Let $b_{+}$, $b_{-}$ and $m$ be positive integers satisfying

(1.5)

$\displaystyle 1\leq b_{\pm}\leq 9,\ 1\leq m\leq b_{+}+b_{-}.$

Then there exists a family of hyperkähler metrics $\hat{h}_{\beta}$ on a $\K 3$ surface which collapse to the standard metric on the closed interval $[0,1]$, i.e.,

(1.6)

$$(\K 3,\hat{h}_{\beta})\xrightarrow{GH}([0,1],dt^{2}),\ \beta\to\infty.$$

Moreover, for each sufficiently large $\beta\gg 1$, there exist a finite set $\mathcal{S}\equiv\{0,t_{1},\ldots,t_{m},1\}\subset[0,1]$ and a continuous surjective map

(1.7)

$$F_{\beta}:\K 3\to[0,1]$$

which is almost distance-preserving, i.e., for some constant $C_{0}>0$ independent of $\beta$

(1.8)

$$\Big||F_{\beta}(p)-F_{\beta}(q)|-d_{\hat{h}_{\beta}}(p,q)\Big|\leq\frac{C_{0}}{\beta},\ \forall p,q\in\K 3,$$

such that the following properties hold.

1. (1)

   (Regular collapsing regions) Denote by $T_{\epsilon}(\mathcal{S})$ the $\epsilon$-tubular neighborhood of $\mathcal{S}$ and $\mathcal{R}_{\epsilon}\equiv[0,1]\setminus T_{\epsilon}(\mathcal{S})$. Then for every $\epsilon\in(0,10^{-2})$ and $k\in\mathbb{N}$, there exists $C_{k,\epsilon}>0$ such that

   (1.9)

   $$\sup\limits_{F_{\beta}^{-1}(\mathcal{R}_{\epsilon})}|\nabla^{k}{\Rm_{\hat{h}_{\beta}}}|\leq C_{k,\epsilon},$$

   and for each $t\in\mathcal{R}_{\epsilon}$, $F_{\beta}^{-1}(t)$ is diffeomorphic to an $S^{1}$-fiber bundle over $\mathbb{T}^{2}$. Furthermore,

   (1.10)

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

2. (2)

   (Bubbling regions) Denote by $F_{\beta}^{-1}(T_{\epsilon}(\mathcal{S}))\equiv\mathcal{S}_{\epsilon}^{-}\cup\bigcup\limits_{j=1}^{m}\mathcal{S}_{\epsilon}^{j}\cup\mathcal{S}_{\epsilon}^{+}$ the components of the singular pre-image. Then the following spaces occur as bubble limits:

   1. (a)

      For each $1\leq j\leq m$, there exists an $x_{\beta,j}\in\mathcal{S}_{\epsilon}^{j}$ such that $F_{\beta}(x_{\beta,j})\rightarrow t_{j}$, $|{\Rm_{\hat{h}_{\beta}}}|(x_{\beta,j})\rightarrow\infty$ as $\beta\rightarrow\infty$, and rescalings of the metrics near $x_{\beta,j}$ converge to Taub-NUT metrics. In fact, it is possible to have several distinct Taub-NUT bubbles coming out of the same component $\mathcal{S}_{\epsilon}^{j}$; see Theorem 1.5 for a more precise statement.

   2. (b)

      There exist $x_{\beta,\pm}\in\mathcal{S}_{\epsilon}^{\pm}$ such that $F_{\beta}(x_{\beta,-})\to 0$, $F_{\beta}(x_{\beta,+})\to 1$, $|{\Rm_{\hat{h}_{\beta}}}|(x_{\beta,\pm})\rightarrow\infty$ as $\beta\rightarrow\infty$, and rescalings of the metrics near $x_{\beta,\pm}$ converge to Tian-Yau metrics on a del Pezzo surface of degree $b_{\pm}$, minus a smooth anti-canonical curve.