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. ∎

This is a consequence of the above construction. To see this, let

(9.160)

$$\mathcal{N}_{w_{1}}^{4}(-T_{1}-1,T_{2}),\ldots,\mathcal{N}_{w_{m}}^{4}(-T_{m}-1,T_{m+1})$$

be the neck regions in (9.159) such that for each $1\leq j\leq m$, the neck region $\mathcal{N}_{w_{j}}^{4}(-T_{j}-1,T_{j+1})$ has exactly $w_{j}$-monopoles which have the same $z$-coordinate. Notice that the degree of the nilmanifold fiber is determined by the ending slope of the Green’s function. Corollary 2.7 implies that the degree of the nilpotent fibers will jump by $w_{j}$ when crossing a singular fiber in $\mathcal{N}_{w_{j}}^{4}(-T_{j}-1,T_{j+1})$. It is also easy to see from the construction that there are $w_{j}$ Taub-NUT bubbles at each singular point $t_{j}\in(0,1),j=1\dots m$. ∎