7. Geometry and regularity of the approximate metric [03IW]

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7. Geometry and regularity of the approximate metric

In this section, we will give a detailed analysis of the geometry of $(\mathcal{M},g_{\beta})$ .

7.1. Notations

Since the arguments in the next sections are very tedious and involved, in this subsection we will list some fixed constants and make necessary conventions which will be frequently used in the later proofs. Throughout the rest of the paper, the notation $a_{j}\to a$ will implicitly mean the limit as $j\rightarrow\infty$ , unless otherwise noted.

7.1.1. Tian-Yau spaces and their asymptotic rates

To start with, for two positive integers

(7.1)

b_{-},b_{+}\in\{1,2,\ldots,9\},

let $(X_{b_{-}}^{4},g_{b_{-}},q_{-})$ and $(X_{b_{{}_{+}}}^{4},g_{b_{+}},q_{+})$ be fixed hyperkähler Tian-Yau spaces with reference points $q_{-}\in X_{b_{-}}^{4}$ and $q_{+}\in X_{b_{+}}^{4}$ such that their degrees are $b_{-}$ and $b_{+}$ respectively. See Section 3 for the definition of a Tian-Yau space and the natural coordinate outside a large compact subset. On $(X_{b_{-}}^{4},g_{b_{-}},q_{-})$ and $(X_{b_{+}}^{4},g_{b_{+}},q_{+})$ , there are diffeomorphisms

(7.2)

\Phi_{\pm}:[\zeta_{0}^{\pm},+\infty)\times\Nil_{b_{\pm}}^{3}\rightarrow X_{b_{\pm}}^{4}\setminus K_{\pm}

between the Gibbons-Hawking space which models over a flat cylinder $\mathbb{T}^{2}\times\mathbb{R}$ . We define the definite constants $D_{0}^{\pm}$ by

(7.3)

\displaystyle D_{0}^{\pm}\equiv\text{The distance between}\ q_{\pm}\ \text{and the level set}\ \{\bm{x}\in X_{b_{\pm}}^{4}|z_{\pm}(\bm{x})=\zeta_{0}^{\pm}\}.

Proposition 3.4 shows that there are some positive constants

(7.4)

\dl>0,\ \dr>0

such that for any $k\in\mathbb{N}$ ,

(7.5)

|\nabla_{g_{\mathcal{C}}^{\pm}}^{k}(\Phi^{*}\omega_{TY}^{\pm}-\omega_{\mathcal{C}}^{\pm})|_{g_{\mathcal{C}}^{\pm}}=O(e^{-\dl z_{\pm}}),

where $\omega_{TY}^{\pm}$ , $\omega_{\mathcal{C}}^{\pm}$ are the Kähler forms on the Tian-Yau spaces $X_{b_{\pm}}^{4}$ ,and the Calabi model spaces respectively.

7.1.2. Some notations about the neck region

Now we fix some parameters in the neck region for the convenience of our discussions in the later sections.

Let $\mathcal{P}_{m_{0}}\equiv\{p_{1},\ldots,p_{m_{0}}\}$ be the set of monopoles on the flat cylinder $(\mathbb{T}^{2}\times\mathbb{R},g_{0})$ with coordinates $(x,y,z)$ such that

(1)

$z(p_{1})=0$ .
(2)

There are definite constants

(7.6) $\iota_{0}>0,\ T_{0}>0$

such that for all $k\neq l$ , we have

(7.7) $\iota_{0}\leq d_{g_{0}}(p_{k},p_{l})\leq T_{0}.$

Around each monopole $p_{m}\in\mathcal{P}_{m_{0}}$ , we define the associated distance function

(7.8)

d_{m}(\bm{x})\equiv d_{g}(p_{m},\bm{x}),\ p_{m}\in\mathcal{P}_{m_{0}},\ \bm{x}\in(\mathcal{M},g).

In our proof, the following notations will also be needed. We fix definite constants

(7.9)

\iota_{0}^{\prime}>0,\ T_{0}^{\prime}>0

such that for all $1\leq m<l\leq m_{0}$ , then in terms of the Gibbons-Hawking metric of the neck region, we have

(7.10)

\iota_{0}^{\prime}\cdot(\beta)^{\frac{1}{2}}\leq d_{g}(p_{m},p_{l})\leq T_{0}^{\prime}\cdot(\beta)^{\frac{1}{2}}.

We have already defined in Section 6 the Gibbons-Hawking metric in the neck region $\mathcal{N}_{m_{0}}^{4}$ . Given a gluing parameter $\beta>0$ , by Theorem 2.6, the defining Green’s function $V_{\beta}$ satisfies the asymptotic property that there are constants

(7.11)

\el>0,\ \er>0

such that for any $k\in\mathbb{N}$ we have

(7.12)

\Big|\nabla^{k}\Big(V_{\beta}-\Big(\frac{2\pi b_{-}}{A}z+\beta\Big)\Big)\Big|=O(e^{\el z}),\qquad z\to-\infty

and

(7.13)

\Big|\nabla^{k}\Big(V_{\beta}-\Big(-\frac{2\pi b_{+}}{A}z+\beta\Big)\Big)\Big|=O(e^{-\er z}),\qquad z\to+\infty.

The following functions defined on $\mathcal{N}_{m_{0}}^{4}$ as well as on the Tian-Yau pieces are crucial in analyzing the rescaled limits and the definition of the weight function in the next section, which naturally comes from the construction of the model metric:

(1)

On the negative part of the neck region, we define the function

(7.14) $L_{-}(\bm{x})\equiv\Big(\frac{2\pi b_{-}}{A}\cdot z(\bm{x})+\beta\Big)^{\frac{1}{2}},\ -T_{-}\leq z(\bm{x})<0$
(2)

On the positive part of the neck region, we define the function

(7.15) $L_{+}(\bm{x})\equiv\Big(-\frac{2\pi b_{+}}{A}\cdot z(\bm{x})+\beta\Big)^{\frac{1}{2}},\ 0\leq z(\bm{x})<T_{+}$
(3)

For $\bm{x}\in\mathcal{M}$ located in the end region of $X_{b_{-}}^{4}$ and satisfy $\zeta_{0}^{-}\leq z_{-}(\bm{x})\leq T_{-}$ , we define

(7.16) $\underline{L}_{-}(\bm{x})\equiv\Big(\frac{2\pi b_{-}}{A}\cdot z_{-}(\bm{x})\Big)^{\frac{1}{2}}$
(4)

For $\bm{x}\in\mathcal{M}$ located in the end region of $X_{b_{+}}^{4}$ and satisfy $\zeta_{0}^{+}\leq z_{+}(\bm{x})\leq T_{+}$ , we define

(7.17) $\underline{L}_{+}(\bm{x})\equiv\Big(\frac{2\pi b_{+}}{A}\cdot z_{+}(\bm{x})\Big)^{\frac{1}{2}}.$

7.1.3. Subdivision of the manifold $\mathcal{M}$

Fix a gluing parameter $\beta>1$ , the manifold $(\mathcal{M},g_{\beta})$ will be divided into the following $9$ regions depending on the different collapsing behaviors of metric $g$ :

		$\displaystyle\I:\ d_{m}(\bm{x})\leq\beta^{-\frac{1}{2}}\ \text{for some}\ 1\leq m\leq m_{0}$
		(in the neck, very close to a monopole point)
		$\displaystyle\II:\ 2\beta^{-\frac{1}{2}}\leq d_{m}(\bm{x})\leq\frac{\iota_{0}^{\prime}}{4}\cdot(\beta)^{\frac{1}{2}}\text{ for some}\ 1\leq m\leq m_{0}$
		(in the neck, not close, but not too far from any monopole point)
		$\displaystyle\III:\ z(\bm{x})\in[-m_{0}T_{0},m_{0}T_{0}]\ \text{and }d_{m}(\bm{x})\geq\frac{\iota_{0}^{\prime}}{2}\cdot(\beta)^{\frac{1}{2}}\text{ for all}\ 1\leq m\leq m_{0}$
		(in a bounded region of the neck, but far from any monopole point)
		$\displaystyle\IV_{-}:\ z(\bm{x})\in[-T_{-}/2,-2m_{0}T_{0}]$
		(in the negative end region of the neck)
		$\displaystyle\IV_{+}:\ z(\bm{x})\in[2m_{0}T_{0},T_{+}/2]$
		(in the positive end region of the neck)
		$\displaystyle\V_{-}:\ \bm{x}\in X_{b_{-}}^{4}\ \text{and}\ 2\zeta_{0}^{-}\leq z_{-}(\bm{x})\leq T_{-}$
		(in the end region of $X_{b_{-}}^{4}$ )
		$\displaystyle\V_{+}:\ \bm{x}\in X_{b_{+}}^{4}\ \text{and}\ 2\zeta_{0}^{+}\leq z_{+}(\bm{x})\leq T_{+}$
		(in the end region of $X_{b_{+}}$ )
		$\displaystyle\VI_{-}:\ \bm{x}\in\overline{B_{D_{0}^{-}}(q_{-})}\subset X_{b_{-}}^{4}$
		(in the bounded part of $X_{b_{-}}^{4}$ )
		$\displaystyle\VI_{+}:\ \bm{x}\in\overline{B_{D_{0}^{+}}(q_{+})}\subset X_{b_{+}}^{4}$
		$\displaystyle\hskip 28.45274pt\mbox{(in the bounded part of $X_{b_{+}}^{4}$)}.$

We note that for $\bm{x}\in\IV_{\pm}$ , we have

(7.18)

\displaystyle T_{0}^{\prime}(\beta)^{\frac{1}{2}}\leq d_{m}(\bm{x})\leq R_{\pm}\text{ for all}\ 1\leq m\leq m_{0},

where

(7.19)		$\displaystyle R_{-}$	$\displaystyle\equiv\sup\Big\{d_{g}(x,p_{1})\Big\|-T_{-}\leq z(\bm{x})\leq 0\Big\},$
(7.20)		$\displaystyle R_{+}$	$\displaystyle\equiv\sup\Big\{d_{g}(x,p_{1})\Big\|0\leq z(\bm{x})\leq T_{+}\Big\}.$

Immediately, there is some constant $C>0$ (independent of $\beta$ ) such that

(7.21)

C^{-1}\beta^{\frac{3}{2}}\leq R_{\pm}\leq C\beta^{\frac{3}{2}}.

Remark 7.1.

Notice that the above regions do not completely cover the manifold $\mathcal{M}$ . However, each gap region shares the geometric behavior with the adjacent regions in the above subdivision. Therefore, the curvature estimates and the rescaled geometries in each gap region will be the same as in the adjacent regions, so we will ignore these gap regions in the following.

7.2. Regularity of the approximate metrics

In this subsection, we prove uniform curvature estimates on $\mathcal{M}$ which will be crucial in showing that certain rescalings of the approximate metric have bounded curvature. We will show two different ways to understand the regularity.

The first way is to directly compute the curvature tensors. Since the Gibbons-Hawking metric has an explicit form in terms of the defining harmonic function, the curvature estimates just follow from straightforward calculations. The following lemma gives sharp curvature estimates for every point on $\mathcal{M}$ .

Lemma 7.2.

The following uniform curvature estimates hold for every point in $\mathcal{M}$ :

(1)

Let $r(\bm{x})$ denote the Euclidean distance to the monopole points, then there exists constants $C>0$ so that such that for each $1\leq m\leq m_{0}$ for every $\bm{x}\in B_{r_{0}}(p_{m})$ with $r_{0}\equiv\frac{1}{2}\InjRad_{g_{0}}(\mathbb{T}^{2})$ , the following curvature estimates hold,

(7.22)

\displaystyle|\Rm|(\bm{x})\leq\begin{cases}C\beta,&0\leq r(\bm{x})<\beta^{-1},\\ \frac{C}{\beta^{2}r(\bm{x})^{3}},&\beta^{-1}\leq r(\bm{x})<r_{0}.\end{cases}

In terms of the intrinsic distance function with respect to the Riemannian metric $g_{j}$ ,

(7.23)

\displaystyle|\Rm|(\bm{x})\leq\begin{cases}C\beta,&0\leq r(\bm{x})<\beta^{-1},\\ \frac{C}{\beta^{\frac{1}{2}}d_{m}(\bm{x})^{3}},&\beta^{-1}\leq r(\bm{x})<r_{0}.\end{cases}

(2)

If $\bm{x}$ is in the neck region but has some definite distance away from the monopoles, the following curvature estimates hold for some uniform constant $C>0$ ,

(7.24)

\displaystyle|\Rm|(\bm{x})\leq\begin{cases}\frac{C}{\beta^{2}z(\bm{x})},&\frac{r_{0}}{10}<|z(\bm{x})|<\beta,\\ \frac{C}{\beta^{3}},&-T_{1}\leq z(\bm{x})\leq-\beta\text{and}\ \beta\leq z(\bm{x})<T_{2}.\end{cases}

(3)

For $\bm{x}\in X_{b_{\pm}}(T_{\pm})\subset\mathcal{M}$ , there is a constant $C$ so that

(7.25)

\displaystyle|\Rm|(\bm{x})\leq\begin{cases}C&d(\bm{x})<\zeta_{\pm}\\ \frac{C}{d(\bm{x})^{2}}&d(\bm{x})\geq\zeta_{\pm},\end{cases}

where $d(\bm{x})$ is the distance to a base point in $X_{b\pm}$ .

(4)

For $\bm{x}\in DZ_{\pm}\subset\mathcal{M}$ , there is a constant $C>0$ so that

(7.26) $\displaystyle|\Rm|(\bm{x})\leq\frac{C}{\beta^{3}}.$

Remark 7.3.

The curvature estimates in Lemma 7.2 are sharp in the following sense. The second estimate in (7.22) and (7.23) corresponds to the curvature behavior of the Taub-NUT metric which is exactly of cubic decay. The curvature estimate in (7.25) is sharp as well because the curvatures decay quadratically in the end of a complete Tian-Yau space.

Proof.

The proof only requires straightforward calculations, so we only sketch the calculations. We use the following formula for the pointwise norm squared of the curvature of a Gibbons-Hawking metric

(7.27)

\displaystyle|\Rm|^{2}=\frac{1}{2}V_{\beta}^{-1}\Delta^{2}(V_{\beta}^{-1}),

see [GW00]. We just need to consider the case of $1$ monopole point located at the origin, the case of several monopole points follows easily from this case. Let $r_{0}\equiv\frac{1}{2}\InjRad_{g_{0}}(\mathbb{T}^{2})$ , then we have the expansion

(7.28)

\displaystyle V_{\beta}(\bm{x})=\frac{1}{2r(\bm{x})}+\beta+h(\bm{x}),\ x\in B_{r_{0}}(0^{3}),

where $h$ is a bounded harmonic function.

First, we estimate the curvature in the case $r(\bm{x})<\frac{1}{\beta}$ . By (7.28),

(7.29)

\displaystyle V_{\beta}^{-1}(\bm{x})=\frac{2r}{1+2r\beta+2rh},

so it follows that

(7.30)

\displaystyle V_{\beta}^{-1}=\frac{2r}{1+2r\beta+2rh}\leq Cr(1-2r\beta+4r^{2}\beta^{2}+8r^{3}\beta^{3})

for $r<\beta^{-1}$ , then

(7.31)

\displaystyle|V_{\beta}^{-1}\Delta^{2}(V_{\beta}^{-1})|\leq Cr\beta^{3},

and the first claimed estimate follows from this.

Before showing the curvature estimates in other regions, we relate the intrinsic distance function $d_{m}(\bm{x})$ and the Euclidean radial function $r(\bm{x})$ . By directly estimating the integral of $\sqrt{V_{\beta}}$ , we have that

(7.32)

\displaystyle\begin{split}\frac{1}{C^{\prime}}\cdot\sqrt{r(\bm{x})}&\leq d_{m}(\bm{x})\leq C^{\prime}\sqrt{r\bm{(}x)},\ \hskip 17.0ptr(\bm{x})<\beta^{-1},\\ \frac{1}{C^{\prime}}\cdot\beta^{\frac{1}{2}}\cdot r(\bm{x})&\leq d_{m}(\bm{x})\leq C^{\prime}\cdot\beta^{\frac{1}{2}}\cdot r(\bm{x}),\ r(\bm{x})\geq\beta^{-1},\end{split}

where $C^{\prime}>0$ is some universal constant. So the first part of the curvature estimate in (7.23) immediately follows.

Next, let $\bm{x}\in B_{r_{0}}(0^{3})$ satisfy $r(\bm{x})\geq\beta^{-1}$ . Substituting (7.28) into (7.27), then similar expansion formula shows that for some uniform constant $C>0$ ,

(7.33)

|\Rm|(\bm{x})\leq\frac{C}{\beta^{2}r^{3}(\bm{x})}.

Correspondingly in terms of the intrinsic distance function, the curvature estimate turns out to be

(7.34)

|\Rm|(\bm{x})\leq\frac{C}{\beta^{\frac{1}{2}}d_{m}(\bm{x})^{3}}.

The above in fact covers the curvature estimates in Region I and Region II.

From now on, we consider the case that $\bm{x}$ is in the neck region satisfying $\frac{r_{0}}{10}\leq|z(\bm{x})|\leq\beta$ . In this case, the harmonic function $V_{\beta}$ has the expansion,

(7.35)

\displaystyle V_{\beta}(\bm{x})=\begin{cases}\frac{2\pi b_{-}z(\bm{x})}{A}+h_{-}(\bm{x})+\beta,&-T_{-}\leq z(\bm{x})\leq-\zeta_{0}\\ h(\bm{x})+\beta,&-\zeta_{0}\leq z(\bm{x})\leq\zeta_{0},\\ -\frac{2\pi b_{+}z(\bm{x})}{A}+h_{+}(\bm{x})+\beta,&\zeta_{0}\leq z(\bm{x})\leq T_{+}.\end{cases}

We apply the above expansion to the curvature formula (7.27), then we obtain the following curvature estimate

(7.36)

|\Rm|(\bm{x})\leq\frac{C}{\beta^{2}z(\bm{x})},

where $C>0$ is a uniform curvature estimate. Similarly, one can calculate that in the damage zones,

(7.37)

|\Rm|(\bm{x})\leq\frac{C}{\beta^{3}}

for some uniform constant $C>0$ . Note that the cutoff function and its derivatives up to third order are uniformly bounded, the curvature of the glued metric is therefore also of order $\beta^{-3}$ in the damage zone region.

Next, we recall from Section 2.2 that for the model spaces, the defining harmonic functions are $V_{-}(\bm{x})=\frac{2\pi b_{-}z(\bm{x})}{A}$ and $V_{+}(\bm{x})=\frac{2\pi b_{+}z(\bm{x})}{A}$ , so (7.27) implies that

(7.38)

\displaystyle|\Rm|(\bm{x})\leq\frac{C}{d(\bm{x})^{2}},

for some uniform constant $C>0$ , so the complete end of the model space has exactly inverse quadratic curvature decay. It follows from Proposition 3.4 that the Tian-Yau metric does also.

∎

The curvature estimates in Lemma 7.2 relies on the explicit formulas of the Gibbons-Hawking ansatz. For the sake of conceptually understanding the collapsing behavior, we introduce the following $\epsilon$ -regularity theorem for collapsed Einstein manifolds due to Naber and the fourth author of this paper (see [NZ16] for more details).

Theorem 7.4 (Naber-Zhang, [NZ16]).

Let $(M^{n},g,p)$ satisfy $\Ric_{g}\equiv\lambda g$ and $|\lambda|\leq n-1$ . Given a manifold $(Z^{k},z^{k})$ with $k=\dim(Z^{k})<n$ , there are uniform constants $\delta_{0}>0$ , $w_{0}>0$ and $C_{0}>0$ which depend only on $n$ and the geometry of $B_{1}(z^{k})$ such that the following property holds: if

(7.39)

d_{GH}(B_{2}(p),B_{2}(z^{k}))<\delta_{0},

then the group $\Gamma_{\delta_{0}}(p)\equiv\Image[\pi_{1}(B_{\delta_{0}}(p))\to\pi_{1}(B_{2}(p))]$ has a nilpotent subgroup $\mathcal{N}$ of index bounded by $w_{0}$ such that $\rank(\mathcal{N})\leq n-k$ .

Furthermore, if $\rank(\mathcal{N})=n-k$ , then $\sup\limits_{B_{1}(p)}|\Rm|\leq C_{0}$ . Conversely, if $\sup\limits_{B_{3}(p)}|\Rm|\leq C_{0}$ , then $\rank(\mathcal{N})=n-k$ .

Remark 7.5.

Given a finitely generated nilpotent group $\mathcal{N}$ , let $\mathcal{N}=\mathcal{N}_{0}\rhd\mathcal{N}_{1}\rhd\ldots\rhd\mathcal{N}_{m}=\{e\}$ be the lower central series with abelian factor groups $\mathcal{N}_{j-1}/\mathcal{N}_{j}$ , where $\mathcal{N}_{j+1}\equiv[\mathcal{N},\mathcal{N}_{j}]$ are the commutator subgroups. Then the nilpotent rank of $\mathcal{N}$ is defined as the sum of the ranks of the abelian factors, i.e.

(7.40)

\rank(\mathcal{N})\equiv\sum\limits_{j=1}^{m}\rank(\mathcal{N}_{j-1}/\mathcal{N}_{j}).

Remark 7.6.

If the Einstein assumption is replaced with bounded Ricci curvature, then the uniform curvature bound can be replaced with bounded $C^{1,\alpha}$ -covering geometry for any $0<\alpha<1$ . This can be used in analyzing the regularity of the damage zones.

In fact, theorem 7.4 has a quick proof in the special case of codimension-1 collapse which exactly applies in our case. For the readers’ convenience, we give the statement and the proof here.

Lemma 7.7.

Let $(M_{j}^{n},g_{j},p_{j})$ be a sequence of Einstein manifolds with $|\Ric_{g_{j}}|\leq\epsilon_{j}\to 0$ such that

(7.41)

(M_{j}^{n},g_{j},p_{j})\xrightarrow{GH}\mathbb{R}^{n-1}

and $\Gamma_{2}(p_{j})\equiv\Image[\pi_{1}(B_{2}(p_{j}))\to\pi_{1}(M_{j}^{n})]$ is of infinite order. Then for any $R>0$ ,

(7.42)

\sup\limits_{B_{R}(p_{j})}|\Rm|\leq\frac{C_{0}(n)}{R^{2}}.

Remark 7.8.

Simple rescaling and contradicting arguments imply theorem 7.4 in the case $k=n-1$ , which is an effective version of the lemma.

Proof.

Let $(\widetilde{M_{j}^{n}},\tilde{g}_{j},\Gamma_{j},\tilde{p}_{j})$ be the Riemannian universal covers of $(M_{j}^{n},g_{j})$ which converge to the limit product space $(\mathbb{R}^{n-1}\times Y,\tilde{d}_{\infty},\Gamma_{\infty},\tilde{p}_{\infty})$ in the equivariant Gromov-Hausdorff topology, where $\Gamma_{j}\equiv\pi_{1}(M_{j}^{n})$ and $\Gamma_{j}\to\Gamma_{\infty}\leq\Isom(\mathbb{R}^{n-1}\times Y)$ . See Section 3 of [FY92] for the precise definition of the equivariant Gromov-Hausdorff convergence. In summary, we have the following diagram

(7.43)

where the covering maps $\pr_{j}:\widetilde{M_{j}^{n}}\to M_{j}^{n}$ converge to a natural projection map $\pr_{\infty}:\mathbb{R}^{n-1}\times Y\to\mathbb{R}^{n-1}$ .

The main part is to prove the claim that $Y$ is isometric to $\mathbb{R}$ .

Applying Cheeger-Colding’s quantitative splitting theorem (see [CC96]), the convergence assumption (7.41) implies that for any fixed $R>0$ , there are harmonic splitting maps $\Phi_{j}\equiv(u_{j}^{(1)},\ldots,u_{j}^{(n-1)}):B_{10R}(p_{j})\to\mathbb{R}^{n-1}$ which realize the Gromov-Hausdorff maps such that

(7.44)

\sum\limits_{\alpha,\beta=1}^{n-1}\fint_{B_{5R}(p_{j})}|\langle\nabla u_{j}^{(\alpha)},\nabla u_{j}^{(\beta)}\rangle-\delta_{\alpha\beta}|+\sum\limits_{\alpha=1}^{n-1}\fint_{B_{5R}(p_{j})}|\nabla^{2}u_{j}^{(\alpha)}|^{2}\to 0.

Let $\widetilde{\Phi}_{j}\equiv(\tilde{u}_{j}^{(1)},\ldots,\tilde{u}_{j}^{(n-1)})$ be the lifted harmonic functions on the universal covers, then the volume comparison theorem implies that

(7.45)

\sum\limits_{\alpha,\beta=1}^{n-1}\fint_{B_{5R}(\tilde{p}_{j})}|\langle\tilde{\nabla}\tilde{u}_{j}^{(\alpha)},\tilde{\nabla}\tilde{u}_{j}^{(\beta)}\rangle-\delta_{\alpha\beta}|+\sum\limits_{\alpha=1}^{n-1}\fint_{B_{5R}(\tilde{p}_{j})}|\tilde{\nabla}^{2}\tilde{u}_{j}^{(\alpha)}|^{2}\to 0.

By the definition of the splitting maps, $Y$ is the Gromov-Hausdorff limit of the level sets of the lifted splitting maps $\widetilde{\Phi}_{j}^{-1}(0^{n-1})$ . Since $\Gamma_{2}(p_{j})\leq\pi_{1}(M_{j}^{n})$ is of infinite order which acts on $\widetilde{M}_{j}^{n}$ isometrically and discretely, the limit space $Y$ must be non-compact.

On other hand hand, notice that $\widetilde{\Phi}_{j}^{-1}(0^{n-1})$ is invariant under the deck transformation group $\Gamma_{j}$ and $\Gamma_{j}$ converge to some limiting group $\Gamma_{\infty}\leq\Isom(\mathbb{R}^{n-1}\times Y)$ such that $\pr_{\infty}$ is given by the quotient $(\mathbb{R}^{n-1}\times Y)/\Gamma_{\infty}=\mathbb{R}^{n-1}$ . Hence $\Gamma_{\infty}\leq\Isom(Y)$ and $\Gamma_{\infty}$ acts homogeneously on $Y$ .

Therefore, by standard arguments, the noncompact homogeneous space $Y$ admits a line (see [CG72] or lemma 2.4 in [NZ16]). The Ricci curvature assumption implies that $Y$ is isometric to $\mathbb{R}$ . This completes the proof of the claim.

The curvature estimate (7.42) immediately follows from the $\epsilon$ -regularity theorem for noncollapsed Einstein manifolds (for example see Section 7 in [CC97]).

∎

7.3. Rescaled geometries

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.

Region $\I$ :

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.

Lemma 7.9.

For every monopole point $p_{m}\in\mathcal{P}_{m_{0}}$ and for every fixed positive constant $\sigma>0$ , we have the following $C^{\infty}$ -convergence

(7.52)

(\mathcal{N}_{m_{0}}^{4},\tilde{g}_{\sigma,\beta},p_{m})\xrightarrow{C^{\infty}}(\mathbb{R}^{4},\tilde{g}_{\sigma,\infty},\tilde{p}_{m,\infty})\ \text{as}\ \beta\to+\infty,

such that $(\mathbb{R}^{4},\tilde{g}_{\sigma,\infty},\tilde{p}_{m,\infty})$ is a Ricci-flat Taub-NUT space with

(7.53)

\tilde{g}_{\sigma,\infty}=G_{\sigma}\cdot g_{\mathbb{R}^{3}}+(G_{\sigma})^{-1}\theta^{2}

and

(7.54)

G_{\sigma}(p)=\frac{1}{2d_{0}(p,0^{3})}+\frac{1}{\sigma^{2}},

where $d_{0}$ is the distance function in the Euclidean space $\mathbb{R}^{3}$ .

Remark 7.10.

When $\sigma\to\infty$ , the above family of Taub-NUT spaces will converge to $\mathbb{R}^{4}$ with the standard Euclidean metric. When $\sigma\to 0$ , the above family of Taub-NUT spaces will converge to $\mathbb{R}^{3}$ with the standard Euclidean metric.

Proof.

To prove this lemma, we need to rescale both the metric and the coordinates. We choose the pull-back region $\pi^{-1}(B_{r_{0}}^{g_{0}}(p_{m}))$ with $r_{0}>0$ defined as the above, then for every $\bm{x}\in\pi^{-1}(B_{r_{0}}^{g_{0}}(p_{m}))$ with $\pi(\bm{x})=(x,y,z)$ ,

(7.55)

V_{\beta}(\bm{x})=\frac{1}{2\sqrt{x^{2}+y^{2}+z^{2}}}+h(\bm{x})+\beta,

where $h$ is a bounded harmonic function on $\mathbb{R}^{3}$ . Let us denote the rescaled coordinates by

(7.56)

\displaystyle\tilde{x}_{\beta}\equiv\gamma_{\beta}\cdot x,\ \tilde{y}_{\beta}\equiv\gamma_{\beta}\cdot y,\ \tilde{z}_{\beta}\equiv\gamma_{\beta}\cdot z,

and we choose

(7.57)

\gamma_{\beta}\equiv\sigma^{2}\cdot\beta.

So the rescaled metrics $\tilde{g}_{\sigma,\beta}$ converge to

(7.58)

\tilde{g}_{\sigma,\infty}=G_{\sigma}\cdot g_{\mathbb{R}^{3}}+(G_{\sigma})^{-1}\theta^{2}

such that

(7.59)

G_{\sigma}(p)=\frac{1}{2d_{0}(p,0^{3})}+\frac{1}{\sigma^{2}},

where $d_{0}$ is the distance function in the Euclidean space $\mathbb{R}^{3}$ . This tells us that $\tilde{g}_{\sigma,\infty}$ is a Taub-NUT metric, and the proof is complete.

∎

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.

Region $\II$ :

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:

(a)

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

(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.

(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.

Region $\III$ :

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.

Region $\IV_{-}$ :

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:

(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 $1\leq m\leq m_{0}$ .
(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.

Region $\IV_{+}$ :

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 $\IV_{-}$

Region $\V_{-}$ :

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:

(a)

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

Region $\V_{+}$ :

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 $\V_{-}$ .

Region $\VI_{-}$ :

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

Region $\VI_{+}$ :

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.

Lemma 7.11.

Let $(\mathcal{M},g_{j})$ be a sequence with gluing parameters $\beta_{j}\to\infty$ . Let $\omega\in\Omega^{1}(\mathcal{M})$ , then are $1$ -forms $\theta_{j}^{x}$ , $\theta_{j}^{y}$ , $\theta_{j}^{z}$ and $\theta_{j}^{c}$ in each circle bundle part with

(7.108)

\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.

Moreover, every $1$ -form $\omega\in\Omega^{1}(\mathcal{M})$ in the circle bundle part can be represented as

(7.109)

\omega=f_{x}\theta_{j}^{x}+f_{y}\theta_{j}^{y}+f_{z}\theta_{j}^{z}+f_{t}\theta_{j}^{t}.

Proof.

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. ∎

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

Lemma 7.12.

Let $(M^{4},g)$ be a Riemannian $4$ -manifold and let $\omega\in\Omega^{1}(M^{4})$ satisfy $\mathscr{D}_{g}\omega=0$ , then

(7.116)

\Delta_{H}\omega=0,

where $\mathscr{D}_{g}\equiv d^{+}+d^{*}$ and $\Delta_{H}$ is the Hodge Laplacian.

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

Lemma 7.13.

In Case (b) of Region $\II$ , the Gromov-Hausdorff map

(7.117)

F_{j}:(\mathcal{M},g_{j},\bm{x}_{j})\longrightarrow(\mathbb{R}^{3},g_{0},\bm{x}_{\infty})

can be given by the rescaled coordinate functions

(7.118)

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

where $\gamma_{j}>0$ is defined in the proof of Lemma 7.11. Moreover, $F\equiv(x_{j},y_{j},z_{j})$ satisfies

(7.119)

\Delta_{\tilde{g}_{j}}x_{j}=\Delta_{\tilde{g}_{j}}y_{j}=\Delta_{\tilde{g}_{j}}z_{j}=0

and satisfy

(7.120)

|\nabla_{\tilde{g}_{j}}x_{j}|_{\tilde{g}_{j}}=|\nabla_{\tilde{g}_{j}}y_{j}|_{\tilde{g}_{j}}=|\nabla_{\tilde{g}_{j}}z_{j}|_{\tilde{g}_{j}}\to 1

and away from the monopoles,

(7.121)

|\nabla_{\tilde{g}_{j}}^{2}x_{j}|_{\tilde{g}_{j}}=|\nabla_{\tilde{g}_{j}}^{2}y_{j}|_{\tilde{g}_{j}}=|\nabla_{\tilde{g}_{j}}^{2}z_{j}|_{\tilde{g}_{j}}\to 0.

Remark 7.14.

The explicitly Gromov-Hausdorff map $F_{j}$ in Lemma 7.13 in fact corresponds to Cheeger-Colding’s quantitative splitting map (see [CC96]).

Proof.

In terms of the original coframes $\{dx,dy,dz,\theta\}$ , the volume form is given by

(7.122)

\dvol_{g_{j}}=V_{\beta_{j}}dx\wedge dy\wedge dz\wedge\theta.

By definition,

(7.123)

\displaystyle*(dx)=dy\wedge dz\wedge\theta,\ *(dy)=-dx\wedge dz\wedge\theta,\ *(dz)=dx\wedge dy\wedge\theta,

which implies

(7.124)

\Delta_{g_{j}}x=\Delta_{g_{j}}y=\Delta_{g_{j}}z=0.

After rescaling, we have that

(7.125)

\Delta_{\tilde{g}_{j}}x_{j}=\Delta_{\tilde{g}_{j}}y_{j}=\Delta_{\tilde{g}_{j}}z_{j}=0.

By Lemma 7.11, the pointwise gradient estimate holds,

(7.126)

|\nabla x_{j}|_{\tilde{g}_{j}}=|\nabla y_{j}|_{\tilde{g}_{j}}=|\nabla z_{j}|_{\tilde{g}_{j}}\to 1.

Now we estimate the Hessian of the harmonic functions $x_{j}$ , $y_{j}$ and $z_{j}$ . It suffices to check it for $x_{j}$ . First, Bochner’s formula gives that

(7.127)

\frac{1}{2}\Delta_{\tilde{g}_{j}}|\nabla x_{j}|_{\tilde{g}_{j}}^{2}=|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}.

Due to Cheeger-Colding (see [CC96]), there exist cutoff functions $\varphi_{j}:\mathcal{M}\to[0,1]$ with

(7.128)

\displaystyle\varphi_{j}(x)=\begin{cases}1,\ x\in B_{R}(p_{j}),\\ 0,\ x\in\mathcal{M}\setminus B_{2R}(p_{j})\end{cases}

and there exists an absolute constant $C_{0}>0$ such that

(7.129)

R|\nabla_{\tilde{g}_{j}}\varphi_{j}|_{\tilde{g}_{j}}+R^{2}|\Delta_{\tilde{g}_{j}}\varphi_{j}|\leq C_{0}.

Integrating (7.127) over $B_{4R}(p_{j})$ ,

(7.130)

\displaystyle\begin{split}\fint_{B_{4R}(p_{j})}\varphi_{j}|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}\dvol_{\tilde{g}_{j}}&=\frac{1}{\Vol_{\tilde{g}_{j}}(B_{4R}(p_{j}))}\int_{B_{4R}(p_{j})}\varphi_{j}|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}\dvol_{\tilde{g}_{j}}\\ &=\frac{1}{\Vol_{\tilde{g}_{j}}(B_{4R}(p_{j}))}\int_{B_{4R}(p_{j})}\frac{1}{2}\varphi_{j}\Delta_{\tilde{g}_{j}}(|\nabla x_{j}|_{\tilde{g}_{j}}^{2}-1)\dvol_{\tilde{g}_{j}}\\ &=\frac{1}{2\Vol_{\tilde{g}_{j}}(B_{4R}(p_{j}))}\int_{B_{4R}(p_{j})}(\Delta_{\tilde{g}_{j}}\varphi_{j})\cdot(|\nabla x_{j}|_{\tilde{g}_{j}}^{2}-1)\dvol_{\tilde{g}_{j}}\rightarrow 0,\end{split}

as $j\rightarrow\infty$ . Therefore, by volume comparison,

(7.131)

\fint_{B_{R}(p_{j})}|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}\dvol_{\tilde{g}_{j}}\to 0,

as $j\rightarrow\infty$ . Let $p_{\infty}\in\mathbb{R}^{3}\setminus\{0^{3}\}$ with $B_{2\bar{s}_{0}}(p_{\infty})\subset\mathbb{R}^{3}\setminus\{0^{3}\}$ and we choose a sequence of geodesic balls $B_{2\bar{s}_{0}}(p_{j})$ such that

(7.132)

(B_{2\bar{s}_{0}}(p_{j}),\tilde{g}_{j})\xrightarrow{GH}(B_{2\bar{s}_{0}}(p_{\infty}),g_{0}).

By Lemma 7.7, the curvatures on $B_{\bar{s}_{0}}(p_{j})$ are uniformly bounded by $C\cdot\bar{s}_{0}^{-2}$ and $C>0$ is an absolute constant. On the other hand, since $\Delta_{\tilde{g}_{j}}x_{j}=0$ , (7.131) can be strengthened to

(7.133)

\sup\limits_{B_{\bar{s}_{0}}(p_{j})}|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}\to 0.

The proof is done.

∎

7. Geometry and regularity of the approximate metric [03IW]

7. Geometry and regularity of the approximate metric

7.1. Notations

7.1.1. Tian-Yau spaces and their asymptotic rates

7.1.2. Some notations about the neck region

7.1.3. Subdivision of the manifold ℳ\mathcal{M}

Remark 7.1.

7.2. Regularity of the approximate metrics

Lemma 7.2.

Remark 7.3.

Proof.

Theorem 7.4 (Naber-Zhang, [NZ16]).

Remark 7.5.

Remark 7.6.

Lemma 7.7.

Remark 7.8.

Proof.

7.3. Rescaled geometries

Lemma 7.9.

Remark 7.10.

Proof.

Lemma 7.11.

Proof.

Lemma 7.12.

Lemma 7.13.

Remark 7.14.

Proof.

7.1.3. Subdivision of the manifold $\mathcal{M}$