform estimate consists of two primary steps: In the first step, we will prove the weighted $C^{1}$ and $C^{2}$ estimates,

Next, based on the above weighted estimate and the weighted Schauder estimate (by Proposition 6.9), we will prove

Step 1. (Weighted $C^{1}$ and $C^{2}$ estimates)

Now we start to prove the estimate (6.66), which will be proved by contradiction. Suppose no such a uniform constant $C>0$ exists. That is, for fixed parameters

there are the following contradicting sequences:

1. (1)

   A sequence of $S^{1}$-invariant Kähler metrics $g_{j}=g_{T_{j}}$ (or $\omega_{j}=\omega_{T_{j}}$) on the neck $\mathcal{M}_{T_{j}}$ constructed in Section 4.1 with $T_{j}\to+\infty$.

2. (2)

   A sequence of $C^{2,\alpha}$-functions $u_{j}\in\mathfrak{A}$ satisfying

   (6.69)		$\displaystyle\frac{\partial u_{j}}{\partial n}\Big|_{\mathcal{M}_{j}}$	$\displaystyle=0,$
   (6.70)		$\displaystyle\|\nabla u_{j}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}+\|\nabla^{2}u_{j}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{j})}$	$\displaystyle=1,$
   (6.71)		$\displaystyle\|\Delta u_{j}\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{j})}$	$\displaystyle\to 0,\quad j\to+\infty.$

So it follows that either $\|\nabla u_{j}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}\geq\frac{1}{2}$ or $\|\nabla^{2}u_{j}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{j})}\geq\frac{1}{2}$. Without loss of generality, we only consider the first case and let $\bm{x}_{j}\in\mathcal{M}_{j}$ satisfy

Now we renormalize the functions $u_{j}$ as follows,

Immediately, $v_{j}(\bm{x}_{j})=0$, $\frac{\partial v_{j}}{\partial n}|_{\mathcal{M}_{j}}=0$ and

So we are led to apply the weighted Schauder estimate in Proposition 6.9, which gives

Moreover, it is straightforward that

We will rescale contradicting spaces $(\mathcal{M}_{j},g_{j})$ around the above reference points $\bm{x}_{j}$ such that the desired contradiction will arise in the limiting space. Let $g_{j}$ be a sequence of contradicting metrics, then 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,

   (6.80)

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

2. (2)

   Rescaling of the solutions:

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

   (6.81)

   $$\tilde{v}_{j}\equiv\kappa_{j}\cdot v_{j}.$$

3. (3)

   Rescaling of the weight functions:

   Denote by $\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)}$ and $\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}$ the weight functions on the rescaled sequence $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})$ and the rescaled limit $(X_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ respectively. So we rescale the weight function $\rho_{j,\delta,\nu,\mu}^{(k+\alpha)}$ by

   (6.82)

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

   Notice that the rescaling factor $\tau_{j}$ depends on $k$ and $\alpha$.

In the following, we study the convergence of the renormalized functions $\tilde{v}_{j}\in\mathfrak{A}$, with respect to the rescaled metrics $\tilde{g}_{j}$, in each region according to the subdivision given in Section 4.3. The main goal is to show $\tilde{v}_{\infty}\equiv 0$ on the rescaled limit $X_{\infty}$ which gives the desired contradiction.

We will produce the desired contradiction in each region of $\bf{I}_{1}$, $\bf{I}_{2}$, $\bf{I}_{3}$ on $\mathcal{M}_{j}$. Before the detailed contradiction arguments, let us determine the rescaling factors in the following way. First, the scaling invariance requires

Now we need to combing the regularity scale analysis in Proposition 4.18 and the choice of the weight function in Definition 4.19. So $\lambda_{j}$, $\tau_{j}$ and $\kappa_{j}$ are determined as follows, which depends on if $|z(\bm{x}_{j})|$ is uniformly bounded: First, if $|z(\bm{x}_{j})|$ is uniformly bounded (corresponding to Region $\bf{I}_{1}$, $\bf{I}_{2}$ and Case (a) of Region $\bf{I}_{3}$), we choose

Next, if $|z(\bm{x}_{j})|\to+\infty$ (corresponding to Case (b) and Case (c) of Region $\bf{I}_{3}$), we choose

In this case, we need to rescale the $z$-coordinate in the meanwhile so that the exponential term shows up in the rescaling factors.

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

In this case, we consider that the reference points $\bm{x}_{j}$ are in Region $\bf{I}_{1}$. According to the discussions in Section 4.3, for any $0<\gamma<1$, $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})$ converges to the following Riemann product in the $C^{2,\gamma}$-topology,

where $\tilde{g}_{\infty}\equiv g_{TN}\oplus g_{\mathbb{C}^{n-2}}$ is the product metric of the Taub-NUT metric $g_{TN}$ and the Euclidean metric $g_{\mathbb{C}^{n-2}}$. Moreover, the rescaled weight function will converge to

where $\Sigma_{0}\equiv\{p_{\infty}\}\times\mathbb{C}^{n-2}\subset\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ for some $p_{\infty}\in\mathbb{C}_{TN}^{2}$, is the Gromov-Hausdorff limit of the lifted divisor $\mathcal{P}\equiv\pi^{-1}(P)\subset\mathcal{M}_{j}$ with respect to the rescaled metrics $\tilde{g}_{j}$ such that and

It is straightforward that, the rescaled functions $\tilde{v}_{j}$ converge to $\tilde{v}_{\infty}$ in the $C^{2,\alpha^{\prime}}$-topology for each $0<\alpha^{\prime}<\alpha$ such that the following properties hold,

1. (1)

   $\|\nabla\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2},\tilde{g}_{\infty})}+\|\nabla^{2}\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2},\tilde{g}_{\infty})}=1$,

2. (2)

   $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$,

3. (3)

   $\Delta_{\tilde{g}_{\infty}}\tilde{v}_{\infty}\equiv 0$ on $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$.

We will prove that $\tilde{v}_{\infty}\equiv 0$ on $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$.

To start with, we will show that $\tilde{v}_{\infty}$ is constant on the Euclidean factor $\mathbb{C}^{n-2}$. Indeed, we write $\bm{x}\equiv(\bm{x}^{\prime},\bm{x}^{\prime\prime})\in\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$, so it suffices to prove that for every $1\leq k\leq 2n-4$, we have

where the partial derivative $\nabla_{k}\tilde{v}_{\infty}(\bm{x})\equiv\frac{\partial\tilde{v}_{\infty}}{\partial x_{k}^{\prime\prime}}(\bm{x}^{\prime},\bm{x}^{\prime\prime})$ is taken in the directions of $\mathbb{C}^{n-2}$. Now for every $1\leq k\leq 2n-4$,

Notice that $\tilde{g}_{\infty}=g_{TN}\oplus g_{\mathbb{C}^{n-2}}$ is a product metric and $\nabla_{k}$ in effect acts on the Euclidean factor $\mathbb{C}^{n-2}$, so $\nabla_{k}$ commutes with both $\Delta_{\mathbb{C}_{TN}^{2}}$ and $\Delta_{\mathbb{C}^{n-2}}$. Therefore,

The weighted bound implies the estimates

Since we have assumed $\nu\in(-1,0)$, so it is straightforward

The above implies that $|\nabla_{k}\tilde{v}_{\infty}|\leq 1$ on $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$. Applying Cheng-Yau’s gradient estimate to the harmonic function $\nabla_{k}\tilde{v}_{\infty}$ on the Ricci-flat manifold $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$, we conclude that $\nabla_{k}\tilde{v}_{\infty}$ is constant on $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$. By (6.92), $\nabla_{k}\tilde{v}_{\infty}\equiv 0$ for every $1\leq k\leq 2n-4$. Therefore, $\tilde{v}_{\infty}$ is constant on the Euclidean factor $\mathbb{C}^{n-2}$.

By the above argument, the limiting function $\tilde{v}_{\infty}$ can be viewed as a harmonic function on the Ricci-flat Taub-NUT space $(\mathbb{C}_{TN}^{2},g_{TN})$. Now applying Bochner’s formula,

Since $\tilde{v}_{\infty}$ satisfies the weighted bound

so we have for any $\bm{x}\in\mathbb{C}_{TN}^{2}\setminus B_{1}(\bm{x}_{\infty})$,

By assumption $\nu\in(-1,0)$, then $|\nabla_{g_{TN}}\tilde{v}_{\infty}|\equiv 0$ on $\mathbb{C}_{TN}^{2}$ and hence $\tilde{v}_{\infty}$ is constant on $\mathbb{C}_{TN}^{2}$. Notice that $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$, so we conclude that $\tilde{v}_{\infty}(\bm{x}_{\infty})\equiv 0$.

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

Now we separate the proof in $3$ cases:

1. (a)

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

   (6.97)

   $$\lambda_{j}\cdot T_{j}^{-\frac{1}{2}}\geq\sigma_{0}.$$

2. (b)

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

   (6.98)

   $$\lambda_{j}\cdot T_{j}^{-\frac{1}{2}}\to 0,\ \lambda_{j}\cdot T_{j}^{\frac{1}{2}}\to\infty.$$

3. (c)

   Assume that there is some $C_{0}>0$ such that $\lambda_{j}\cdot T_{j}^{\frac{1}{2}}\leq C_{0}$.

Case (a):

In this case, the rescaled limit is the Riemann product $\mathbb{C}_{TN,\sigma}^{2}\times\mathbb{C}^{n-2}$, where $\mathbb{C}_{TN,\sigma}^{2}$ is the Taub-NUT space and the length of the circle fiber at infinity equals $\sigma\in[\sigma_{0},1]$. The remainder of the proof is the same as that in Region $\bf{I}_{1}$, so we omit it.

Case (b):

In this case, the rescaled spaces $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})$ converge to the product Euclidean space $(\mathbb{R}^{3}\times\mathbb{C}^{n-2},g_{0},\bm{x}_{\infty})$ in the pointed Gromov-Hausdorff topology, i.e.,

where the metric $g_{0}$ is the standard Euclidean metric on $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$. In this rescaled limit, the limiting reference point $\bm{x}_{\infty}$ satisfies $d_{g_{0}}(\bm{x}_{\infty},\Sigma_{0^{3}})=1$ and $\Sigma_{0^{3}}\equiv\{0^{3}\}\times\mathbb{C}^{n-2}\subset\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ is the singular slice. Moreover, the convergence keeps curvatures uniformly bounded away from the singular slice $\Sigma_{0^{3}}$. By passing to the local universal covers, in fact one can show that, away from $\Sigma_{0^{3}}\subset\mathbb{R}^{3}\times\mathbb{C}^{n-2}$, the rescaled contradicting functions $\tilde{v}_{j}$ converge to $\tilde{v}_{\infty}$ in the $C^{2,\alpha^{\prime}}$-topology for each $0<\alpha^{\prime}<\alpha<1$, such that the following properties hold,

1. (1)

   $\|\nabla\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathbb{R}^{3}\times\mathbb{C}^{n-2})}+\|\nabla^{2}\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathbb{R}^{3}\times\mathbb{C}^{n-2})}=1$,

2. (2)

   $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$,

3. (3)

   $\Delta_{\tilde{g}_{\infty}}\tilde{v}_{\infty}\equiv 0$ in $(\mathbb{R}^{3}\times\mathbb{C}^{n-2})\setminus\Sigma_{0^{3}}$,

where the limiting weight function is

Our goal is to show that $\tilde{v}_{\infty}\equiv 0$ on $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$, which consists of the following ingredients:

First, we will prove that $\tilde{v}_{\infty}$ in fact globally harmonic in $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$. To show the singular slice $\Sigma_{0^{3}}$ is removable, for each $q\in\Sigma_{0^{3}}$, we take a unit ball $B_{1}(q)\subset\Sigma_{0^{3}}$, and for any $r\in(0,1)$, we choose the tubular neighborhood $T_{r}(B_{1}(q))\subset\mathbb{R}^{3}\times\mathbb{C}^{n-2}$. Notice that $\nabla\tilde{v}_{\infty}$ satisfies the uniform estimate

integrating the above weighted bound, then for any $\bm{x}\in T_{r}(B_{1}(q))\setminus B_{1}(q)$,

By Lemma 6.5, $B_{1}(q)$ is a removable singular set in $T_{r}(B_{1}(q))$ and hence $\tilde{v}_{\infty}$ is harmonic in $T_{r}(B_{1}(q))$.

Next, we will show that $\tilde{v}_{\infty}$ is constant in $\mathbb{C}^{n-2}$. It is straightforward that for each $1\leq k\leq 2n-4$, the partial derivative $\nabla_{k}\tilde{v}_{\infty}\equiv\frac{\partial}{\partial x_{k}^{\prime\prime}}\tilde{v}_{\infty}(\bm{x}^{\prime},\bm{x}^{\prime\prime})$ satisfies

The weighted condition implies that $\nabla_{k}\tilde{v}_{\infty}$ satisfies the uniform estimate,

Since we have assumed $\nu\in(-1,0)$, Lemma 6.6 implies that $|\nabla_{k}\tilde{v}_{\infty}|\equiv 0$ on $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ and hence $\tilde{v}_{\infty}$ is constant in $\mathbb{C}^{n-2}$. Therefore, $\tilde{v}_{\infty}$ can be viewed as a harmonic function in the Euclidean space $(\mathbb{R}^{3},g_{\mathbb{R}^{3}})$. By assumption, $\tilde{v}_{\infty}$ satisfies

Since $\nu\in(-1,0)$, applying the standard Liouville theorem for sublinear growth harmonic functions on a Euclidean space, we conclude that $\tilde{v}_{\infty}$ is a constant. The last step is to use the renormalization $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$, then $\tilde{v}_{\infty}\equiv 0$.

Case (c):

The rescaled limit is the cylinder $(Q,g_{c})\equiv(D\times\mathbb{R},g_{D}\oplus dz^{2})$, where $(D,g_{D})$ is a closed Calabi-Yau manifold. The limiting solutions $\tilde{v}_{\infty}$ satisfies

1. (1)

   $\|\nabla\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(Q)}+\|\nabla^{2}\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^{0}(Q)}=1$,

2. (2)

   $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$,

3. (3)

   $\Delta_{\tilde{g}_{\infty}}\tilde{v}_{\infty}\equiv 0$ in $Q\setminus P$,

where the limiting weight function is

Similar to Case (b), first we need to extend the limiting function $\tilde{v}_{\infty}$ across the singular set $P$. Integrating $\nabla\tilde{v}_{\infty}$ around $P$, we have that $\tilde{v}_{\infty}$ satisfies the growth estimate

Since we have assumed $\nu\in(-1,0)$, so Lemma 6.5 implies that the singular set $P$ is removable. Now we have obtained that $\tilde{v}_{\infty}$ is harmonic on $Q$ and satisfies

for $|z(\bm{x})|$ large. Therefore, $\tilde{v}_{\infty}\equiv 0$ on $Q$ which completes the proof of Case (c).

Region $\bf{I}_{3}$ (the cylindrical bubble and the boundary behavior):

In this region, the rescaling factors of the metrics $g_{j}$ are chosen such that the rescaled Gromov-Hausdorff limit is the cylinder $Q\equiv D\times\mathbb{R}$. Let $\zeta_{j}\equiv z(\bm{x}_{j})$, then there are two different cases to analyze which depends on if the convergence keeps curvatures uniformly bounded.

1. (a)

   Assume that there is some $\zeta_{0}>0$ such that $|z_{j}|\leq\zeta_{0}$.

2. (b)

   Assume that $z_{j}$ satisfies

   (6.109)

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

3. (c)

   Assume that $z_{j}$ satisfies

   (6.110)

   $\displaystyle c_{0}\leq\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\leq 1.$

Case (a):

So the rescaled spaces $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})$ converge to the cylinder $(Q,g_{c})=(D^{2n}\times\mathbb{R},g_{D^{2n}}\oplus dz^{2})$ and the sequence has uniformly bounded geometry away from $Q\setminus\mathcal{P}$. Moreover, the weight function in the rescaled limit space is

The rest of the proof is the same as Case (c) in Region II.

Case (b) in Region $\bf{I}_{3}$

In this case, the reference point $\bm{x}_{j}$ satisfies

In addition, we also need to perform the coordinate change centered at the reference point $\bm{x}_{j}$,

In the following, we only consider the case $z_{j}\ll 0$. It is shown in Section 4.3 that the rescaled limit is isometric to a cylinder $Q=D\times\mathbb{R}$ with a product metric

Moreover, as $T_{j}\to+\infty$, the rescaled weight function limits to

Now the growth condition implies that the limiting function $\tilde{v}_{\infty}$ satisfies

By the choice of the parameter $\delta$ in (6.12),

Applying Lemma 6.7, for every $\bm{x}\in Q$,

So the proof of Case (b) is done.

Case (c) in Region $\bf{I}_{3}$

In this case, the reference point $\bm{x}_{j}$ is close to the boundary such that Neumann boundary condition plays a crucial role. Precisely, the scale condition is given by the following: there is some $c_{0}>0$ such that

We can assume that $z_{j}\ll 0$ and passing to a subsequence, there is some constant $\mathfrak{c}_{0}\in[c_{0},1]$ such that

For the convenience of the computations, we will perform the coordinate change centered at the boundary slice, that is,

We have computed in Section 4.3 that the limit of the rescaled spaces $(\mathcal{M}_{T},\tilde{g}_{j},\bm{x}_{j})$ is the Calabi model space $(\mathcal{C}^{n}_{-},g_{\mathcal{C}^{n}_{-}},\bm{x}_{\infty})$. Moreover, the limiting weight function is

where

Since $\mathfrak{c}_{0}\in[c_{0},1]$, so the limiting function $\tilde{v}_{\infty}$ satisfies

In the following, we will prove that $\tilde{v}_{\infty}$ is vanishing everywhere in the Calabi space $\mathcal{C}_{-}^{n}$ such that the contradiction arises.

To see this, recall that the incomplete Calabi model space $(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}})$ is diffeomorphic to the topological product $[w_{0},+\infty)\times Y^{2n-1}$, where $Y^{2n-1}\equiv\{\rho=\rho_{0}\}$ is with respect to the boundary slice in the Calabi model (see Section 5 for detailed discussions on it). The above structure leads to a natural coordinate representation $\bm{x}=(w,\bm{y})\in\mathcal{C}_{-}^{n}$ for each point in the Calabi model space such that the boundary of $\mathcal{C}_{-}^{n}$ is given by $\{w=w_{0}\}$, where the coordinate $w$ is the natural moment map coordinate.

Denote by $\Sigma_{Y^{2n-1}}=\{\Lambda_{k}\}_{k=0}^{\infty}$ the spectrum of the fiber $Y^{2n-1}$ with respect to the induced Riemannian metric. Let $\{\varphi_{k}\}_{k=0}^{\infty}$ be the orthonormal basis with respect to the $L^{2}$-inner product on $Y^{2n-1}$, such that for each $k\in\mathbb{N}$,

If $\delta>0$ is chosen sufficiently small, applying Proposition 5.14, then $\tilde{v}_{\infty}$ has the expansion

where the function $\mathcal{D}_{k}(w)$ has some definite exponential decaying rate (see Lemma 5.4 and Lemma 5.7 for the accurate rates).

Now we apply the Neumann condition to show that $\kappa=0$ and $c_{k}=0$ for all $k\in\mathbb{N}$. In fact,

Integrating (6.127) over the boundary slice $\{w=w_{0}\}$,

which implies

Next, for each fixed $k\in\mathbb{Z}_{+}$, multiplying $\varphi_{k}$ on the both sides of (6.127) and integrating over $Y^{2n-1}$,

The conclusion $c_{k}=0$ follows from the claim