We start with a flat orbifold $\mathbb{T}^{4}/\mathbb{Z}_{2}$ given by the quotient of a flat $4$-torus by the involution $x\mapsto-x$. It has 16 orbifold singularities. One can resolve these singularities by gluing 16 Eguchi-Hanson spaces onto $X$, which are complete hyperkähler ALE metrics defined on the cotangent bundle of $S^{2}$. By varying the flat structure on $\mathbb{T}^{4}/\mathbb{Z}_{2}$ and the gluing parameters, one obtains an open set in the moduli space of all hyperkähler metrics on the $\K 3$ surface where the areas of the exceptional curves are small. As these areas go to zero the corresponding hyperkähler metrics naturally converge back to the flat orbifold $\mathbb{T}^{4}/\mathbb{Z}_{2}$, and the Eguchi-Hanson spaces appear as bubbles under rescaling. For a rigorous proof we refer readers to [LS94], [Don12] and the references therein. This is a typical example of singularity formation in the non-collapsing situation. In general the Gromov-Hausdorff limit will be an orbifold hyperkähler $\K 3$ surface, and the bubbles are ALE gravitational instantons, classified by Kronheimer in [Kro89].

In [Fos16] Foscolo constructed a family of hyperkähler metrics on a $\K 3$ surface that collapses to the flat orbifold $\mathbb{T}^{3}/\mathbb{Z}_{2}$. The collapse has bounded curvature away from finitely many points, and is given by shrinking the fibers of an $S^{1}$-fibration. In the simplest case, curvature blow-up occurs at the 8 singular points of $\mathbb{T}^{3}/\mathbb{Z}_{2}$, where the bubbles are given by complete hyperkähler spaces with cubic volume growth, which in this case are ALF-$D_{2}$ spaces. See [CC15, Min11] for a partial classification of hyperkähler ALF spaces. Let us also point out that the results of [Fos16] have motivated the study of codimension-$1$ collapse of $G_{2}$-manifolds to $3$-dimensional Calabi-Yau manifolds in [FHN17].

In [GW00], Gross and Wilson constructed a family of hyperkähler metrics on the $\K 3$ surface which collapse to a singular metric $d_{\infty}$ on a topological sphere $X_{\infty}^{2}\approx S^{2}$. One starts from an elliptic $\K 3$ surface, i.e., a $\K 3$ surface that admits a holomorphic fibration over $\mathbb{C}P^{1}$ with the general fibers being smooth elliptic curves. Moreover we assume the generic situation when there are exactly 24 singular fibers of type $I_{1}$. Using a combination of a gluing construction and Yau’s estimates, [GW00] gave a fairly satisfactory picture describing the metric asymptotic behavior when the area of the fibers goes to zero. Away from the singular fibers, the metric is modeled on the Green-Shapere-Vafa-Yau hyperkähler semi-flat metrics [GSVY90], whose restrictions to the fibers are exactly flat; in a neighborhood of each singular fiber the metric is modeled on the Ooguri-Vafa metric (see [GW00] and [OV96]). The latter is an incomplete hyperkähler metric constructed using the Gibbons-Hawking ansatz which we will recall in Section 2. When we rescale near the singular point of any singular fiber, the complete bubble that we obtain is $\mathbb{C}^{2}$ endowed with the Taub-NUT metric, which is Kähler with respect to the standard complex structure on $\mathbb{C}^{2}$ and has cubic volume growth (see [LeB91, NTU63, Tau04]).

Notice that the limit metric $d_{\infty}$ on the topological sphere $X_{\infty}^{2}$ is non-smooth at the $24$ points corresponding to the singular fibers, but every tangent cone at $X_{\infty}^{2}$ is in fact isometric to $\mathbb{R}^{2}$. Away from the singular points, $d_{\infty}$ gives a Riemannian metric on $X_{\infty}^{2}$ which satisfies a real Monge-Ampère equation, an adiabatic limit of the Calabi-Yau equation. By hyperkähler rotation, this family of hyperkähler metrics also describes the geometry of the Calabi-Yau metrics on a polarized family of $\K 3$ surfaces approaching a large complex structure limit.

Here we start with two complete noncompact hyperkähler $4$-manifolds with cylindrical ends. These were constructed by Tian and Yau [TY90] by removing smooth fibers from rational elliptic surfaces, and were proved in [Hei12] to converge to their $\mathbb{R}\times\mathbb{T}^{3}$ flat asymptotic models at an exponential rate. Such spaces are known as $\ALH$-spaces or half-$\K 3$ surfaces in the literature. It is then possible to glue together two $\ALH$ spaces to obtain a family of hyperkähler metrics on $\K 3$ which degenerates by developing a long neck modeled on $\mathbb{T}^{3}$ times an interval (see [CC16] for a rigorous proof). If we rescale these metrics so that the rescaled diameter equals $1$, then the Gromov-Hausdorff limit is the unit interval and the bubbles are the Tian-Yau asymptotically cylindrical metrics at each endpoint. Gluing of asymptotically cylindrical geometric structures is a very familiar construction in geometry, see for example [Flo91, KS01] for anti-self-dual metrics in dimension $4$, and [Kov03] for holonomy $G_{2}$ metrics in dimension $7$.