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.