Soon after Yau’s proof of the Calabi Conjecture showed that the (smooth $4$–manifold underlying a complex) K3 surface carries Kähler Ricci-flat metrics, Gibbons and Pope [18] suggested the construction (further explored by Page in [36]) of explicit approximately Ricci-flat metrics on Kummer surfaces. They considered the quotient of a flat $4$–torus by an involution and resolved the $16$ orbifold singularities by gluing in copies of the Eguchi–Hanson metric. This Kummer construction was later made rigorous by Topiwala [39] and LeBrun–Singer [26] using twistor methods. Much more recently Donaldson [16] gave a different proof using analysis, closest to the approach taken in the current paper.

From a broader perspective the Kummer construction furnishes the prototypical example of the appearance of orbifold singularities in non-collapsing sequences of Einstein $4$–manifolds. In [1, Theorem C] Anderson showed that a sequence of Einstein $4$–manifolds $(M_{i},g_{i})$ with a uniform lower bound on volume and upper bounds on diameter and Euler characteristic converges (up to subsequences) to an Einstein $4$–orbifold $M_{\infty}$ with finitely many singular points. The formation of orbifold singularities is modelled on complete Ricci-flat ALE spaces which appear as rescaled limits, or “bubbles”, of the sequence $(M_{i},g_{i})$ around points that approach one of the singularities of the orbifold $M_{\infty}$.

In the Ricci-flat case collapsing can also occur. Anderson [2, Theorem II] showed that every sequence of Ricci-flat metrics $(M,g_{i})$ of unit volume but unbounded diameter collapses everywhere, i.e. $\text{inj}_{g_{i}}(x)\rightarrow 0$ for all $x\in M$. The collapse is in the sense of Cheeger–Gromov outside finitely many points $x_{1},\dots,x_{n}$, i.e. $\text{inj}_{g_{i}}(x)\rightarrow 0$ and $\text{inj}_{g_{i}}(x)^{2}|\text{Rm}_{g_{i}}|_{g_{i}}(x)\leq\epsilon_{0}$ for all $x\in M\setminus\{x_{1},\dots,x_{n}\}$, for a universal constant $\epsilon_{0}>0$. In fact, Cheeger and Tian [10, Theorems 0.1 and 0.8] have shown that the collapse occurs with bounded curvature away from a definite number of points.

Contrary to the case of orbifold singularities, almost nothing is known about the structure of the singular points arising in collapsing sequences of Ricci-flat metrics [3, §6]. One would expect that the geometry around these points is modelled on complete Ricci-flat manifolds with non-maximal volume growth. A simple example of the expected phenomena was suggested by Page [37] in 1981. Consider the Kummer construction of Ricci-flat metrics on the K3 surface along a family of split tori $T^{4}=T^{3}\times S^{1}_{\ell}$ with a circle factor of length $\ell\rightarrow 0$. We can then think of the $2$–spheres arising in the resolution of the $16$ singularities of $T^{4}/\mathbb{Z}_{2}$ as coming in pairs aligned along the collapsing circle over each of the $8$ singular points of $T^{3}/\mathbb{Z}_{2}$. If we now rescale the sequence of Kähler Ricci-flat metrics on the K3 surface by $\ell^{-2}$ around one of these pairs, Page suggests, in the limit $\ell\rightarrow 0$ we should obtain a complete Ricci-flat (hyperkähler) metric on a noncompact space which at infinity looks like $(\mathbb{R}^{3}\times S^{1})/\mathbb{Z}_{2}$.

In this paper we regard this example as a simple case of a more general construction of sequences of Ricci-flat metrics on the K3 surface that collapse to a $3$–dimensional limit. In this more general construction, the $4$–torus $T^{3}\times S^{1}_{\ell}$ is replaced by a non-trivial circle bundle over a (punctured) $3$–torus and the role of Page’s “periodic but nonstationary gravitational instanton” is played by other ALF gravitational instantons.

A gravitational instanton is a complete hyperkähler $4$–manifold with decaying curvature at infinity. Since every hyperkähler manifold is in particular Ricci-flat, gravitational instantons have constrained volume growth: the volume of a geodesic ball of radius $r$ grows at most as $r^{4}$. Gravitational instantons of maximal volume growth are the ALE spaces constructed and classified by Kronheimer [25] following earlier work of Eguchi–Hanson, Gibbons–Hawking and Hitchin. We have seen how ALE spaces arise as models for the formation of orbifold singularities of non-collapsed sequences of Einstein $4$–manifolds. In [34] Minerbe showed that if $\operatorname{Vol}\big(B_{r}(p)\big)=O(r^{a})$ for some $3\leq a<4$ and all $p$, then we must have $a=3$. Gravitational instantons of cubic volume growth are called ALF. By [34] the (unique) end of an ALF space looks like a circle fibration over the complement of a ball in $\mathbb{R}^{3}$ or $\mathbb{R}^{3}/\mathbb{Z}_{2}$ with fibres of asymptotically finite length. If the base of the circle fibration at infinity is $\mathbb{R}^{3}$ (respectively, $\mathbb{R}^{3}/\mathbb{Z}_{2}$) then we say that the ALF space is of cyclic (dihedral) type, since the boundary of large geodesic balls is diffeomorphic to $S^{3}/\Gamma$, where $\Gamma\subset SU(2)$ is a cyclic group in the first case and a binary dihedral group in the second.

The prototypical example of an ALF space of cyclic type is the Taub–NUT metric on $\mathbb{R}^{4}$. This metric is explicit and the circle fibration at infinity is induced by the Hopf projection $S^{3}\rightarrow S^{2}$. The first example of an ALF metric of dihedral type was found by Atiyah–Hitchin [5] by studying moduli spaces of magnetic monopoles on $\mathbb{R}^{3}$, i.e. the solutions of the dimensional reduction of the Yang–Mills self-duality equations from $4$ to $3$ dimensions. The Atiyah–Hitchin manifold is diffeomorphic to the complement of a Veronese $\mathbb{R}\mathbb{P}^{2}$ in $\mathbb{S}^{4}$ and the metric is explicitly given in terms of elliptic integrals.

Recently ALF gravitational instantons have been the focus of intense research with the aim of constructing and classifying examples. Minerbe [35] classified ALF spaces of cyclic type. These are all explicitly given by the Gibbons–Hawking construction of hyperkähler $4$–manifolds with a triholomorphic circle action [19]. Most (if not all) the known methods of construction of hyperkähler metrics have been applied to the dihedral ALF case: twistor methods [23, 12], hyperkähler quotient constructions [14], gauge-theoretic constructions as in the case of the Atiyah–Hitchin manifold [12], Kummer-type constructions [8] and complex Monge–Ampère methods [6, 7]. For example, Page’s “periodic but nonstationary” gravitational instantons of [37], more commonly known as $D_{2}$ ALF spaces, were first constructed rigorously by Hitchin [23] using twistor methods and more recently by Biquard–Minerbe [8] using an extension of the Kummer construction to non-compact spaces. The $D_{2}$ ALF spaces can also be thought of as the moduli spaces of centred charge $2$ $SO(3)$ monopoles on $\mathbb{R}^{3}$ with two singularities endowed with their natural $L^{2}$–metric [13]. The fact that all these constructions yield equivalent families of ALF metrics was shown only recently by Chen–Chen [11].

Despite this rich theory of ALF gravitational instantons, until now it has remained unclear how they can appear as models for the formation of singularities in collapsing sequences of hyperkähler metrics on the K3 surface. The aim of this paper is to exploit singular perturbation methods to construct examples of Ricci-flat metrics on the K3 surface collapsing to a $3$–dimensional limit and exhibit ALF gravitational instantons as the “bubbles” appearing in the process.

We refer to Theorem 6.15 for a more precise statement.

In [20] Gross–Wilson studied hyperkähler metrics on elliptic K3 surfaces with fibres of small size. They considered the generic case when all singular fibres ($24$ of them) are of Kodaira type $I_{1}$ (i.e. a pinched torus). The hyperkähler metric is approximated by a semi-flat metric on the locus of the smooth fibres and by a certain (incomplete) explicit hyperkähler metric, the Ooguri–Vafa metric, in the neighbourhood of each singular fibre. As the size of the fibres converges to zero, the K3 surface collapses to a metric on $S^{2}$ (the base of the elliptic fibration) with $24$ singular points. To the knowledge of the author, besides Gross–Wilson’s work, Theorem 1.1 is the only study of collapsing sequences of hyperkähler metrics on the K3 surface.

Now, one way to make precise Page’s observations in [37] about the Kummer construction for a degenerating family of tori is to consider a gluing construction in which one glues $8$ copies of the $D_{2}$ ALF space to the $\mathbb{Z}_{2}$ quotient of the trivial circle bundle $T^{3}\times S^{1}$ over the flat $3$–torus. The proof of Theorem 1.1 is also based on a gluing construction. In order to allow for more general ALF spaces to appear as rescaled limits, the main idea is to replace $T^{3}\times S^{1}$ with an (incomplete) background hyperkähler metric on a non-trivial circle bundle over a punctured $3$–torus. The tool to construct such a background metric is the Gibbons–Hawking construction of hyperkähler metrics with a triholomorphic $S^{1}$ symmetry, i.e. an isometric circle action that also preserves the $2$–sphere of complex structures compatible with the metric. The $S^{1}$–invariant hyperkähler metrics we seek are explicitly given in terms of a positive harmonic function $h$ on $T^{3}$ with prescribed singularities at a finite number of points. For most configurations of punctures the harmonic function $h$ becomes negative somewhere. However, by multiplying $h$ by a small number $\epsilon>0$ (which geometrically corresponds to making the circle fibres have small length) it is possible to construct highly collapsed hyperkähler metrics $g^{\textup{gh}}_{\epsilon}$ outside of an arbitrarily small neighbourhood of the punctures. Furthermore, the construction of this background metric can be made invariant under the action of an involution.

The key observation now is that the asymptotic model of any ALF metric (up to a double cover in the dihedral case) can be written in Gibbons–Hawking coordinates. By choosing the configuration of punctures appropriately it is then possible to glue in copies of ALF spaces to extend the Gibbons–Hawking metric $g^{\textup{gh}}_{\epsilon}$ to an approximately hyperkähler metric $g_{\epsilon}$: close to a fixed point of the $\mathbb{Z}_{2}$–action on $T^{3}$ we glue in an ALF space of dihedral type (this explains why we need $8$ of them in Theorem 1.1); close to a puncture which is not a fixed point of the $\mathbb{Z}_{2}$–action we glue in an ALF space of cyclic type. The Euler characteristic constraint in the statement of Theorem 1.1 is necessary for the resulting $4$–manifold to have the same Euler characteristic as the K3 surface, but it can also be reinterpreted as the necessary and sufficient condition for the existence of the harmonic function $h$ in the first place.

The approximate solution $g_{\epsilon}$ is then deformed into an exact hyperkähler metric by means of the Implicit Function Theorem. Since some of the ALF spaces are not biholomorphic to their asymptotic model outside a compact set, it is necessary to set up the problem as a gluing problem for hyperkähler structures, rather than the most standard procedure (as in the classical Kummer construction) of first constructing a complex surface using complex geometry and then solving a complex Monge–Ampère equation on this given complex manifold.

We leave aside for future work the question of understanding the relation between the metric degenerations described in this paper and degenerations of a compatible complex structure on the K3 surface. Similarly, it would be very interesting to understand to what extent the collapsing behaviour exhibited in this paper is typical of an arbitrary sequence of Ricci-flat metrics on the K3 surface collapsing to a $3$–dimensional limit.

We give instead an application of our gluing construction to the theory of minimal surfaces. It is well known that holomorphic submanifolds of a Kähler manifold minimise volume in their homology class. A classical problem in minimal surface theory is to understand to what extent area minimising surfaces (and more generally stable minimal surfaces) in Kähler manifolds must be (anti)holomorphic. For example, in 1993 Yau asked whether it is possible to classify all stable minimal $2$–spheres in a simply connected Kähler Ricci-flat manifold [42, Question 64]. In [32] Micallef showed that every stable minimal surface in a flat $4$–torus must be holomorphic for some complex structure compatible with the metric. For some time there was hope to prove a similar result in the case of the K3 surface endowed with a hyperkähler metric. Eventually, Micallef–Wolfson [31] showed that this is not the case. A simple application of our gluing construction allows us to give an alternative (simpler) counterexample: there exist hyperkähler metrics on the K3 surface that admit a strictly stable minimal sphere which is not holomorphic with respect to any complex structure compatible with the metric, cf. Theorem 7.1.

The standard approach in the Kummer construction of Kähler Ricci-flat metrics on the K3 surface is to proceed in two steps. First one constructs a complex surface with vanishing first Chern class by blowing up the singularities of a flat orbifold $T^{4}/\mathbb{Z}_{2}$. On this given complex manifold one then constructs a Kähler Ricci-flat metric by solving a complex Monge-Ampère equation. Some of the building blocks we are going to use in the gluing construction of this paper are not biholomorphic to their asymptotic model outside a compact set. This will force us to adopt a different strategy and glue hyperkähler structures all together. In this initial preliminary section we explain how Donaldson [15] suggested an approach to this problem, based on the notion of definite triples.

Recall that the space of $2$–forms on an oriented $4$–dimensional vector space carries a natural non-degenerate bilinear form of signature $(3,3)$.

Given a triple $\bm{\underline{\omega}}$ of $2$–forms on $(M,\mu_{0})$ we consider the matrix $Q\in\Gamma\big(M,\text{Sym}^{2}(\mathbb{R}^{3})\big)$ defined by

$\bm{\underline{\omega}}$ is a definite triple if and only if $Q$ is a positive definite matrix.

To every definite triple $\bm{\underline{\omega}}$ we associate a volume form $\mu_{\bm{\underline{\omega}}}$ by

and the new matrix $Q_{\bm{\underline{\omega}}}=\left(\det{Q}\right)^{-\frac{1}{3}}Q$ which satisfies (2.2) with $\mu_{\bm{\underline{\omega}}}$ in place of $\mu_{0}$. Note that the volume form $\mu_{\bm{\underline{\omega}}}$ and the matrix $Q_{\bm{\underline{\omega}}}$ are independent of the choice of volume form $\mu_{0}$. We refer to $\mu_{\bm{\underline{\omega}}}$ and $Q_{\bm{\underline{\omega}}}$ as the associated volume form and intersection matrix of the definite triple $\bm{\underline{\omega}}$.

Now, let $(M^{4},\mu_{0})$ be an oriented $4$–dimensional manifold. It is well known that the choice of a $3$–dimensional positive definite subspace of $\Lambda^{2}T^{\ast}_{x}M$ for all $x\in M$ is equivalent to the choice of a conformal class on $M$. Thus every definite triple defines a Riemannian metric $g_{\bm{\underline{\omega}}}$ by requiring that $\text{Span}(\omega_{1},\omega_{2},\omega_{3})|_{x}=\Lambda^{+}T^{\ast}_{x}M$ for all $x\in M$ and $\operatorname{dv}_{g_{\bm{\underline{\omega}}}}=\mu_{\bm{\underline{\omega}}}$.

The metric $g_{\bm{\underline{\omega}}}$ associated to a hyperkähler triple is hyperkähler, in the sense that it has holonomy contained in $Sp(1)\simeq SU(2)$.

In this section we collect known results about gravitational instantons of type ALF, with an emphasis on their asymptotic geometry. ALF gravitational instantons will appear as local models for the geometry of high curvature regions in sequences of hyperkähler metrics on K3 collapsing to a $3$–dimensional limit.

A gravitational instanton is a complete hyperkähler $4$–manifold $(M,g)$ with decaying Riemannian curvature at infinity. The minimum requirement (automatically satisfied for rescaled limits of Einstein metrics on $4$–manifolds with bounded Euler characteristic by the Chern–Gauss–Bonnet formula) is that $(M,g)$ has finite energy $\|\text{Rm}\|_{L^{2}}$. In order to say something about the structure of gravitational instantons it has often been necessary to strengthen this finite energy assumption to faster than quadratic curvature decay $|\text{Rm}|=O(r^{-2-\epsilon})$, $\epsilon>0$ (or a slightly weaker finite weighted energy assumption). Note however that there are examples of gravitational instantons which do not satisfy this stronger decay assumption [21, Theorem 1.5].

Since hyperkähler manifolds are in particular Ricci-flat, gravitational instantons have only one end and constrained volume growth: the volume of a geodesic ball of radius $r$ can grow at most as $r^{4}$ and at least linearly. An initial rough classification of gravitational instantons can be given in terms of their volume growth. The gravitational instantons of maximal volume growth are the ALE spaces classified by Kronheimer [25] following earlier work of Eguchi–Hanson, Gibbons–Hawking and Hitchin. Under the assumption of faster than quadratic curvature decay (or a slightly weaker finite weighted energy assumption) Minerbe [34, Theorem 0.1] has shown that if we assume $\operatorname{Vol}\big(B_{r}(p)\big)=O(r^{a})$ for some $3\leq a<4$ and all $p$, then $a=3$. Minerbe also described the asymptotic geometry of gravitational instantons of cubic volume growth and faster than quadratic curvature decay: they are all ALF spaces, in the following sense.

We are interested in refining these asymptotics. In order to describe a more precise model for the end of an ALF gravitational instanton it is necessary to recall the explicit construction of $4$–dimensional hyperkähler metrics with a triholomorphic circle action known as the Gibbons–Hawking ansatz. We will use this same ansatz later in the paper to construct (incomplete) hyperkähler metrics on circle bundles over a punctured $3$–torus.

In this section we use the Gibbons–Hawking ansatz (3.3) to construct families $\{g^{\textup{gh}}_{\epsilon}\}_{\epsilon>0}$ of (incomplete) hyperkähler metrics on circle bundles over a punctured $3$–torus modulo an involution. The parameter $\epsilon>0$ essentially determines the length of the circle fibres. As $\epsilon\rightarrow 0$ the metric $g^{\textup{gh}}_{\epsilon}$ collapses to the flat orbifold $T^{3}/\mathbb{Z}_{2}$ with bounded curvature away from the punctures. The family $\{g^{\textup{gh}}_{\epsilon}\}_{\epsilon>0}$ will serve as the model for a family of hyperkähler metrics on the K3 surface collapsing to a $3$–dimensional limit in the region where the collapsing occurs with bounded curvature. In the next section we will use ALF gravitational instantons as models for high curvature regions to extend the metric $g^{\textup{gh}}_{\epsilon}$ to a complete almost hyperkähler metric.