Recall that by Theorem 9.7, $\mathcal{M}$ is diffeomorphic to the $\K 3$ surface.

First, we consider the simpler case that there is only one cluster of monopoles, i.e., $m=1$. Without loss of generality, one can assume that all the monopoles in the neck region are located on the same torus fiber of $\mathbb{T}^{2}\times\mathbb{R}$.

We start the proof by describing the hyperkähler metrics $\hat{h}_{\beta}$ and the continuous map $F_{\beta}:\K 3\to[0,1]$. Given any sufficiently large parameter $\beta\gg 1$, denote by $g_{\beta}$ the approximate metric which is almost Ricci-flat and determined by the approximate triple constructed in Section 6 such that

(9.136)

$$C^{-1}\beta^{\frac{3}{2}}\leq\diam_{g_{\beta}}(\mathcal{M})\leq C\beta^{\frac{3}{2}}$$

for some constant $C>0$ independent of $\beta$. By Theorem 9.7, there is a hyperkähler metric $\hat{g}_{\beta}$ such that

(9.137)

$$\|\hat{g}_{\beta}-g_{\beta}\|_{C^{0,\alpha}(\mathcal{M})}\leq Ce^{-\delta_{0}\beta}$$

for some $C>0$ and $\delta_{0}>0$ independent of $\beta$. Let $\hat{h}_{\beta}$ be the rescaling of the hyperkähler metric $\hat{g}_{\beta}$ with $\diam_{\hat{h}_{\beta}}(\mathcal{M})=1$. Denote by $h_{\beta}$ the rescaling of $g_{\beta}$ with $\diam_{h_{\beta}}(\mathcal{M})=1$, then

(9.138)

$$\|\hat{h}_{\beta}-h_{\beta}\|_{C^{0,\alpha}(\mathcal{M})}\leq Ce^{-\frac{\delta_{0}\beta}{2}}.$$

Now we are ready to define the map $F_{\beta}:\mathcal{M}\to[0,1]$. First, recalling the notation in Section 6, we extend the function $z$ on the neck region to $\mathcal{M}$ as follows

(9.139)

$\displaystyle\tilde{z}(\bm{x})=\begin{cases}\zeta_{0}^{-}-2T_{-}&\bm{x}\in X^{4}_{b_{-}}\setminus\{z_{-}\geq\zeta_{0}^{-}\}\\ z_{-}(\bm{x})-2T_{-}&\bm{x}\in X^{4}_{b_{-}}\cap\{\zeta_{0}^{-}\leq z_{-}\leq T_{-}\}\\ z(\bm{x})&\bm{x}\in\mathcal{N}(T_{-},T_{+})\\ 2T_{+}-z_{+}(\bm{x})&\bm{x}\in X^{4}_{b_{+}}\cap\{\zeta_{0}^{+}\leq z_{+}\leq T_{+}\}\\ 2T_{+}-\zeta_{0}^{+}&\bm{x}\in X^{4}_{b_{+}}\setminus\{z_{+}\geq\zeta_{0}^{+}\}\\ \end{cases},$

and then define

(9.140)

$\displaystyle F_{\beta}(\bm{x})=\frac{\tilde{z}(\bm{x})-\zeta_{0}^{-}+2T_{-}}{2(T_{+}+T_{-})-\zeta_{0}^{-}-\zeta_{0}^{+}}.$

Then it follows directly from the gluing construction that there is some point $t_{1}\in(0,1)$ such that $F_{\beta}^{-1}(t_{1})$ is a singular $S^{1}$-bundle over $\mathbb{T}^{2}$ with exactly $(b_{-}+b_{+})$ vanishing circles. In fact, the vanishing circles occur at the monopoles of the neck region $\mathcal{N}_{m_{0}}^{4}$ constructed in Section 6.1 which is a Gibbons-Hawking space over $\mathbb{T}^{2}\times\mathbb{R}$. Moreover, for each $t\in(0,t_{1})\cup(t_{1},1)$, the fiber $F_{\beta}^{-1}(t)$ is diffeomorphic to a Heisenberg nilmanifold with

(9.141)

$\displaystyle\deg(F_{\beta}^{-1}(t))=\begin{cases}b_{-},&t\in(0,t_{1}),\\ b_{+},&t\in(t_{1},1).\end{cases}$

By the explicit construction in Section 6, there is some uniform constant $C_{0}>0$ such that for each regular fiber,

(9.142)

$\displaystyle C_{0}^{-1}\beta^{-1}\leq\diam_{\hat{h}_{\beta}}(F_{\beta}^{-1}(t))\leq C_{0}\beta^{-1},\ C_{0}^{-1}\beta^{-2}\leq\diam_{\hat{h}_{\beta}}(S^{1})\leq C_{0}\beta^{-2}.$

With these diameter estimates, we are ready to prove the uniform curvature estimates by applying theorem 7.4. Fix any $\epsilon\in(0,10^{-2})$, let $\beta>0$ sufficiently large such that

(9.143)

$$\diam_{\hat{h}_{\beta}}(F_{\beta}^{-1}(t))<\frac{\delta_{0}\cdot\epsilon}{10},$$

where $\delta_{0}>0$ is the dimensional constant in theorem 7.4. Now for a ball around each regular point $B_{\epsilon}(x)\subset F_{\beta}^{-1}([0,1]\setminus T_{2\epsilon}(\mathcal{S}))$ with $\mathcal{S}\equiv\{0,t_{1},1\}$, then

(9.144)

$$\Gamma_{\delta_{0}\epsilon}(x)\equiv\Image[\pi_{1}(B_{\delta_{0}\epsilon}(x))\to B_{\epsilon}(x)]\cong\pi_{1}(\Nil^{3})$$

and hence $\rank(\Gamma_{\delta_{0}\epsilon}(x))=3$. Then by theorem 7.4,

(9.145)

$$\sup\limits_{B_{\epsilon/2}(x)}|\Rm_{\hat{h}_{\beta}}|\leq C_{0,\epsilon},$$

where $C_{0,\epsilon}>0$ depends only on $\epsilon$ and is independent of $\beta$. The higher order curvature estimates can be proved by considering a local universal cover and applying the standard regularity theory for non-collapsing Einstein metrics. This completes (1) of Theorem 1.1.

Now we proceed to prove (2). We still apply theorem 7.4 to prove curvatures blowing-up behavior around the singular fiber. In fact, if $x\in T_{\epsilon/2}(F_{\beta}^{-1}(t_{1}))$, it suffices to show $\sup\limits_{B_{\epsilon/2}(x)}|\Rm_{\hat{h}_{\beta}}|\to\infty$ as $\beta\to\infty$. In fact, notice that

(9.146)

$$\Gamma_{\epsilon/2}(x)\equiv\Image[\pi_{1}(B_{\epsilon/2}(x))\to B_{1/10}(x)]\cong\mathbb{Z}\oplus\mathbb{Z}$$

and hence $\rank(\Gamma_{\epsilon/2}(x))=2<3$. Therefore, theorem 7.4 implies that

(9.147)

$$\sup\limits_{B_{\epsilon/2}(x)}|\Rm_{\hat{h}_{\beta}}|\to\infty$$

as $\epsilon\to 0$.

The next part is to prove the classification of the bubble limits in (2) of statement of the theorem. Fix the gluing parameter $\beta\gg 1$, we analyze the curvature behavior of the approximate metric $g_{\beta}$ in the gluing construction at the scale such that

(9.148)

$$C^{-1}\beta^{\frac{3}{2}}\leq\diam_{g_{\beta}}(\mathcal{M},g)\leq C\beta^{\frac{3}{2}}.$$

There are two cases to analyze.

First, let the reference point $\bm{x}_{\beta}$ be a curvature maximum point of a Tian-Yau piece. It follows directly from the construction that, as $\beta\to+\infty$, the curvature $|\Rm_{g_{\beta}}|(\bm{x}_{\beta})$ is uniformly bounded but not going to $0$. So $(\mathcal{M},g_{\beta},\bm{x}_{\beta})$ converges to a complete hyperkähler Tian-Yau space $(X^{4},g_{TY},\bm{x}_{\infty})$ in the pointed $C^{k}$-topology for any $k\in\mathbb{Z}_{+}$. We will show that $(\mathcal{M},\hat{g}_{\beta},\bm{x}_{\beta})$ also converges to the same Tian-Yau space $(X^{4},g_{TY},\bm{x}_{\infty})$ in the pointed $C^{k}$-topology for any $k\in\mathbb{Z}_{+}$. In fact, by Theorem 9.7,

(9.149)

$$\|\hat{g}_{\beta}-g_{\beta}\|_{C^{0,\alpha}(\mathcal{M})}\leq Ce^{-\delta\beta},$$

which implies that $(\mathcal{M},\hat{g}_{\beta},x_{\beta})$ converges to the same Tian-Yau space $(X^{4},g_{TY},\bm{x}_{\infty})$ in the pointed $C^{0,\alpha}$-topology. The stronger convergence follows from a regularity result for non-collapsed Einstein metrics in [AC92]. Since the rescaling factor $\beta^{\frac{3}{2}}$ is much smaller than exponential, so the bubble limit of $(\mathcal{M},\hat{h}_{\beta})$ around $\bm{x}_{\beta}$ is a complete hyperkähler Tian-Yau space.

Next, we consider the case in which the reference point $\bm{x}_{\beta}$ is very close to one of monopoles, i.e. $\bm{x}_{\beta}\in B_{\beta^{-\frac{1}{2}}}(p_{m})$ in terms of the metric $\hat{h}_{\beta}$, where

(9.150)

$$p_{m}\in\mathcal{P}_{b_{-}+b_{+}}\equiv\{p_{1},\ldots,p_{b_{-}+b_{+}}\}.$$

Applying Lemma 7.9, then

(9.151)

$$(\mathcal{M},\beta\cdot g_{\beta},\bm{x}_{\beta})\longrightarrow(\mathbb{R}^{4},g_{TN},\bm{x}_{\infty}),$$

where $g_{TN}$ is the Taub-NUT metric and the convergence is with respect to the pointed $C^{k}$-topology for any $k\in\mathbb{Z}_{+}$. Applying the error estimate (9.149) and the same arguments as the above, $(\mathcal{M},\beta\cdot h_{\beta},\bm{x}_{\beta})$ converges to $(\mathbb{R}^{4},g_{TN},\bm{x}_{\infty}),$ in the pointed $C^{k}$-topology for any $k\in\mathbb{Z}_{+}$. This implies that in terms of the hyperkähler metric $\hat{h}_{\beta}$, we have the pointed $C^{k}$-convergence for any $k\in\mathbb{Z}_{+}$,

(9.152)

$$(\mathcal{M},\beta^{4}\cdot\hat{h}_{\beta},\bm{x}_{\beta})\longrightarrow(\mathbb{R}^{4},g_{TN},\bm{x}_{\infty}).$$

So the proof of (2) is done.

The above completes the proof in the case with 1 singular point of convergence in the interior of the interval. Next we are in a position to give a generalization of the gluing construction in Section 6 to produce multiple singular points of convergence in the interior of the interval.

First, we fix two hyperkähler Tian-Yau spaces $(X_{b_{-}}^{4},g_{b_{-}},p_{-})$ and $(X_{b_{+}}^{4},g_{b_{+}},p_{+})$ with $b_{-},b_{+}\in\{1,\ldots,9\}$. Let $\{w_{j}\}_{j=1}^{m}$ be positive integers satisfying

(9.153)

$$w_{1}+\ldots+w_{m}=b_{-}+b_{+}.$$

For each $1\leq j\leq m$, we choose the neck region $\mathcal{N}_{w_{j}}^{4}$ as a Gibbons-Hawking space over a finite flat cylinder $(\mathbb{T}^{2}\times[-T_{j},T_{j+1}],g_{0})$ with $w_{j}$-monopoles. As in the construction of Section 7, each pair of monopoles in $\mathcal{N}_{w_{j}}^{4}$ has a definite and bounded distance. Now let $G_{j}:\mathbb{T}^{2}\times\mathbb{R}\to\mathbb{R}$ be a global sign-changing Green’s function which satisfies

(9.154)

$$-\Delta_{g_{0}}G_{j}=2\pi\sum\limits_{s=1}^{w_{j}}\delta_{p_{s}}$$

and there are constants $\beta_{j}^{-},\beta_{j}^{+}\in\mathbb{R}$ and $k_{j}^{-}>0$, $k_{j}^{+}<0$ such that

(9.155)

$\displaystyle\begin{split}&|\nabla_{g_{0}}^{k}(G_{j}-(k_{j}^{-}z+\beta_{j}^{-}))|\leq C_{k}e^{\lambda_{1}z},\ z<-100\beta,\\ &|\nabla_{g_{0}}^{k}(G_{j}-(k_{j}^{+}z+\beta_{j}^{+}))|\leq C_{k}e^{-\lambda_{1}z},\ z>100\beta,\\ &k_{j}^{-}=-k_{j}^{+}=\frac{\pi w_{j}}{\Area(\mathbb{T}^{2})}.\end{split}$

Note that the first step of gluing is to modify the above Green’s function by adding a linear function, i.e. let

(9.156)

$$V_{j}\equiv G_{j}+(\ell_{j}z+\beta_{j})$$

such that two adjacent neck regions have compatible slopes, that is,

(9.157)

$\displaystyle\begin{split}k_{j+1}^{-}+\ell_{j+1}&=k_{j}^{+}+\ell_{j}\\ k_{1}^{-}+\ell_{1}&=\frac{2\pi b_{-}}{A}.\end{split}$

Immediately, we have $k_{1}^{+}+\ell_{1}=\frac{2\pi(b_{-}-w_{1})}{A}$ where $A=\Area(\mathbb{T}^{2})$. Eventually, one can check that at the right end of the last neck region $\mathcal{N}_{w_{m}}^{4}$,

(9.158)

$$k_{m}^{+}+\ell_{m}=\frac{2\pi b_{-}-\sum\limits_{j=1}^{m}w_{j}}{A}=-\frac{2\pi b_{+}}{A}.$$

Applying the construction in Section 6, we obtain a manifold

(9.159)

$$\mathcal{M}=X_{b_{-}}^{4}(T_{1})\bigcup_{\Psi_{1}}\mathcal{N}_{w_{1}}^{4}(-T_{1}-1,T_{2})\bigcup_{\Psi_{2}}\ldots\bigcup_{\Psi_{m}}\mathcal{N}_{w_{m}}^{4}(-T_{m}-1,T_{m+1})\bigcup_{\Psi_{m+1}}X_{b_{+}}^{4}(T_{m+1}+1),$$

where the attaching maps $\Psi_{1},\dots\Psi_{m}$ are chosen analogously to $\Psi_{-}$, and $\Psi_{m+1}$ is chosen analogously to $\Psi_{+}$. Furthermore, there is an approximate hyperkähler triple $\bm{\omega}^{\mathcal{M}}$ on $\mathcal{M}$ which is hyperkähler away from the damage zones, and satisfies the conclusions of Proposition 6.4. The weight function on $\mathcal{M}$ is defined in an analogous way to (8.1), and the arguments in the previous sections are easily modified to prove the existence of a hyperkähler metric $\hat{g}_{\beta}$, close to $g_{\beta}$.

Next, choose the parameters so that $\beta_{j}=\beta$. The parameters $T_{j}$ are then all proportional to $\beta$, and the diameter of the neck region $\mathcal{N}^{4}_{w_{j}}(-T_{j}-1,T_{j+1})$ in the metric $\hat{g}_{\beta}$ is proportional to $\beta^{3/2}$. Therefore, for the sequence of unit diameter hyperkähler metrics $\hat{h}_{\beta}$, these neck regions limit to nontrivial intervals, and thus there are exactly $m$ distinct singular points of convergence $t_{j}\in(0,1),j=1\dots m,$ in the interior of the interval. The analysis of the regular collapsing regions and the bubbling regions is the same as above. ∎