In this section we recall the known gluing constructions of hyperkähler metrics on $\K 3$ surfaces in the literature.

The main result of this paper gives a new gluing construction in which a family of hyperkähler metrics on a $\K 3$ surface collapses to a unit interval and generically the collapse happens along a $3$-dimensional Heisenberg nilmanifold (i.e., a nontrivial $S^{1}$-bundle over $\mathbb{T}^{2}$). Part of our motivation was an attempt to understand the hyperkähler metric degenerations corresponding to Type II complex structure degenerations of polarized $\K 3$ surfaces. A guiding example is when we have a family of quartic $\K 3$ surfaces $Z_{t}$ in $\mathbb{C}P^{3}$ defined by the equation $tq+f_{1}f_{2}=0$, where $q$ is a general quartic and $f_{1}$ and $f_{2}$ are general quadrics. So the general fiber is a smooth $\K 3$ surface while the central fiber is a union of two quadric surfaces $X_{1}$ and $X_{2}$, intersecting transversally along an elliptic curve defined by $f_{1}=f_{2}=0$. We would like to understand the behavior of the Ricci-flat metrics on $Z_{t}$ in the cohomology class of $2\pi c_{1}(\mathcal{O}(1)|_{Z_{t}})$ as $t$ tends to zero.

In general, given a del Pezzo surface $M$ and a smooth anti-canonical curve $D\subset M$, Tian-Yau proved in [TY90] the existence of a hyperkähler metric on $M\setminus D$, with interesting asymptotic geometry at infinity. Namely, outside a compact set the manifold is diffeomorphic to $N\times[0,\infty)$, where $N$ is an $S^{1}$-bundle over $D$ of degree $d=c_{1}(X)^{2}$, and the metric is modeled on a doubly-warped product so that as we move towards infinity the $S^{1}$-fibers shrink in size while the base torus $D$ expands. The volume growth rate of the hyperkähler metric is $4/3$ and the curvature decays quadratically. We call such a hyperkähler metric a Tian-Yau metric throughout this paper, and for more precise details we refer to Section 3. The proof in [TY90] uses the Calabi ansatz in a neighborhood of infinity and then solves a Monge-Ampère equation, so it is not a priori clear whether these metrics are unique or canonical in a suitable sense. Nevertheless they provide candidates for the bubble limits of the degeneration that we would like to understand.

Motivated by the above Type II degeneration picture, our basic idea was to glue together two Tian-Yau metrics to obtain hyperkähler $\K 3$ surfaces. However, an easy topological consideration shows that one cannot naively glue the ends of the two Tian-Yau metrics together to even match the topology of a $\K 3$ surface. Geometrically, even though the end of a Tian-Yau metric is toplogically cylindrical, the metric itself is not. So we need to construct a neck region that approximates the Tian-Yau ends on both sides.

A key novel ingredient of this paper is exactly to construct such a transition region. It is an incomplete hyperkähler $4$-manifold that can be viewed as a doubly-periodic cousin of the Ooguri-Vafa metric. Recall that the Ooguri-Vafa metric is the metric arising from the Gibbons-Hawking ansatz applied to a harmonic function on $S^{1}\times\mathbb{R}^{2}$ with a pole on $S^{1}\times\{0\}$; or equivalently, a harmonic function on $\mathbb{R}\times\mathbb{R}^{2}$, periodic in the first variable, and with poles on $\mathbb{Z}\times\{0\}\subset\mathbb{R}\times\{0\}$. Our neck metric is instead constructed by applying the Gibbons-Hawking ansatz to a harmonic function on the flat cylinder $\mathbb{T}^{2}\times\mathbb{R}$ with finitely many poles (which we call the monopole points) in $\mathbb{T}^{2}\times\mathbb{R}$. This is equivalent to a harmonic function on $\mathbb{R}^{2}\times\mathbb{R}$, doubly periodic in the first and second variables, and with poles on lattices. For more details of this construction we refer to Section 2. Here we point out that in analogy with the Ooguri-Vafa case, the resulting metric is incomplete because $\mathbb{T}^{2}\times\mathbb{R}$ is parabolic, hence admits no globally positive harmonic functions; moreover, the two ends of this neck metric indeed match up closely with the ends of Tian-Yau metrics.

Our main theorem says that it is in fact possible to “glue together” two Tian-Yau metrics with a suitable neck region as above to construct families of Ricci-flat metrics on $\K 3$ with non-trivial nilpotent collapsing structure.

For the precise definition of a $3$-dimensional Heisenberg nilmanifold, see Section 2.1. These are $S^{1}$-bundles over $\mathbb{T}^{2}$, and thus they have a degree which is only well-defined up to sign. However, if one specifies a projection to an oriented $\mathbb{T}^{2}$, then the degree is a well-defined integer. We denote by $\Nil_{b}^{3}$ a $3$-dimensional nilmanifold of degree $b$, where we will always have a certain projection to an oriented $\mathbb{T}^{2}$ in mind. Let $t_{0}=0$, $t_{m+1}=1$, and let $d_{j}$ be the degree of a nilpotent fiber $\Nil_{d_{j}}^{3}$ on the interval $(t_{j}+\epsilon,t_{j+1}-\epsilon)$, $j=0,\dots,m$, with $d_{0}=b_{-}$ and $d_{m}=-b_{+}$. The following Domain Wall Crossing Theorem describes the possible jumps of the degrees of the nilmanifolds upon crossing the singular regions.

We will adopt the general description of a hyperkähler metric in terms of a triple of three symplectic forms due to Donaldson [Don06], a description which was also used, for example, in [FLS17, Fos16]. Namely, we will not directly construct a Ricci-flat metric on $\mathcal{M}$. Instead we will glue together triples of symplectic forms, which we will then perturb to obtain a hyperkähler triple, which will then yield in particular a Ricci-flat Kähler metric.

Let $M^{4}$ be an oriented $4$-manifold with a volume form $\dvol_{0}$. A triple of $2$-forms $\bm{\omega}=(\omega_{1},\omega_{2},\omega_{3})$ is called a definite triple if the matrix $Q=(Q_{ij})$ defined by

(1.12)

$$\frac{1}{2}\omega_{i}\wedge\omega_{j}=Q_{ij}\dvol_{0}$$

is positive. Given a definite triple $\bm{\omega}$, the associated volume form is defined as

(1.13)

$$\dvol_{\bm{\omega}}=(\det(Q))^{\frac{1}{3}}\dvol_{0},$$

which is independent of the choice of volume form $\dvol_{0}$. We denote by $Q_{\bm{\omega}}\equiv(\det(Q))^{-\frac{1}{3}}Q$ the renormalized matrix with unit determinant. Furthermore, a definite triple $\bm{\omega}=(\omega_{1},\omega_{2},\omega_{3})$ is called a hyperkähler triple if $d\omega_{1}=d\omega_{2}=d\omega_{3}=0$ and the renormalized coefficient matrix satisfies

(1.14)

$\displaystyle Q_{\bm{\omega}}=\Id.$

Note that equation (1.14) is equivalent to

(1.15)

$$\frac{1}{2}\omega_{i}\wedge\omega_{j}=\frac{1}{6}\delta_{ij}(\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}),$$

for every $1\leq i\leq j\leq 3$.

Each definite triple $\bm{\omega}$ defines a Riemannian metric $g_{\bm{\omega}}$ such that each $\omega_{j}$ is self-dual with respect to $g_{\bm{\omega}}$. If moreover $\bm{\omega}=(\omega_{1},\omega_{2},\omega_{3})$ is a hyperkähler triple, then $g_{\bm{\omega}}$ is a hyperkähler metric. Furthermore, $\omega_{2}+i\omega_{3}$ is a holomorphic $2$-form with respect to the complex structure determined by $\omega_{1}$. See Section 2 for more concrete examples of such triples.

Since in our construction each piece of the manifold $\mathcal{M}$ is hyperkähler, hence carries a hyperkähler triple, we will first show that there exists a glued triple of closed $2$-forms $\bm{\omega}$ on $\mathcal{M}$ which is very close to being a hyperkähler triple, i.e.,

(1.16)

$$\|Q_{\bm{\omega}}-\Id\|_{C^{0}(\mathcal{M})}<\epsilon$$

for some sufficiently small constant $\epsilon>0$. A precise statement will be proved in Section 6. Starting from such a glued approximately hyperkähler triple $\bm{\omega}$, the goal of the perturbation procedure is to find a triple of closed $2$-forms $\bm{\theta}=(\theta_{1},\theta_{2},\theta_{3})$ such that $\underline{\bm{\omega}}\equiv\bm{\omega}+\bm{\theta}$ is an actual hyperkähler triple on $\mathcal{M}$. Precisely, we will solve the system

(1.17)

$$\frac{1}{2}(\omega_{i}+\theta_{i})\wedge(\omega_{j}+\theta_{j})=\delta_{ij}\dvol_{\bm{\omega}+\bm{\theta}},$$

which is equivalent to

(1.18)

$$\frac{1}{2}(\omega_{i}+\theta_{i})\wedge(\omega_{j}+\theta_{j})=\frac{1}{6}\delta_{ij}\sum\limits_{j=1}^{3}(\omega_{j}+\theta_{j})^{2}.$$

We expand equation (1.18):

(1.19)

$$\frac{1}{2}(\omega_{i}\wedge\omega_{j}+\omega_{i}\wedge\theta_{j}+\omega_{j}\wedge\theta_{i}+\theta_{i}\wedge\theta_{j})=\frac{1}{6}\delta_{ij}\sum\limits_{j=1}^{3}\Big(\omega_{j}^{2}+\theta_{j}^{2}+2\omega_{j}\wedge\theta_{j}\Big).$$

Now split $\bm{\theta}$ into its self-dual and anti-self-dual parts with respect to $g_{\bm{\omega}}$, $\bm{\theta}=\bm{\theta}^{+}+\bm{\theta}^{-}$. We define a matrix $A=(A_{ij})$ by $\theta^{+}_{i}=\sum\limits_{j=1}^{3}A_{ij}\omega_{j}$ and also define the matrix $S_{\bm{\theta}^{-}}=(S_{ij})$ by

(1.20)

$$\frac{1}{2}\theta_{i}^{-}\wedge\theta_{j}^{-}=S_{ij}\dvol_{\bm{\omega}},\ 1\leq i\leq j\leq 3.$$

Then in terms of the volume form $\dvol_{\bm{\omega}}$ given by the approximately hyperkähler triple $\bm{\omega}$, the expansion (1.19) can be rewritten as the matrix equation

(1.21)

$\displaystyle\begin{split}&(Q_{\bm{\omega}}+Q_{\bm{\omega}}A^{T}+AQ_{\bm{\omega}}+AQ_{\bm{\omega}}A^{T})+S_{\bm{\theta}^{-}}\\ &=\frac{1}{3}\Id\cdot\Big(\Tr(Q_{\bm{\omega}})+\Tr(AQ_{\bm{\omega}}A^{T})+\Tr(S_{\bm{\theta}^{-}})+\Tr(AQ_{\bm{\omega}})+\Tr(Q_{\bm{\omega}}A^{T})\Big).\end{split}$

For any $3\times 3$ real matrix $B$ denote by

(1.22)

$$\TF(B)=B-\frac{1}{3}\Tr(B)\Id$$

the trace-free part of $B$. Then we get

(1.23)

$$\TF(Q_{\bm{\omega}}A^{T}+Q_{\bm{\omega}}A+AQ_{\bm{\omega}}A^{T})=\TF(-Q_{\bm{\omega}}-S_{\bm{\theta}^{-}}).$$

For simplicity, in our context, we always identify a $3\times 3$-matrix with a triple of self-dual $2$-forms. Then observe that a solution of following gauge-fixed system

(1.24)

$\displaystyle d^{+}\bm{\eta}+\bm{\xi}=\mathfrak{F}_{0}\Big(\TF(-Q_{\bm{\omega}}-S_{d^{-}\bm{\eta}})\Big),\ d^{*}\bm{\eta}=0,$

is also a solution of (1.23). Here $\mathfrak{F}_{0}$ denotes the local inverse near zero to the local diffeomorphism $\mathfrak{G}_{0}:\mathscr{S}_{0}(\mathbb{R}^{3})\to\mathscr{S}_{0}(\mathbb{R}^{3})$ on the space of trace-free symmetric $3\times 3$-matrices defined by

(1.25)

$\displaystyle\mathfrak{G}_{0}(A)=\TF(Q_{\bm{\omega}}A^{T}+AQ_{\bm{\omega}}+AQ_{\bm{\omega}}A^{T}).$

Moreover, $\bm{\eta}=(\eta_{1},\eta_{2},\eta_{3})$ with $\eta_{j}\in\Omega^{1}(\mathcal{M})$, $\bm{\xi}=(\xi_{1},\xi_{2},\xi_{3})$ with $\xi_{i}\in\mathcal{H}^{+}(\mathcal{M})$ (the space of self-dual harmonic $2$-forms with respect to $g_{\bm{\omega}}$), and $d^{\pm}\bm{\eta}$ is the self-dual or anti-self-dual part of $d\bm{\eta}=\bm{\theta}-\bm{\xi}$ with respect to $g_{\bm{\omega}}$, respectively.

The linearization of the elliptic system (1.24) at $\bm{\eta}=0$ is

(1.26)

$$\mathscr{L}=(\mathscr{D}\oplus\Id)\otimes\mathbb{R}^{3}:(\Omega^{1}(\mathcal{M})\oplus\mathcal{H}^{+}(\mathcal{M}))\otimes\mathbb{R}^{3}\longrightarrow(\Omega^{0}(\mathcal{M})\oplus\Omega^{2}_{+}(\mathcal{M}))\otimes\mathbb{R}^{3},$$

where

(1.27)

$$\mathscr{D}\equiv d^{*}+d^{+}:\Omega^{1}(\mathcal{M})\longrightarrow(\Omega^{0}(\mathcal{M})\oplus\Omega^{2}_{+}(\mathcal{M}))$$

is a Dirac-type operator.

In order to solve the elliptic system (1.24) we will use the implicit function theorem (see Lemma 9.3). This requires finding a bounded right inverse to the linearized operator $\mathscr{L}$.

The first step is to define certain weighted Banach norms whose setup requires a careful understanding of the collapsing geometries in our construction. The construction of a suitable weight function will be more complicated than in almost all known gluing constructions because our neck region is topologically nontrivial. Geometrically, a crucial step is to assign to each $p\in\mathcal{M}$ an appropriate scale $r_{p}$ such that the geodesic ball $B_{r_{p}}(p)$ captures enough geometric information and curvature remains uniformly bounded for most points in $B_{r_{p}}(p)$. This is equivalent to finding a canonical rescaling factor for each $p\in\mathcal{M}$ which reflects the collapsing behavior of the glued metric near $p$. This leads to a decomposition of $\mathcal{M}$ (see Section 7.1 and 7.3) into 9 different regions, labeled $\I$, $\II$, $\III$, $\IV_{\pm}$, $\V_{\pm}$, $\VI_{\pm}$, with each of these regions exhibiting different regularity and convergence behaviors.

Second, in order to prove uniform boundedness of the right inverse by contradiction (Proposition 9.2), we then need to analyze the kernel of the linearized operator in suitable weighted spaces on the building blocks of the gluing construction. Here a crucial ingredient is a new Liouville theorem for half-harmonic 1-forms, i.e., $1$-forms $\phi$ satisfying

(1.28)

$$d^{*}\phi=0,\ \ d^{+}\phi=0,$$

on a Tian-Yau space, see Theorem 5.1.

In Section 2, we give some background on the Gibbons-Hawking ansatz. We also define the Heisenberg nilmanifolds, and describe the Calabi model space used in the Tian-Yau construction from the Gibbons-Hawking perspective. Lastly, we construct a harmonic function whose associated Gibbons-Hawking ansatz defines the neck region $\mathcal{N}$.

The Calabi model space will be described in Section 3 from a complex geometric perspective, which will be used to obtain the precise asymptotic behavior of the complete hyperkähler Tian-Yau spaces. The main result is Proposition 3.4 which roughly states that a complete Tian-Yau space is exponentially asymptotic to a Calabi model space, up to any arbitrary order of derivatives.

In Section 4, we will establish a Liouville type theorem for harmonic functions which says that any harmonic function of sufficiently small exponential growth on a complete hyperkähler Tian-Yau space has to be a constant. To prove this, the asymptotics proved in Section 3 will be crucial. These will allow us to reduce our problem to a question about harmonic functions on the Calabi model space, which will be solved using separation of variables. Specifically, the Laplace equation on the model space will be reduced to certain linear ODEs, and the quantitative analysis of the harmonic functions reduces to some delicate estimates for Hermite functions. The estimates in Section 4 do not require any advanced theory in hypergeometric functions. Instead the ODE solutions are defined by exponential integrals, which has an advantage that all the calculations in Section 4 are in fact self-contained. This step is already quite involved because it requires us to develop some new elliptic theory on a model space which is a doubly warped product rather than just a cylinder.

Using this work, we will then prove the aforementioned Liouville theorem for half-harmonic 1-forms on a complete hyperkähler Tian-Yau space in Section 5 (see Theorem 5.1). This will later be crucial in the proof of the key uniform estimate (Proposition 9.2) for our gluing construction. An important observation here is that equation (1.28) is equivalent to the $(0,1)$-component of $\phi$ satisfying $\bar{\partial}^{*}\phi^{0,1}=\bar{\partial}\phi^{0,1}=0$ for any choice of compatible integrable complex structure, which will allow us to invoke tools from complex geometry. More precisely, since the equation $\bar{\partial}\phi^{0,1}=0$ depends only on the complex structure, we may study it using a smooth Kähler metric on the compactified Fano manifold. The Kodaira vanishing theorem then ensures that $\phi^{0,1}=\bar{\partial}f$ for some function $f$. Such an $f$ is unique only modulo holomorphic functions, so a priori there is no growth estimate on $f$. However, using the construction of the Tian-Yau metrics and basic elliptic estimates we may find a solution $f$ satisfying some growth estimates. The equation $\bar{\partial}^{*}\phi^{0,1}=0$ then implies that $f$ is harmonic with respect to the Tian-Yau metric. However at this point we are not yet able to conclude using the Liouville theorem for harmonic functions that $f$ must be constant because our growth estimate for $f$ is too weak. We overcome this problem by using separation of variables on the Calabi model space. This final step hinges on the fact that the asymptotic decay rate of the complex structure of the Tian-Yau manifold relative to the Calabi model is much faster than the asymptotic decay rate of the hyperkähler metric.

In Section 6 we will complete the construction of the neck region and construct a closed almost hyperkähler triple on a manifold $\mathcal{M}$. We will also describe some topological invariants of $\mathcal{M}$. Note that at this stage it would be difficult to show directly that $\mathcal{M}$ is actually diffeomorphic to $\K 3$, a fact which will follow easily once we have shown it admits a hyperkähler metric.

Section 7 will focus on the geometry the manifold $\mathcal{M}$. We provide two different methods for analyzing the curvature of the approximate metric. One way is to apply some new types of $\epsilon$-regularity theorems for collapsed Einstein manifolds due to Naber and the fourth author [NZ16], which will yield curvature control once a simple topological condition is verified (see theorem 7.4 in Section 7). On the other hand, in Lemma 7.2 we will also give a direct curvature estimate in the collapsed regions. The remainder of this section will then describe all of the rescaled Gromov-Hausdorff limits for basepoints in each of the various regions.

In Sections 8 and 9, we will set up the weight analysis package and prove the main technical theorems. The first step is to define weighted Hölder spaces which are consistent with the different collapsing behaviors in different regions of $\mathcal{M}$. The remainder of Section 8 will then consist of proving the weighted Schauder estimate in Proposition 8.3. Existence of a bounded right inverse to the linearized operator will be proved in Section 9 (see Proposition 9.4), where we will then complete the proofs of Theorems 1.1 and 1.5.

We would like to thank Olivier Biquard, Simon Donaldson, Lorenzo Foscolo, Mark Gross, Mark Haskins, and Xiaochun Rong for various helpful comments and discussions. We would also like to thank Bobby Acharya and David Morrison for pointing out to us some interpretations of our work from the physical perspective.