1 Introduction↩︎

Gravitational instantons were first introduced by S. Hawking [1] in the study of Euclidean quantum gravity. In [1] they were roughly described as “solutions of the classical Einstein equations which are non-singular on some section of complexified spacetime and in which the curvature dies away at large distances”. In the literature, there does not seem to be a consistent definition of gravitational instantons. To fix terminology, we will use the following definition in this paper.

Gravitational instantons are examples of Riemannian Einstein metrics. Notice that in dimensions less than 4, an Einstein metric must have constant sectional curvature, and hence is locally given by the standard space forms. This fact is related to the classical uniformization theorem in dimension 2. It also underpins the development of Ricci flow as a tool to study the geometry of 3-manifolds. In dimensions 4 and above, Einstein metrics are more flexible and they may form singularities in families. Gravitational instantons provide local singularity models for Einstein metrics in dimension 4, thus they are expected to play an important role in the understanding of 4-dimensional geometry.

Gravitational instantons have been extensively studied in both mathematics and physics. Two fundamental questions in this area involve the construction and classification of gravitational instantons. These boil down to understanding the geometry of a difficult nonlinear PDE system. The classification problem can be further divided into two questions.

(A) Classify the asymptotic structures of gravitational instantons.

This divides gravitational instantons into discrete families, each characterized by distinct asymptotic behavior. An intriguing feature in the study of gravitational instantons is the existence of many different model ends, such as \(\mathop{\mathrm{ALE}}\), \(\mathop{\mathrm{AF}}\), etc.

Given a \(\sharp\) model end \((M_0, g_0)\), we say that a gravitational instanton \((M, g)\) is \(\sharp\) if there exists a compact set \(K \subset M\) such that \(M \setminus K\) can be smoothly identified with \(M_0\), and \[\sup_{M \setminus K} \bigl| \nabla_{g_0}^k (g - g_0) \bigr|_{g_0} \leq C_k \, \rho^{-k-\epsilon},\] where \(\rho\) denotes the distance function with respect to \(g_0\).

(B) Classify gravitational instantons with a given asymptotic structure.

This involves understanding the bulk geometry of gravitational instantons. Gravitational instantons with a given asymptotic structure can form a continuous moduli space.

As we will explain in Section 3, we can divide gravitational instantons into three Types.

1. Type \(\mathop{\mathrm{I}}\): locally (hyper-)Kähler (\(\mathop{\mathrm{W}}^+\equiv 0\), i.e., anti-self-dual).

2. Type \(\mathop{\mathrm{II}}\): locally Hermitian but not locally Kähler (\(\mathop{\mathrm{W}}^+\) has two distinct eigenvalues everywhere).

3. Type \(\mathop{\mathrm{III}}\): not locally Hermitian (\(\mathop{\mathrm{W}}^+\) has three distinct eigenvalues generically).

Type \(\mathop{\mathrm{I}}\) and Type \(\mathop{\mathrm{II}}\) gravitational instantons are intimately related to complex geometry and can be viewed as gravitational instantons equipped with special structures. On the other hand, most known examples of gravitational instantons are all of Type \(\mathop{\mathrm{I}}\) or \(\mathop{\mathrm{II}}\).

A major recent achievement in this field is the essential classification of hyperkähler and Hermitian gravitational instantons due to the work by many people in the past decade or so. These involve intriguing interactions between Riemannian geometry, complex geometry, PDEs, and also ideas from physics. On the other hand, systematic examples of Type III gravitational instanton have only been constructed very recently. In this article we will survey these progress and discuss some open questions for future study.

Acknowledgments: The authors have benefited from discussions and collaborations with many colleagues on this topic over the past few years, especially Lars Andersson, Gao Chen, Simon Donaldson, Lorenzo Foscolo, Hans-Joachim Hein, Shaosai Huang, Marcus Khuri, Claude LeBrun, Hongyi Liu, John Lott, Jeff Viaclovsky, Gilbert Weinstein, and Ruobing Zhang. They are also grateful for the invitation to the program “Einstein 4-Manifolds and Gravitational Instantons” at the Simons Center for Geometry and Physics in January 2026, where part of this article is written.

2 General Riemannian geometric consequences↩︎

Let \((M, g)\) be a gravitational instanton. Since it has vanishing Ricci curvature, one may deduce some basic properties of \((M, g)\) by appealing to general theory on complete Riemannian manifolds with non-negative Ricci curvature.

Since \(g\) is not flat, by the Cheeger-Gromoll splitting theorem we know that \(M\) has exactly one end. Next, it follows from the Bishop inequality that for any \(p\in M\), the ratio \[\nu(g; r):=\mathop{\mathrm{Vol}}(B(p,r))/\omega_4r^4\] is a non-increasing function of \(r\in (0, \infty)\), where \(\omega_4\) denotes the volume of a unit ball in \(\mathbb{R}^4\). The limit \[\nu_\infty:=\lim_{r\rightarrow\infty} \nu(g;r)\in [0, 1]\] is independent of \(p\). In particular, for all \(r>0\) we have \[\mathop{\mathrm{Vol}}(B(p,r))\leq \nu_\infty \omega_4 r^4.\] Furthermore, \(\nu_\infty=1\) if and only if \((M, g)\) is isometric to the standard flat \(\mathbb{R}^4\).

In general, by Gromov’s precompactness theorem, one obtains asymptotic cones of \((M, g)\) at infinity.

By assumption, the rescaled sequence has locally uniformly bounded curvature away from \(O\). Note, however, that a priori the limit may depend on the choice of the sequence \(\{\lambda_j\}\), so the asymptotic cone need not be unique. Moreover, a priori, an asymptotic cone is not known to be a genuine metric cone. These are the central difficulties involved in the classification of asymptotic structures of gravitational instantons; they are both confirmed in the special case of hyperkähler and Hermitian gravitational instantons, see Theorem 4 and Theorem 13.

We say that \((M, g)\) is non-collapsed if \(\nu_\infty>0\). This means that for a fixed \(p\in M\), the unit ball around \(p\) with respect to the rescaled metric \[g_{\lambda}:=\lambda^{-2}g\] has volume bounded below by \(c>0\) (i.e., the volume is non-collapsing) for all \(\lambda\geq0\). In this case, it follows from the work of Bando-Kasue-Nakajima [2] that \((M, g)\) is \(\mathop{\mathrm{ALE}}\) (Asymptotically Locally Euclidean), i.e., there is a flat orbifold \((\mathbb{R}^4/\Gamma, g_0)\) for some \(\Gamma\subset SO(4)\), whose end can be smoothly identified with \(M\setminus K\) for some compact set \(K\), and there is a \(\delta>0\) such that for all \(k\in\mathbb{Z}_{\geq0}\), we have \[|\nabla^k_{g_0}(g-g_0)|_{g_0}\leq C_k r^{-\delta-k}\] for some \(C_k>0\). Here \(r=d(p, \cdot)\) denotes the distance from \(p\) on \(M\), and it is comparable to the standard radial function on \(\mathbb{R}^4/\Gamma\). In particular, \((M, g)\) has a unique asymptotic cone, given by \(\mathbb{R}^4/\Gamma\).

At the other extreme, a result of Calabi and Yau [3], [4] shows that \(g\) has at least linear volume growth, that is, there exists \(c>0\) such that \[\mathop{\mathrm{Vol}}(B(p, r))\geq cr \;\text{for all} \; r\geq 1.\] There is a class of gravitational instantons, namely \(\mathop{\mathrm{ALH}}\) gravitational instantons (see Section 4.2), which has precisely linear volume growth.

There are some general topological constraints on \(M\) arising from the fact that \(g\) has nonnegative Ricci curvature. A result of Anderson [5] implies that \[b_1(M) \leq 3,\] and that \(M\) has a finite fundamental group if \((M, g)\) is non-collapsed. A result of Shen–Sormani [6] implies that \[b_3(M) = 0.\]

If \(M\) is non-collapsed, then it has finite topological type (i.e., \(M\) is diffeomorphic to the interior of a manifold with boundary), since it is \(\mathop{\mathrm{ALE}}\). When \(M\) is collapsed, then by [7] the end of \(M\) admits an \(\mathcal{N}\) (i.e., nilpotent)-structure. Roughly speaking, the latter means that, locally around each point, up to a finite cover there is an effective action of a nilpotent Lie group, which satisfies suitable compatibility conditions.

3 Riemannian geometry in dimension four↩︎

Recall that the Riemann curvature tensor of a general Riemannian metric \(g\) can be viewed as a self-adjoint endomorphism of the bundle of 2-forms \[\mathop{\mathrm{Riem}}: \Lambda^2\rightarrow \Lambda^2.\] The standard representation theory of \(SO(n)\) decomposes \(\mathop{\mathrm{Riem}}\) into the sum of three components: one involving the scalar curvature \(S\), one involving the trace-free Ricci curvature \(\overset{\circ}{\mathop{\mathrm{Ricci}}}\), and one involving the Weyl curvature \(W\). In dimension four, since \(SO(4)\) is a double cover of the product \(SO(3)\times SO(3)\), we obtain a refined decomposition. Assuming \(M\) is oriented, the bundle \(\Lambda^2\) splits as the direct sum of the bundle of self-dual and anti-self-dual 2-forms \[\Lambda^2=\Lambda^+\oplus \Lambda^-.\] The unit sphere in \(\Lambda_p^+\) can be identified with the set of orthogonal almost complex structures on \(T_pM\) that are compatible with the given orientation.

Accordingly, we may write \(\mathop{\mathrm{Riem}}\) as \[\mathop{\mathrm{Riem}}=\begin{pmatrix} \mathop{\mathrm{W}}^++\frac{\mathop{\mathrm{Scal}}}{12} & \overset{\circ}{\mathop{\mathrm{Ricci}}}\\ \overset{\circ}{\mathop{\mathrm{Ricci}}} & \mathop{\mathrm{W}}^-+\frac{\mathop{\mathrm{Scal}}}{12} \end{pmatrix}.\] At each point we may view \(\mathop{\mathrm{W}}^+\) as a trace-free symmetric \(3\times 3\) matrix.

For convenience of discussion, we divide 4-dimensional oriented Ricci-flat metrics into three classes.

1. Type \(\mathop{\mathrm{I}}\): \(\mathop{\mathrm{W}}^+\equiv0\) (i.e., \(g\) is anti-self-dual).

2. Type \(\mathop{\mathrm{II}}\): \(\mathop{\mathrm{W}}^+\) has exactly one eigenvalue of multiplicity two everywhere.

3. Type \(\mathop{\mathrm{III}}\): \(\mathop{\mathrm{W}}^+\) has three distinct eigenvalues generically.

It is not completely obvious that there are exactly three Types as above. The non-trivial point is that if \(\mathop{\mathrm{W}}^+\) has exactly two distinct eigenvalues on an open domain, then this must hold everywhere. This was proved in [10]. Using the terminology originating in the Petrov classification of algebraic types, the Type \(\mathop{\mathrm{I}}\), \(\mathop{\mathrm{II}}\), \(\mathop{\mathrm{III}}\) cases are also called Types \(O, D, I\) respectively. Sometimes, Type \(\mathop{\mathrm{III}}\) is also called algebraically general, while Type \(\mathop{\mathrm{I}}\) and \(\mathop{\mathrm{II}}\) are called algebraically special.

As we shall explain below, in the first two cases, the Ricci-flat metrics are closely related to complex geometry. This fact leads to many deep results for Type \(\mathop{\mathrm{I}}\) and \(\mathop{\mathrm{II}}\) gravitational instantons. At present, we have a relatively satisfactory theory in these two cases.

3.1 Type \(\mathop{\mathrm{I}}\) case↩︎

A Ricci-flat metric \(g\) satisfies \(\mathop{\mathrm{W}}^+=0\) if and only if the induced Levi-Civita connection on \(\Lambda^+\) is flat. In this case, the metric \(g\) has local holonomy contained in \(SU(2)\subset SO(4)\). Note that under the double cover map \[SO(4)\rightarrow SO(3)\times SO(3),\] \(SU(2)\) maps onto one of the \(SO(3)\) factors. Having local holonomy contained in \(SU(2)\) is the same as saying that \(g\) is locally hyperkähler. Namely, locally there is a triple of orthogonal complex structures \(J_\alpha \;(\alpha=1, 2, 3)\) satisfying the quaternion relations \[J_1J_2=-J_2J_1=J_3\] and satisfying \[\nabla_g J_\alpha=0.\] The set of parallel compatible complex structures is naturally parametrized by the unit 2-sphere \(\mathbb{S}^2\): for each \(\boldsymbol{u}=(u_1, u_2, u_3)\in \mathbb{S}^2\), we associate the complex structure \[J_{\boldsymbol{u}}:=u_1J_1+u_2J_2+u_3J_3.\] Locally, there is a triple of parallel (hence closed) 2-forms \({\boldsymbol{\omega}}=\{\omega_{\alpha}|\alpha=1, 2, 3\}\) (where \(\omega_\alpha=g(J_\alpha, \cdot)\)), which gives rise to an orthonormal basis of \(\Lambda^+\) at each point, i.e., \[\label{eqn:hyperkahler}\frac{1}{2}\omega_\alpha\wedge \omega_\beta=\delta_{\alpha\beta} \cdot \mu\tag{5}\] where \(\mu=\mathop{\mathrm{dvol}}_g\). Conversely, it is also straightfoward to see that a locally hyperkähler metric \(g\) is naturally a Ricci-flat metric of Type \(\mathop{\mathrm{I}}\) with respect to the orientation defined by the complex structures \(J_\alpha\).

There is a characterization of the hyperkähler condition in terms of a differential system involving only the triple \({\boldsymbol{\omega}}\). Suppose that on a 4-manifold, we are given a triple of closed 2-forms \({\boldsymbol{\omega}}=\{\omega_1, \omega_2,\omega_3\}\) and a volume form \(\mu\) such that 5 holds, then there is a unique hyperkähler metric \(g\) such that \({\boldsymbol{\omega}}\) arises from the above construction. Such a triple \({\boldsymbol{\omega}}\) is usually referred to as a hyperkähler triple.

If we fix a choice of the complex structure, say \(J_1\), then we can interpret the hyperkähler condition in terms of a complex Monge-Ampère equation. This established a connection with complex geometry. We may fix a Kähler form \(\omega:=\omega_1\) and a holomorphic 2-form \(\Omega:=\omega_2+\sqrt{-1}\omega_3\), and they satisfy the equation \[\label{eqn:complex32MA}\omega^2=\frac{1}{2}\Omega\wedge\bar\Omega.\tag{6}\] From this one can also see that a 4-dimensional Ricci-flat metric is locally hyperkähler if and only if it is locally Kähler.

In practice, it is often straightforward to write down the holomorphic 2-form \(\Omega\) using complex/algebraic geometry and obtaining a hyperkähler triple reduces to finding \(\omega\). Locally, \(\omega\) has a Kähler potential \(\varphi\), which solves a Monge-Ampère type PDE. On compact manifolds, the existence of Kähler-Einstein metrics is completely understood in terms of algebraic geometric data, due to the work of Yau [11], Aubin [12], and Chen-Donaldson-Sun [13]. In the noncompact setting, the connection to complex algebraic geometry is also important in the construction and classification of Type \(\mathop{\mathrm{I}}\) gravitational instantons.

3.2 Type \(\mathop{\mathrm{II}}\) case↩︎

Now suppose that \(g\) is a Ricci-flat metric of Type \(\mathop{\mathrm{II}}\). Locally, we can choose a self-dual 2-form \(\omega\), which is the eigen-form associated to the non-repeated eigenvalue of \(\mathop{\mathrm{W}}^+\) and is normalized to have unit norm. Then \(\omega\) defines an orthogonal almost complex structure \(J\) such that \(\omega=g(J\cdot, \cdot)\). A crucial observation by Derdziński [10] is that \(J\) is integrable. Furthermore, if we denote \[\lambda:=2\sqrt{6}|\mathop{\mathrm{W}}^+|_g,\] then the conformal metric \[g_K:=\lambda^{2/3}g\] is Kähler with respect to \(J\). Since \(g\) is Ricci-flat, \(g_K\) has vanishing Bach tensor. In particular, \(g_K\) is extremal in the sense of Calabi as observed by LeBrun [14], i.e., \[\mathbb{K}:=J\nabla_{g_K}\mathop{\mathrm{Scal}}_{g_K}\] is a non-zero Hamiltonian Killing field, where the scalar curvature of \(g_K\) satisfies \[|\mathop{\mathrm{Scal}}_{g_K}|=\lambda^{1/3}.\] It then follows that \[\mathbb{K}=-J\nabla_g \lambda^{-1/3}\] is also a Killing field for the original Ricci-flat metric \(g\). In particular, a Type \(\mathop{\mathrm{II}}\) Ricci-flat metric is locally conformal to a Kähler metric and admits non-trivial continuous symmetries. Notice that the choice of \(\omega\) is determined up to sign, so, after passing to a double cover if necessary, we may assume \(g\) is globally conformal to a Kähler metric.

Conversely, note that for a Kähler metric in 4 dimensions, \(\mathop{\mathrm{W}}^+\) always has a repeated eigenvalue \(-\frac{\mathop{\mathrm{Scal}}}{12}\), and the Weyl curvature is conformally invariant. Therefore, if a Ricci-flat metric is locally conformal to a Kähler metric (for some complex structure), then it must be either Type \(\mathop{\mathrm{I}}\) or Type \(\mathop{\mathrm{II}}\). Moreover, by the Riemannian Goldberg-Sachs theorem [15], if a Ricci-flat metric is Hermitian with respect to an integrable almost complex structure, then it is either Type \(\mathop{\mathrm{I}}\) or Type \(\mathop{\mathrm{II}}\). It is Type \(\mathop{\mathrm{I}}\) precisely when \(W^+=0\).

For simplicity of discussion, we introduce the following terminologies in this paper

The above discussion then implies that being Kähler implies Type \(\mathop{\mathrm{I}}\), and being Hermitian implies Type \(\mathop{\mathrm{II}}\). Conversely, for simply-connected manifolds, Type \(\mathop{\mathrm{I}}\) implies being Kähler, and Type \(\mathop{\mathrm{II}}\) implies being Hermitian.

4 Type \(\mathop{\mathrm{I}}\) gravitational instantons↩︎

In this section, we discuss Type \(\mathop{\mathrm{I}}\) gravitational instantons. Notice that a Ricci-flat metric of Type \(\mathop{\mathrm{I}}\) is in general not globally hyperkähler. It is the case when the manifold is simply-connected. We focus primarily on the case of hyperkähler gravitational instantons and briefly discuss the general Type \(\mathop{\mathrm{I}}\) case in Section 4.8.

4.1 Gibbons-Hawking ansatz↩︎

The ansatz of Gibbons-Hawking [16] provides an explicit construction of 4-dimensional hyperkähler metrics admitting a continuous symmetry. It also plays an important role in describing model ends for gravitational instantons.

Let \(V\) be a positive harmonic function defined on an open set \(Q\subset \mathbb{R}^3\). If \(H_2(Q; \mathbb{R})=0\), then we can find a 1-form \(\theta\) such that \[*dV=d\theta,\] where \(*\) denotes the Hodge star operator on \(\mathbb{R}^3\). Consider the product space \(P=Q\times \mathbb{S}^1\) equipped with the Riemannian metric \(g\) given by \[g=V\sum_{\alpha=1}^3dx_\alpha^{2}+V^{-1}(dt+\theta)^{ 2},\] where \(x_\alpha\)’s are standard coordinates on \(\mathbb{R}^3\), and \(t\) denotes the standard coordinate on \(\mathbb{S}^1=\mathbb{R}/\mathbb{Z}\). Clearly \(g\) is invariant under the obvious free \(\mathbb{S}^1\) rotation in the \(\mathbb{S}^1\) factor.

It is straightforward to check that \(g\) is a hyperkähler metric. For example, one can write down an \(\mathbb{S}^1\)-invariant triple of closed self-dual 2-forms \({\boldsymbol{\omega}}=(\omega_1, \omega_2, \omega_3)\), by setting \[\begin{align} \omega_1=Vdx_1\wedge dx_2+dx_2\wedge \theta,\\ \omega_2=Vdx_2\wedge dx_3+dx_1\wedge \theta, \\ \omega_3=Vdx_3\wedge dx_1+dx_3\wedge \theta. \end{align}\]

Conversely, if we are given a 4-dimensional hyperkähler metric \(g\) with a non-vanishing Killing field \(\xi\), which is tri-holomorphic (i.e., holomorphic with respect to all the complex structures \(J_\alpha\)), then locally \(g\) must arise from the above construction. Indeed, one can recover the coordinate \(x_\alpha\) as moment maps with respect to the hyperkähler form \(\omega_\alpha\) and recover the function \(V^{-\frac{1}{2}}\) as the length of \(\xi\) and \(\theta\) as the metric dual 1-form of \(\xi\). The key point is that the presence of an \(\mathbb{S}^1\) symmetry allows us to reduce the nonlinear PDE governing the hyperkähler structure to the Laplacian equation on \(\mathbb{R}^3\), which is a simple linear PDE.

There are certain variants of the above construction which will be important later. First, we can let \(Q\) be a domain in a general parallelizable flat 3 manifold and we may allow \(Q\) to have non-trivial topology. The above construction still applies if we can choose a global parallel orthonormal frame of \(Q\) provided the function \(V\) satisfies the integrality condition that \[\label{eqn:integrality32condition}\frac{1}{2\pi}\int_{C} *dV\in \mathbb{Z}, \;\text{for all 2-cycles}\; C\in H_2(U;\mathbb{Z}).\tag{7}\] In this case, we may replace \(P\) by a principal \(U(1)\) bundle \[\pi: P\rightarrow Q,\] and replace the 1-form \(dt+\theta\) by a 1-form \(\Theta\) on \(P\) such that \(-\sqrt{-1}\Theta\) is a connection 1-form whose curvature is \(*dV\). One can check that gauge equivalent connections lead to isometric metrics; thus, \(g\) depends only on the gauge equivalence class of \(\Theta\). Different gauge equivalence classes differ by tensoring with a flat connection, and the set of isomorphism classes of flat connections is given by \(H^1(Q;\mathbb{R})/H^1(Q; \mathbb{Z})\).

Next, we want to include the case when the \(\mathbb{S}^1\) action has fixed points. For this purpose we need to allow \(V\) to be \(+\infty\) on a discrete subset of \(Q\). By Bôcher’s theorem, locally \(V\) must be of the form \(v+br^{-1}\) for a smooth harmonic function \(v\) across the singularity and a positive number \(b\), where \(r\) is the local distance function on the flat \(\mathbb{R}^3\). Away from the singularity, for \(V\) to satisfy the integrality condition 7 , \(b\) must be of the form \(\frac{k}{2}\) for some non-negative integer \(k\).

In the simplest situation we can take \(Q=\mathbb{R}^3\) and \[V=\frac{1}{2r}.\] Then we recover the standard flat hyperkähler metric on \(\mathbb{R}^4\). Fixing a choice of a compatible complex structure we may identify \(\mathbb{R}^4\) with \(\mathbb{C}^2\), then the \(\mathbb{S}^1\) action is given by \[\label{eqn:S132action}\lambda.(z_1, z_2)=(\lambda^{-1} z_1, \lambda z_2).\tag{8}\] The projection map \(\pi:\mathbb{C}^2\rightarrow \mathbb{R}^3\) in this case is the Hopf fibration, which can be explicitly expressed as \[(z_1, z_2)\mapsto (\text{Re}(z_1z_2), \text{Im}(z_1z_2), \frac{1}{2}(|z_1|^2-|z_2|^2)).\] If we instead take \[V=\frac{k}{2r}, \;\;k\in \mathbb{Z}_{>0},\] then we get the flat orbifold \(\mathbb{C}^2/\mathbb{Z}_k\), given by the quotient of \(\mathbb{C}^2\) by an action of \(\mathbb{Z}_k\) of the form \[(z_1, z_2)\mapsto (\zeta_k z_1, \zeta_k^{-1}z_2).\] For a general function \[V=v+\frac{k}{2r}\] as above, one obtains an orbifold hyperkähler metric; the metric is smooth if and only if \(k = 1\).

Next we explain some important families of hyperkähler metrics obtained using the Gibbons-Hawking ansatz.

4.2 Asymptotic models↩︎

We now introduce several families of asymptotic model ends for gravitational instantons. Throughout this section, unless otherwise specified, we focus on the end at infinity of the spaces under consideration.

We begin with the four standard model ends: \(\mathop{\mathrm{ALE}}\), \(\mathop{\mathrm{ALF}}\), \(\mathop{\mathrm{ALG}}\), and \(\mathop{\mathrm{ALH}}\), which are asymptotically locally products of the form \(\mathbb{T}^k \times \mathbb{R}^{4-k}\) for \(k = 0,1,2,3\), respectively. The terminology \(\mathop{\mathrm{ALF}}\) originally stands for “Asymptotically Locally Flat,” whereas \(\mathop{\mathrm{ALG}}\) and \(\mathop{\mathrm{ALH}}\) are not derived from abbreviations.

4.2.1 \(\mathop{\mathrm{ALE}}\) models↩︎

For a finite subgroup \(\Gamma \subset SO(4)\) acting freely on \(S^3\), an \(\mathop{\mathrm{ALE}}\) (asymptotically locally Euclidean)-\(\Gamma\) model end is given by the flat cone \(\mathbb{R}^4/\Gamma\). Such finite subgroups \(\Gamma\) have been completely classified.

After identifying \(\mathbb{R}^4 \simeq \mathbb{C}^2\), we may assume that \(\Gamma\) is a subgroup of \(U(2) \subset SO(4)\). In particular, an \(\mathop{\mathrm{ALE}}\) model end is always Kähler with respect to some choice of complex structure. It is hyperkähler if, in addition, \(\Gamma\) can be chosen to lie in \(SU(2)\). In this case, \(\Gamma\) is one of the ADE groups, which are classified as follows:

• \(A_k(k\geq 0)\): this is the cyclic group of order \(k+1\), generated by the diagonal matrix \[\begin{pmatrix} \zeta_{k+1}& 0 \\ 0 & \zeta_{k+1}^{-1} \end{pmatrix}.\]

• \(D_k(k\geq 3)\): this is the binary dihedral group generated by \[\begin{pmatrix} \zeta_{k-2}& 0 \\ 0 & \zeta_{k-2}^{-1} \end{pmatrix}, \;\;\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.\]

• \(E_k(k=6,7, 8)\): these correspond to the binary tetrahedral, octahedral, icosahedral groups.

The complete classification of finite subgroups of \(U(2)\) acting freely on \(S^3\) can be found in [22].

Notice that the \(\mathop{\mathrm{ALE}}\)-\(A_k\) model end admits an \(\mathbb{S}^1\) symmetry, so it is given by the Gibbons-Hawking ansatz. Indeed, one can take \(Q=\mathbb{R}^3\setminus B_1\) and \[V=\frac{k+1}{2r}.\]

4.2.2 \(\mathop{\mathrm{ALF}}\) models↩︎

An \(\mathop{\mathrm{ALF}}\)-\(A_k\;(k\in \mathbb{Z})\) model end is given by the Gibbons-Hawking ansatz applied to \(Q=\mathbb{R}^3\) and the harmonic function \[V=\frac{k+1}{2r}+T\] for some \(T>0\). In particular, it is hyperkähler. When \(k\geq 0\), it is given by the \(\mathbb{Z}_{k+1}\) quotient of the Taub-NUT metric.

An \(\mathop{\mathrm{ALF}}\)-\(D_k(k\in \mathbb{Z})\) model end is the quotient of an \(\mathop{\mathrm{ALF}}\)-\(A_{2k-5}\) end by a further \(\mathbb{Z}_2\) action. The \(\mathbb{Z}_2\) action can be described in the Gibbons-Hawking ansatz as an involution \(\boldsymbol{x}\mapsto -\boldsymbol{x}\) on \(\mathbb{R}^3\) and \(t\mapsto -t\). An \(\mathop{\mathrm{ALF}}\)-\(D_k\) model end is also hyperkähler.

A general \(\mathop{\mathrm{ALF}}\) model end is a finite free isometric quotient of an \(\mathop{\mathrm{ALF}}\)-\(A_k\) end for some \(k\). It is not necessarily hyperkähler.

Note that an \(\mathop{\mathrm{ALF}}\) model end has cubic volume growth. At infinity, it is a locally asymptotically flat \(\mathbb{S}^1\)-fibration. The asymptotic cone is \(\mathbb{R}^3\) in the \(A_k\) case, \(\mathbb{R}^3/\mathbb{Z}_2\) in the \(D_k\) case, and more generally an orbifold of the form \(\mathbb{R}^3/\Gamma\).

4.2.3 \(\mathop{\mathrm{ALG}}\) models↩︎

For \(\beta\in (0, 1]\), take a 2-dimensional flat cone \(\mathbb{C}_\beta\) with angle \(2\pi\beta\), the cotangent bundle \(T^*\mathbb{C}_\beta\) admits a canonical flat metric. Indeed, the metric is hyperkähler. Take a lattice sub-bundle, we obtain a flat end, which we denote by \(\mathop{\mathrm{ALG}}_\beta\).

An \(\mathop{\mathrm{ALG}}_\beta\) model end is always Kähler. It is hyperkähler exactly when the lattice sub-bundle is preserved by the monodromy \(M_\beta\), which is the case exactly when \(\beta\in \{\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{6}, \frac{5}{6}, 1\}\) (see for example [23]).

An \(\mathop{\mathrm{ALG}}\) model end has quadratic volume growth. At infinity it is a locally asymptotically flat \(T^2\) fibration, and the asymptotic cone is exactly \(\mathbb{C}_\beta\).

4.2.4 \(\mathop{\mathrm{ALH}}\) models↩︎

An \(\mathop{\mathrm{ALH}}\) model end is given by \([0, \infty)\times F\) for a compact flat 3-manifold \(F\). It is hyperkähler if \(F\) is a flat torus.

An \(\mathop{\mathrm{ALH}}\) model end has linear volume growth, with asymptotic cone given by the ray \([0, \infty)\).

Next we introduce two two exceptional families of hyperkähler model ends. They are given by the Gibbons-Hawking ansatz applied to slightly more general flat 3-manifolds.

4.2.5 \(\mathop{\mathrm{ALH}}^*\) models↩︎

Consider the product \[Q=\mathbb{T}\times [1, \infty)\] for an oriented flat 2-torus \(\mathbb{T}=\mathbb{R}^2/\Lambda\). Denote by \(g_{\mathbb{T}}+dz^2\) the product metric on \(Q\) and fix the orthonormal frame \(e_1=\partial_x, e_2=\partial_y, e_3=\partial_z\), where \(\{x, y\}\) are the standard coordinates on \(\mathbb{R}^2\). Denote by \(A\) the area of \(\mathbb{T}\). Consider the harmonic function \[V=\frac{2\pi kz}{A}\] for \(k\in \mathbb{Z}_{>0}\). The integrality condition 7 is satisfied, so we can apply the Gibbons-Hawking ansatz here. Fix a choice of \(\Theta\), then we obtain an \(\mathop{\mathrm{ALH}}^*_k\) model end. As is explained in [24], for a different choice of \(\Theta\), the hyperkähler metric only differs by a diffeomorphism.

Since \(V\) is itself invariant under the translation along \(\mathbb{T}\), \(g\) admits extra symmetries. Indeed, there is a tri-holomorphic isometric action of the 3-dimensional Heisenberg group on \((P, g)\), and the action is transitive on the level set of \(z\).

An \(\mathop{\mathrm{ALH}}^*\) model end exhibits interesting multi-scale Riemannian geometry: as \(z\) tends to infinity, the size of the \(\mathbb{S}^1\) orbits shrinks, whereas the size of the base \(\mathbb{T}^2\) expands. See Figure 1. It is easy to see that the volume growth rate is given by \(\frac{4}{3}\), and the asymptotic cone is the ray \([0, \infty)\).

The \(\mathop{\mathrm{ALH}}^*\) model ends admit several interesting descriptions from the viewpoint of complex geometry. On the one hand, one may choose a compatible complex structure on \((P,g)\) such that the natural projection \(P \to \mathbb{T}\) is holomorphic. In this case, \(P\) is biholomorphic to the complement of the zero section in the disk bundle inside the total space of a holomorphic line bundle \(L\) of degree \(k\) over \(\mathbb{T}\). Thus the complete end of \(P\) can be compactified complex-analytically by adding the zero section of \(L\) (the holomorphic structure on \(L\) depends on the choice of the connection \(\Theta\)).

From this perspective, the construction fits into a special case of the Calabi ansatz, which produces special Kähler metrics on a Hermitian line bundle \((L,h)\) over a compact Kähler manifold, of the form \[\sqrt{-1}\,\partial\bar{\partial} F(|\xi|_h^2).\] The general Calabi ansatz extends to higher dimensions; see, for example [25]–[28].

On the other hand, we can choose another compatible complex structure on \((P, g)\) so that the natural projection map \(P\rightarrow \mathbb{S}^1\times (1, \infty)\) is holomorphic, for some totally geodesic circle \(\mathbb{S}^1\subset \mathbb{T}\). Notice that \(\mathbb{S}^1\times [1, \infty)\) can be holomorphically identified with a punctured disk \(\Delta^*\). Then \(P\) becomes an elliptic fibration over \(\Delta^*\), with monodromy of type \(I_k\). From standard theory the complete end of \(P\) can be compactified into a smooth complex surface which is an elliptic fibration over \(\Delta\), with a singular fiber at \(0\), of Type \(I_k\) in terms of the Kodaira classification.

4.2.6 \(\mathop{\mathrm{ALG}}^*\) models↩︎

Similarly we consider the flat product \[Q=\mathbb{S}^1\times (\mathbb{R}^2\setminus B_1),\] where \(B_1\) denotes the standard unit ball in \(\mathbb{R}^2\). Denote by \(\ell\) the length of \(\mathbb{S}^1\). Take the harmonic function \[V=\frac{k\log r}{\ell}\] for \(k\in\mathbb{Z}_{>0}\), where \(r\) is the standard radial function on \(\mathbb{R}^2\). Again by fixing a \(U(1)\) connection \(-\sqrt{-1}\Theta\) the Gibbons-Hawking construction gives rise to a \(U(1)\) invariant hyperkähler metric \(g\) on a \(U(1)\) bundle \(P\rightarrow Q\) of degree \(k\). This is what we call an \(\mathop{\mathrm{ALG}}^*\)-\(I_k\) model end.

An \(\mathop{\mathrm{ALG}}^*\)-\(I_k^*\) model end is a free \(\mathbb{Z}_2\) quotient of an \(\mathop{\mathrm{ALG}}^*\)-\(I_k\) model end, where the \(\mathbb{Z}_2\) action covers the standard involution on \(\mathbb{R}^2\) and on \(\mathbb{S}^1\).

As above, an \(\mathop{\mathrm{ALG}}^*\)-\(I_k\) model end admits a tri-holomorphic isometric action of the 3-dimensional Heisenberg group. On the \(\mathbb{Z}_2\)-quotient, however, only a 2-dimensional torus action descends. The level sets of \(r\) are infranil manifolds, namely circle bundles over a Klein bottle. The geometry also exhibits a multi-scale structure: when viewed as a torus fibration, the fibers degenerate as \(r \to \infty\).

One can check that the volume growth is quadratic. The asymptotic cone is \(\mathbb{R}^2\) in the \(\mathop{\mathrm{ALG}}^*\)-\(I_k\) case, and \(\mathbb{R}^2/\mathbb{Z}_2 = \mathbb{C}_\pi\) in the \(\mathop{\mathrm{ALG}}^*\)-\(I_k^*\) case.

From the viewpoint of complex geometry, one may choose a compatible complex structure on \(P\) such that the projection map \(P \to \mathbb{R}^2 \setminus B_1\) is holomorphic. Identifying \(\mathbb{R}^2 \setminus B_1\) holomorphically with the punctured disk \(\Delta^*\), the space \(P\) becomes an elliptic fibration over \(\Delta^*\), with monodromy of type \(I_k\) or \(I_k^*\). Moreover, it admits a compactification to an elliptic fibration over \(\Delta\).

4.3 Constructions of examples↩︎

There are diverse constructions of examples of hyperkähler gravitational instantons in both the mathematics and physics literature.

\(\mathop{\mathrm{ALE}}\)-\(A_k\) \((k\geq 1)\) hyperkähler gravitational instantons can be constructed using the Gibbons-Hawking ansatz, as given in Example 3. General \(\mathop{\mathrm{ALE}}\) hyperkähler gravitational instantons were constructed by Kronheimer [29] using the technique of hyperkähler quotients (see [30]).

\(\mathop{\mathrm{ALF}}\)-\(A_k\) \((k\geq 1)\) hyperkähler gravitational instantons can be constructed using the Gibbons-Hawking ansatz, as given in Example 3. \(\mathop{\mathrm{ALF}}\)-\(D_k\) \((k\geq 0)\) gravitational instantons can be constructed by the twistor theory and generalized legendre tranform (see for example [31]–[36]) and the gluing method (see for example) [37]–[40]. \(\mathop{\mathrm{ALF}}\)-\(D_{0}\) gravitational instanton (up to scaling) is famously known as the Atiyah-Hitchin metric, which was originally constructed as the moduli space of centered charge two \(SU(2)\) monopoles on \(\mathbb{R}^3\) (see [41]). It has fundamental group \(\mathbb{Z}_2\), and its double cover is an \(\mathop{\mathrm{ALF}}\)-\(D_1\) gravitational instanton; general \(\mathop{\mathrm{ALF}}\)-\(D_1\) gravitational instantons were constructed by Dancer [42] using Nahm’s equation and hyperkähler quotients. \(\mathop{\mathrm{ALF}}\)-\(D_2\) case was first speculated by Page [43] as a periodic but non-stationary gravitational instanton, and this was rigorously constructed by Hitchin [31] using the twistor theory.

Examples of \(\mathop{\mathrm{ALG}}, \mathop{\mathrm{ALH}}, \mathop{\mathrm{ALG}}^*, \mathop{\mathrm{ALH}}^*\) hyperkähler gravitational instantons can be constructed using solutions to the complex Monge-Ampère equation by the work of Tian-Yau-Hein [26], [44]. These involve abstract PDE method and the metrics do not have explicit formulae in general. In [44] these were produced on the complement of anti-canonical fiber \(D\) in a rational elliptic surface \(X\). More specifically, in terms of the Kodaira classification of singular fibers, the metrics are

The strategy of the construction goes as follows. First it is easy to write down an explicit holomorphic volume form \(\Omega\) on \(X\setminus D\). Then one can write down a background Kähler metric \(\tilde{\omega}\) on \(X\setminus D\) which agrees with an explicit semi-flat hyperkähler metric near \(D\) and which has the asymptotics determined by the Kodaira type of \(D\). Finally one solves the complex Monge-Ampère equation 6 using \(\tilde{\omega}\) as a background metric, i.e., by writing \(\omega=\tilde{\omega}+\sqrt{-1}\partial\bar\partial\phi\) and solving the equation for \(\phi.\)

Special examples of these can be constructed using the gluing technique (see for example [37], [38]). Examples of \(\mathop{\mathrm{ALH}}^*\) hyperkähler gravitational instantons can also be constructed on the complement of an anti-canonical divisor \(D\) in a del Pezzo surface \(X\) (see [26], [27]), where the background metric is a constructed using the Calabi ansatz, noticing that a neighborhood of \(D\) is modeled on the total space of a positive line bundle over \(D\).

There are also many works in the literature which aim to construct hyperkähler gravitational instantons (and hyperkähler metrics in general dimensions too) using gauge theory. These include for example the moduli space of singular monopoles and the Hitchin moduli space of Higgs bundles, see for example [34], [35], [41], [45]–[64] and the references therein.

4.4 Classification Results A: Asymptotic Structure↩︎

From the above discussion, we see that there is a wide variety of asymptotic models for hyperkähler gravitational instantons. Nevertheless, a complete classification of their asymptotic ends is now available. Notice that a flat hyperkähler gravitational instanton must be a product \(\mathbb{R}^k\times \mathbb{T}^{4-k}\), so we may focus on the non-flat case.

In the non-collapsed case it follows from the classical result of Bando-Kasue-Nakajima [2] that \((M, g)\) is \(\mathop{\mathrm{ALE}}\). The proof of Theorem 4 is mainly concerned with the collapsed case. It relies on the interaction between the general collapsing theory of Cheeger-Fukaya-Gromov [7] in Riemannian geometry and the analysis of the hyperkähler equation 5 . Below we describe some key steps in the argument.

We always assume \((M, g)\) is collapsed. The first major step in the proof is to classify the asymptotic cones of \((M, g)\). Recall that an asymptotic cone \((\mathcal{C}, O)\) is a pointed Gromov-Hausdorff limit of \((M_j, g_j, p_j):=(M, \lambda_j^2g, p)\) for a sequence \(\lambda_j\rightarrow0\). A priori, \(\mathcal{C}\) may depend on the choice of the sequence \(\{\lambda_j\}\), and it may not even be a genuine metric cone. These were only known [65], [66] to hold under the assumption that \((M,g)\) has faster-than-quadratic curvature decay, using general Riemannian geometric arguments. The following result establishes these facts for hyperkähler gravitational instantons.

We give a sketch of the proof of Theorem 5 here. Let \((\mathcal{C}, O)\) be an asymptotic cone. By the quadratic curvature decay assumption, we know that locally away from \(O\), the convergence to \(\mathcal{C}\) is collapsing with uniformly bounded curvature. General Cheeger-Fukaya-Gromov theory implies that \(\mathcal{C}^\circ:=\mathcal{C}\setminus \{0\}\) is locally given by the quotient of a smooth Riemannian manifold by a compact Lie group and the collapsing is locally along an infranil fibration, possibly with singular fibers. In our special setting of 4-dimensional hyperkähler metrics one can show that \(\mathcal{C}^\circ\) is indeed a smooth Riemannian manifold, and the collapsing is locally along a nilpotent fibration. In particular, the fibers are either tori or three dimensional Heisenberg manifolds.

The hyperkähler structure on \(M\) gives a special affine structure on \(\mathcal{C}^\circ\) (the precise meaning depends on the dimension of \(\mathcal{C}^\circ\)). Take any \(q\in \mathcal{C}^\circ\) and a sequence \(q_j\in (M_j, g_j)\) converging to \(q\). One can find \(r>0\) such that the universal cover \(\widetilde{B}_j\) of \(B(q_j, r)\) is locally volume non-collapsing, and after passing to subsequences it converges to a 4-dimensional smooth limit manifold \(\widetilde{B}_\infty\) equipped with a hyperkähler metric. Moreover, the action of the deck transformation group \(G_j\) on \(\widetilde{B}_j\) converges to the tri-holomorphic isometric action of a Lie group \(G_\infty\) of positive rank, and \(B(q, r)\) is given by the quotient \(\widetilde{B}_\infty/G_\infty\). To understand the global geometry of \(\mathcal{C}\) one needs to analyze the singularity behavior at \(O\). For general Einstein equation, this is a challenging question in the volume collapsing situation. Notice that there is no a priori knowledge about the topology of \(\mathcal{C}\) around \(O\). The main observation in [23] is that, in our hyperkähler setting, the singularity at \(O\) can be controlled by the special affine structure in a punctured neighborhood of \(O\).

Let us focus on the case when the dimension of \(\mathcal{C}^\circ\) is 3. In this case \(G_\infty\) is generated by a single tri-holomorphic Killing field. As a converse to the Gibbons-Hawking ansatz, the limit Riemannian metric \(g_\infty\) on \(\mathcal{C}^\circ\) is locally of the form \[g_\infty=V(dx^2+dy^2+dz^2),\] where the coordinates \(x, y, z\) can be chosen to be globally well-defined up to a translation and a simultaneous rescaling by a constant \(C\neq 0\), and \(V>0\) is a harmonic function with respect to the flat metric \[g^\flat:=V^{-1}g_\infty.\]

The analysis essentially reduces to the study of a positive harmonic function on an (incomplete) flat 3-manifold. A crucial step is to show that, near \(O\), the metric \(g^\flat\) can be completed by adding exactly one point. This requires a growth estimate for \(V^{-1}\) near the singularity.

To obtain such an estimate, observe that if \(V^{-1}\) were to grow too quickly, then \(V\) would decay correspondingly fast and hence extend across the singularity as a harmonic function in the sense of metric measure spaces. The maximum principle would then imply that \(V\) is bounded near the singularity, leading to a contradiction.

If one knew that the metric completion of \(g^\flat\) were a flat 3-dimensional cone (i.e., \(\mathbb{R}^3\) or \(\mathbb{R}^3/\mathbb{Z}_2\)), then Bôcher’s theorem would imply that \(V\) has the form \[\label{eqn:harmonic32function32local32form} V = \frac{c}{r} + v,\tag{9}\] for some \(c > 0\) and a smooth harmonic function \(v\) extending across the singularity. In practice, however, such a removable singularity theorem is not available (see Question 4.13 in [23]).

Instead, [23] exploits the additional structure coming from the special affine coordinates \(\{x,y,z\}\), whose monodromy lies in \(\mathbb{R}^3 \rtimes \mathbb{Z}_2\), together with the classification of ends of flat manifolds by Eschenburg-Schroeder [67].

In our setting, one can further show that the constant \(c\) must vanish. This uses the Gauss-Bonnet-Chern theorem, the finite curvature energy of \((M,g)\), and a result on topological lower bounds for symplectic fillings of spherical 3-manifolds [68]. As a consequence, the singularity is a 3-dimensional smooth orbifold with a unique tangent cone given by flat \(\mathbb{R}^3\) or \(\mathbb{R}^3/\mathbb{Z}_2\).

A similar analysis applies when \(\mathcal{C}^\circ\) has dimension \(2\), where the singularity of \(\mathcal{C}\) is described by a singular special Kähler structure, and the possible monodromies can be classified.

Using the local analysis together with the completeness of \(\mathcal{C}\), one can further show that, in general, \(\mathcal{C}\) must be isometric to \(\mathbb{R}^k \times \mathbb{T}^{3-k}\) for \(k = 1,2,3\), or to a \(\mathbb{Z}_2\)-quotient thereof. By the rigidity of these spaces, it follows that there is a unique asymptotic cone, which is a metric cone belonging to the list in Theorem 5.

The second major step in proving Theorem 4 is to create local symmetries on the end of \(M\). The Cheeger-Fukaya-Gromov theory yields an \(\mathcal{N}\)-structure on the end. In our setting, this means, roughly speaking, that there exists a local free action of a nilpotent group \(\mathcal{N}\) on \(M \setminus K\), for some compact set \(K \subset M\), such that under rescaling to the asymptotic cone \(\mathcal{C}\), the \(\mathcal{N}\)-orbits correspond precisely to the collapsing fibers.

The following result reduces the analysis of the asymptotic behavior of \(g\) to the \(\mathcal{N}\)-invariant case.

The proof of Theorem 6 involves two main ingredients. First, it was shown in [7] that by averaging over the \(\mathcal{N}\)-orbits, one can construct an \(\mathcal{N}\)-invariant Riemannian metric \(\widetilde{g}\) satisfying 10 . However, in general, this averaging procedure destroys the special structures associated with \(g\).

Secondly, a quantitative implicit function theorem is established in [23] to further perturb \(\widetilde{g}\) to a new metric that is hyperkähler. This step relies crucially on Theorem 5 and is inspired by related gluing constructions for hyperkähler metrics on K3 surfaces (see [24], [56]).

To complete the proof of Theorem 4, it remains to classify \(\mathcal{N}\)-invariant hyperkähler ends. We illustrate this in the case where the asymptotic cone \(\mathcal{C}\) is \(\mathbb{R}^3\). In this situation, the \(\mathcal{N}\)-structure is given by an \(\mathbb{S}^1\)-fibration, and the metric \(\widetilde{g}\) is locally described by the Gibbons-Hawking ansatz.

As explained above, the orbit space \(\mathcal{Q}\) of the local \(\mathbb{S}^1\)-action inherits a special affine structure. In this setting, one can show that \(\mathcal{Q}\) may be identified with the complement of a compact set in \(\mathbb{R}^3\). One consequence of Theorem 6 is that the length of the \(\mathbb{S}^1\)-orbits, given by \(V^{-1/2}\), satisfies \[r^{-\delta} \leq V^{-1/2} \leq r^{\delta}\] for all \(\delta > 0\). It follows that \(V\) must have the form \[V = c + \frac{k+1}{2r} + O(r^{-2}),\] for some constants \(c \in \mathbb{R}\) and \(k \in \mathbb{Z}\). Consequently, \(\widetilde{g}\), and hence \(g\), is asymptotic to the \(\mathop{\mathrm{ALF}}\)-\(A_k\) model end.

We remark that a partial classification of gravitational instantons was obtained by Chen-Chen [69] under the additional assumption that the curvature decays at the rate \(O(r^{-2-\epsilon})\) for some \(\epsilon > 0\). In this case, the metric is locally asymptotic to a flat torus bundle, and only the four families \(\mathop{\mathrm{ALE}}\), \(\mathop{\mathrm{ALF}}\), \(\mathop{\mathrm{ALG}}\), and \(\mathop{\mathrm{ALH}}\) can occur.

The arguments in [69] are based on the approach of Minerbe [70], which studies the holonomy of short geodesic loops at infinity via ODE comparison. By contrast, the approach developed in [23] is quite different: it makes use of the full strength of Cheeger-Fukaya-Gromov theory. This has the advantage of, on the one hand, accommodating possible multi-scale collapsing phenomena at infinity, and on the other hand, highlighting the role of the hyperkähler equation as an elliptic PDE.

4.5 Classification Results B: Global Structure↩︎

To classify hyperkähler gravitational instantons, Theorem 4 reduces the problem to understanding all hyperkähler gravitational instantons \((M,g)\) with a prescribed asymptotic model. There are several aspects to consider:

• (Complex geometry) For a choice of compatible complex structure \(J\), describe the metric \(g\) in terms of the complex-geometric data of \((M,J)\).

• (Torelli problem) Characterize the metric \(g\) in terms of cohomological data (i.e., periods).

4.5.1 Complex geometry↩︎

To understand the complex geometry of gravitational instantons, we recall the longstanding compactification conjecture of Yau [71].

This conjecture is known to fail in general; a counterexample is given by the 4-dimensional complete hyperkähler metrics constructed in [72]. The following result confirms a version of Conjecture 7 in complex dimension \(2\).

Theorem 8 follows from Theorem 4, together with several works that treat each asymptotics separately, which we will describe below. Given Theorem 8, a natural question is to further classify the hyperkähler metric in terms of more refined complex-geometric data; this will also be discussed below.

In the \(\mathop{\mathrm{ALE}}\) case, by the classical work of Kronheimer [73], we know that for any choice of compatible complex structure \(J\), \((M,J)\) is given by the minimal resolution of a partial smoothing of a complex singularity \(\mathbb{C}^2/\Gamma \subset \mathbb{C}^3\) for some \(\Gamma \subset SU(2)\). These spaces can be compactified into a projective orbifold by adding a natural orbifold divisor at infinity. The metric \(g\) is, up to scaling, characterized by the choice of a Kähler class on \((M,J)\). More generally, it belongs to the class of asymptotically conical Ricci-flat Kähler metrics in arbitrary dimension; see [25], [74], [75].

In the \(\mathop{\mathrm{ALF}}\)-\(A_k\) case, it was proved by Minerbe [76] that all such metrics arise from the Gibbons-Hawking ansatz, and are precisely the multi-Taub-NUT metrics. For any choice of compatible complex structure \(J\), \((M,J)\) is given by the minimal resolution of a partial smoothing of an \(A_k\) singularity. In particular, one must have \(k \geq 1\).

In the general \(\mathop{\mathrm{ALF}}\) case, it was proved by Chen-Chen [77] that for any choice of \(J\), \((M,J)\) admits a compactification to a rational ruled surface. In the \(\mathop{\mathrm{ALF}}\)-\(D_k\) case, the hyperkähler metrics can be classified in terms of their twistor spaces, and are precisely those constructed by Cherkis-Hitchin-Ivanov-Kapustin-Lindström-Roček [31]–[36]. In particular, one must have \(k \geq 0\).

For the remaining four families, it is known that for some choice of compatible complex structure \(J\), \((M,J)\) can be compactified into a rational elliptic surface by adding an anti-canonical fiber. This is proved by Chen-Chen [69], [78] in the \(\mathop{\mathrm{ALG}}\) and \(\mathop{\mathrm{ALH}}\) cases, by [79] in the \(\mathop{\mathrm{ALG}}^*\) case, and by [27], [80], [81] in the \(\mathop{\mathrm{ALH}}^*\) case. In particular, in the \(\mathop{\mathrm{ALG}}\) case one must have \(\beta \in \{\tfrac12, \tfrac13, \tfrac23, \tfrac14, \tfrac34, \tfrac16, \tfrac56\}\); in the \(\mathop{\mathrm{ALG}}^*\) case one must have \(\mathop{\mathrm{ALG}}^*\text{-}I_k^*\) with \(k \in \{1,2,3,4\}\); and in the \(\mathop{\mathrm{ALH}}^*\) case one must have \(k \in \{1,\dots,9\}\).

Furthermore, the hyperkähler metrics can be obtained via a generalized Hein construction by incorporating additional parameters into [44], except possibly in the \(\mathop{\mathrm{ALH}}\) case (see the discussion in [78]. A common strategy in these compactification results is to construct a holomorphic function realizing \((M,J)\) as an elliptic fibration over \(\mathbb{C}\). Such a function \(f\) can be readily constructed on the asymptotic model for a suitable choice of \(J\). One then grafts \(f\) to \(M\) and uses weighted Fredholm theory to first produce a harmonic function \(\widetilde{f}\) asymptotic to \(f\), and subsequently prove that \(\widetilde{f}\) is in fact holomorphic.

In the \(\mathop{\mathrm{ALH}}^*\) case, it is proved in [27] (see also [82]) that, for a different choice of complex structure \(I\), the manifold \((M,I)\) admits a compactification to a weak del Pezzo surface, and the hyperkähler metric arises from the construction in [26].

This compactification is more delicate, since the natural \(I\)-holomorphic functions on the asymptotic model exhibit super-exponential growth at infinity. As a result, the strategy based on weighted Fredholm theory encounters difficulties, due to the fact that in the construction of [26], the decay rate of the metric is much slower than that of the complex structure. In [27], a new approach is introduced, based on the more robust \(L^2\)-estimates in several complex variables pioneered by Hörmander. This method also admits higher-dimensional generalizations, relating asymptotically Calabi metrics to weak Fano manifolds; see [27], [28].

4.6 Torelli type theorems↩︎

Torelli-type theorems for hyperkähler gravitational instantons are motivated by the corresponding results for hyperkähler metrics on the K3 manifold. Recall that the K3 manifold \(\mathcal{K}\) is, by definition, the underlying oriented smooth 4-manifold of a complex K3 surface. It is known that \(\mathcal{K}\) is simply connected, and the cup product on \(H^2(\mathcal{K}; \mathbb{Z})\) has signature \((3,19)\). Denote by \[\mathcal{D} = \mathop{\mathrm{O}}(3,19)/(\mathop{\mathrm{O}}(3)\times \mathop{\mathrm{O}}(19))\] the Grassmannian of positive 3-dimensional subspaces of \(H^2(\mathcal{K}; \mathbb{R})\). There is a natural action of the automorphism group \(\mathop{\mathrm{Aut}}(H^2(\mathcal{K}; \mathbb{Z}))\) on \(\mathcal{D}\).

Denote by \(\widehat{\mathcal{M}}\) the set of hyperkähler metrics on \(\mathcal{K}\) with unit diameter. Then we have a period map \[\mathcal{P}: \widehat{\mathcal{M}}\rightarrow \mathcal{D}; g\mapsto \mathbb{H}^+_g,\] where \(\mathbb{H}^+_g\subset H^2(\mathcal{K}, \mathbb{R})\) denotes the three dimensional subspace of \(H^2(\mathcal{K}; \mathbb{R})\) represented by the span of the hyperkähler triple \(\{\omega_1, \omega_2, \omega_3\}\). The Torelli theorem in this case states that

• (Injectivity) \(g_1\) and \(g_2\) are isometric, i.e., \(g_2=\varphi^*g_1\) for \(\varphi\in \text{Diff}(\mathcal{K})\), if and only if \(\mathcal{P}(g_1)=\gamma.\mathcal{P}(g_2)\) for some \(\gamma\in \mathop{\mathrm{Aut}}(H^2(\mathcal{K}; \mathbb{Z}))\).

• (Surjectivity) \(\mathcal{P}\) is surjective onto the open subset \(\mathcal{D}^\circ\subset\mathcal{D}\) of non-degenerate periods. Here a 3-dimensional positive subspace \(V\) of \(H^2(\mathcal{K}; \mathbb{R})\) is said to be non-degenerate if for any \(\delta\in H_2(\mathcal{K}; \mathbb{Z})\) with \(\delta.\delta=-2\), there is an element \(v\in V\) such that \(\int_{\delta}v\neq0\).

Torelli-type results for hyperkähler gravitational instantons have been established by Kronheimer [73] in the \(\mathop{\mathrm{ALE}}\) case, by Chen-Chen [77], [78] in the \(\mathop{\mathrm{ALF}}\) case, by Chen-Viaclovsky-Zhang [83] and Lee-Lin [84] in the \(\mathop{\mathrm{ALG}}\) and \(\mathop{\mathrm{ALG}}^*\) cases, and by Collins-Jacob-Lin [81] and Lee-Lin [84] in the \(\mathop{\mathrm{ALH}}^*\) case.

In the \(\mathop{\mathrm{ALE}}\) case, the underlying smooth manifold \(\mathcal{E}\) is the minimal resolution of \(\mathbb{C}^2/\Gamma\) for some \(\Gamma\). The period domain \(\mathcal{D}\) is given by the Grassmannian of all 3-dimensional subspaces of \(H^2(\mathcal{E}; \mathbb{R})\), together with a suitable non-degeneracy condition. In the \(\mathop{\mathrm{ALF}}\) case, the situation is similar, but there is an additional continuous parameter corresponding to the asymptotic length of the circle fibration.

In the remaining cases, the statements are more intricate, and we refer to the references above for precise formulations. The proofs of the Torelli theorems in these settings rely crucially on the compactification to rational elliptic surfaces discussed above, as well as on gluing constructions involving K3 manifolds, which we describe below.

4.7 Connections to compactfied moduli space of K3↩︎

Denote by \[\mathcal{M}:=\widehat{\mathcal{M}}/\text{Diff}(\mathcal{K})\] the space of isometry classes of unit diameter hyperkähler metrics on \(\mathcal{K}\). There is a natural Gromov-Hausdorff compactification \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\). There is a subset \(\overline{\mathcal{M}}^{\partial}\) consisting of non-collapsed Gromov-Hausdorff limits, and the complement \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\setminus \overline{\mathcal{M}}^{\partial}\) consists of collapsed Gromov-Hausdorff limits. Hyperkähler gravitational instantons are closely related to understanding \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\) and the singularity formation of hyperkähler metrics in \(\mathcal{M}\).

The gluing construction allows one to realize certain hyperkähler gravitational instantons as bubble limits, i.e., rescaled pointed Gromov-Hausdorff limits, of degenerating hyperkähler metrics on \(\mathcal{K}\). A prototypical example is provided by the metric Kummer construction.

One begins with a flat 4-torus \(\mathbb{T}^4\) and considers its \(\mathbb{Z}_2\)-quotient by the standard involution. The quotient \(\mathbb{T}^4/\mathbb{Z}_2\) is then a hyperkähler orbifold with 16 singularities, each locally modeled on the flat cone \(\mathbb{R}^4/\mathbb{Z}_2\). One can resolve these singularities by gluing in 16 copies of the Calabi-Eguchi-Hanson space, suitably rescaled so that the totally geodesic \((-2)\)-spheres have small area. A quantitative implicit function theorem is then applied to deform the resulting metric to a hyperkähler metric on \(\mathcal{K}\).

As the areas of these \((-2)\)-spheres tend to zero, the metrics converge in the Gromov-Hausdorff sense to the orbifold limit. In particular, \(\mathbb{T}^4/\mathbb{Z}_2\) lies in \(\overline{\mathcal{M}}^{\partial}\), and the Calabi-Eguchi-Hanson space appears as a bubble limit of hyperkähler metrics on \(\mathcal{K}\).

More generally, gluing constructions for \(\mathop{\mathrm{ALE}}\) hyperkähler gravitational instantons are well studied (see, for example, [85]–[87]). The upshot is that, given any hyperkähler orbifold, there exists a gluing construction that realizes it as an element of \(\overline{\mathcal{M}}^\partial\), and \(\mathop{\mathrm{ALE}}\) gravitational instantons arise as bubble limits. See also [88] for determining these bubbles limits in case of algebraic degenerations of polarized K3 surfaces, in terms of period mappings.

However, not all \(\mathop{\mathrm{ALE}}\) gravitational instantons can arise in this way, since \(\mathcal{K}\) has fixed topology. For example, if an \(\mathop{\mathrm{ALE}}\)-\(A_k\) gravitational instanton bubbles off, then \(k \leq 16\). It is therefore an interesting problem to determine which \(\mathop{\mathrm{ALE}}\) gravitational instantons can occur; equivalently, this amounts to determining which \(ADE\) singularities can appear on an orbifold \(K3\) surface.

There are extensive results on constructing examples of collapsing hyperkähler metrics on \(\mathcal{K}\). Since \(\mathbb{T}^4/\mathbb{Z}_2\) lies in \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\) and clearly admits collapse to lower-dimensional spaces of the form \(\mathbb{T}^k/\mathbb{Z}_2\) for \(k = 1,2,3\), it follows that these limits also belong to \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\). More systematic constructions require gluing non-\(\mathop{\mathrm{ALE}}\) hyperkähler gravitational instantons.

More general three-dimensional limits in \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\) were constructed by Foscolo [89], where the nontrivial bubble limits are given by \(\mathop{\mathrm{ALF}}\)-\(A_k\) or \(D_k\) gravitational instantons. Again, there are bounds on \(k\) arising from the fixed topology.

Two-dimensional limits in \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\) can be obtained by collapsing the fibers of an elliptic \(K3\) surface, beginning with the work of Gross-Wilson [90], which considers an elliptic \(K3\) surface with \(24\) singular fibers of Kodaira type \(I_1\). In this setting, away from the singular fibers the metrics are modeled on semi-flat metrics introduced by Greene-Shapere-Vafa-Yau [91], while the nontrivial bubble limits are given by the Taub-NUT metrics.

An interesting feature is the presence of an intermediate scale at which one observes an incomplete model metric, namely the Ooguri-Vafa metric. This metric can be constructed by applying the Gibbons-Hawking ansatz to the Green’s function on a domain in \(\mathbb{S}^1 \times \mathbb{R}^2\) (note that this space is non-parabolic, so the Green’s function cannot be globally positive).

The case of collapsing general elliptic \(K3\) surfaces was studied in [92] (see also earlier work [87], [93]), where the bubble limits around singular fibers with finite monodromy are shown to be \(\mathop{\mathrm{ALG}}\) gravitational instantons.

A one-dimensional limit \((X_\infty, d_\infty)\) in \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\) must be isometric to the unit interval \([0,1]\). However, general theory also endows \(X_\infty\) with additional structure, namely a renormalized limit measure \(\nu_\infty\). As shown in [94], there is a canonical affine structure on \(X_\infty\), and with respect to an affine coordinate \(z\) one can write the metric as \[g_\infty = c_1 L \, dz^2,\] and the renormalized limit measure as \[\nu_\infty = c_2 L \, dz,\] where \(L\) is a piecewise linear function of \(z\) and \(c_1, c_2 > 0\) are constants.

The case where \(L\) is constant was constructed by Chen-Chen [78] by gluing two \(\mathop{\mathrm{ALH}}\) hyperkähler gravitational instantons. From the complex-geometric point of view, this corresponds to doubling a rational elliptic surface to obtain an elliptic \(K3\) surface.

The case of non-constant \(L\) was constructed by Hein-Sun-Viaclovsky-Zhang [24] via a gluing construction of two \(\mathop{\mathrm{ALH}}^*\) gravitational instantons with a nontrivial neck region. This neck region can be viewed as a counterpart of the Ooguri-Vafa metric, and can be constructed by applying the Gibbons-Hawking ansatz to the Green’s function on a domain in \(\mathbb{T}^2 \times \mathbb{R}\).

From the complex-geometric point of view, it is shown in [95] that this models Type \(\mathrm{II}\) degenerations of polarized \(K3\) surfaces. In fact, a similar picture extends to the so-called small complex structure limits of higher-dimensional Calabi-Yau manifolds, although the construction of the neck region becomes significantly more involved. A similar neck region was subsequently used in [83] to show that \(\mathop{\mathrm{ALG}}^*\) gravitational instantons can arise as bubble limits of hyperkähler metrics on \(\mathcal{K}\).

As a converse to the above gluing constructions, the collapsing theory developed in [23] provides a description of families of hyperkähler metrics in \(\mathcal{M}\) that undergo collapse. First, the possible limits in \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}} \setminus \overline{\mathcal{M}}^\partial\) are classified: they are either a flat \(T^3/\mathbb{Z}_2\), a singular special Kähler metric on \(S^2\) (with integral monodromy, as observed in [96]), or the unit interval \([0,1]\).

Furthermore, after passing to a subsequence, the collapse over the regular set of the limit occurs along a nilpotent fiber bundle, and the metrics can be perturbed to hyperkähler metrics that are invariant under the nilpotent group action. The analysis is similar to that carried out in the proof of Theorem 4.

The collapsing theory developed in [23] also provides a framework for controlling the period map \(\mathcal{P}\) as one approaches the boundary of \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\). In particular, a strategy (see [23], Section 7) was proposed to study the relationship between \(\overline{\mathcal{M}}^{\mathop{\mathrm{GH}}}\) and certain Satake compactifications of \(\mathop{\mathrm{Aut}}(H^2(\mathcal{K}; \mathbb{Z})) \setminus \mathcal{D}\). It is also possible to show that any degenerating sequence in \(\mathcal{M}\) is given by a suitable generalization of the known gluing constructions, based on the analysis of bubble limits and classification of gravitational instantons (with possible orbifold singularities).

Foundational work in this direction was carried out by Odaka-Oshima [87], [97], [98], where precise conjectural relationships were formulated and partially established. The techniques and strategy of [23] were subsequently applied in [99] to give a simple proof of the surjectivity of the period map for \(K3\) surfaces, and later in [100] to complete the proof of a conjecture from [87]. It is plausible that these can also be applied to the study of moduli spaces of gravitational instantons (see Problem 25). For example, it is tempting to give a new proof of the Torelli theorem for hyperkähler gravitational instantons, as in [99].

4.8 General Type \(\mathop{\mathrm{I}}\) case↩︎

As mentioned earlier, a Type \(\mathrm{I}\) gravitational instanton \((M,g)\) need not be globally hyperkähler.

If \(\pi_1(M)\) is finite, which is the case when \((M,g)\) is non-collapsed, then the universal cover \((\widetilde{M}, \widetilde{g})\) is a hyperkähler gravitational instanton, and one may appeal to the classification theorem discussed above. The problem then reduces to studying free actions of finite groups on hyperkähler gravitational instantons. This is completely understood in the non-collapsed case by the results of Souvaina [101] and Wright [102]. In a suitable complex structure, such spaces arise as smoothings of \(\mathbb{C}^2/\Gamma\) for \(\Gamma \subset U(2)\).

When \(\pi_1(M)\) is infinite, the situation is more subtle. Its universal cover carries a complete hyperkähler metric, but can not have quadratic curvature decay, unless \((M, g)\) is flat. At present, all known such examples are flat, so given by a quotient of \(\mathbb{R}^4\). See Question 5.

5 Type \(\mathop{\mathrm{II}}\) gravitational instantons↩︎

In this section we discuss Type II gravitational instantons. Such a metric \((M, g)\) is locally Hermitian but not locally Kähler. In the following we will focus on the case when \((M, g)\) is Hermitian. As explained before, in general one may need to pass to a double cover.

As in Section 3.2 we denote by \(g_K=\lambda^{2/3}g\) the associated Kähler metric. A straightforward computation yields the equation \[\label{eqn:scalar32curvature32equation}6\mathop{\mathrm{Scal}}_{g_K}\Delta_{g_K}\mathop{\mathrm{Scal}}_{g_K}-12|\nabla_{g_K}\mathop{\mathrm{Scal}}_{g_K}|_{g_K}^2+\mathop{\mathrm{Scal}}_{g_K}^3=0.\tag{11}\] Since the curvature of \(g\) decays at infinity, we know that the function \[|\mathop{\mathrm{Scal}}_{g_K}|=\lambda^{1/3}=(2\sqrt{6}|\mathop{\mathrm{W}}^{+}|_g)^{1/3}\] tends to zero at infinity. A simple maximum principle argument then implies that \(\mathop{\mathrm{Scal}}_{g_{K}}\) must be everywhere positive, and thus \[\mathop{\mathrm{Scal}}_{g_K}=\lambda^{1/3}.\]

5.1 LeBrun-Tod ansatz↩︎

Given a Hermitian gravitational instanton \((M,g)\), one can locally perform a Kähler reduction to obtain an associated conformal Kähler metric \(g_K\) with respect to the vector field \(\mathbb{K}\). The corresponding moment map is given by \(\xi = \mathop{\mathrm{Scal}}_{g_K}\). Let \(x + i y\) be a local holomorphic coordinate on the complex quotient. Then, away from the zero set of \(\mathbb{K}\), the metric \(g_K\) can be locally expressed via the LeBrun-Tod ansatz [103], [104].

Denote by \(\eta\) the dual 1-form to \(\mathbb{K}\), and set \[W:=|\mathbb{K}|_{g_K}^{-2}.\] Then \(g_K\) can be written in the form \[\label{eq:lebrun-tod32kahler32metric} g_K=Wd\xi^2+W^{-1}\eta^2+We^v(dx^2+dy^2),\tag{12}\] with \[\label{eqn:linearized32Toda} (We^v)_{\xi\xi}+W_{xx}+W_{yy}=0.\tag{13}\] Here, \(\eta\) satisfies that \[d\eta=(We^v)_{\xi} dxdy+W_xdyd\xi+W_yd\xi dx,\] and the complex structure is given by \[J:d\xi\mapsto -W^{-1}\eta,\;dx\mapsto -dy.\] One can check (see [105]) that equation 13 together with the condition that \(g_K\) is conformal to a Ricci-flat metric are equivalent to that \(v\) satisfies a twisted \(SU(\infty)\) Toda equation \[\label{eq:twisted32toda32equation} (e^v)_{\xi\xi}+v_{xx}+v_{yy}=-\xi We^v,\tag{14}\] and \(W\) is determined by \[\label{eq:W32in32lebrun-tod} W=\frac{12}{\xi^3}-\frac{6v_\xi}{\xi^2}.\tag{15}\] By setting \[\varrho:=1/\xi,\;V:=\xi^2W,\;u:=v-4\log\xi,\] one can also rewrite the above ansatz as \[\label{eq:lebrun-tod32ricci32flat32metric} g=\varrho^2 g_K=V(d\varrho^2+e^u(dx^2+dy^2))+V^{-1}\eta^2,\tag{16}\] with \(u\) satisfying the usual \(SU(\infty)\) Toda equation \[\label{eq:toda32equation} (e^u)_{\varrho\varrho}+u_{xx}+u_{yy}=0,\tag{17}\] and \(V\) given by \[\label{eq:V32in32lebrun-tod} V=-12\varrho+6\varrho^2u_{\varrho}.\tag{18}\]

5.2 Asymptotic models↩︎

In this subsection, we introduce some Ricci-flat model ends which are Hermitian.

5.2.1 \(\mathop{\mathrm{ALE}}\) models↩︎

For a finite subgroup of \(\Gamma\subset U(2)\) acting freely on \(S^3\), we have the standard flat Kähler metric \(g_\Gamma\) on \((\mathbb{C}^2\setminus\{0\})/\Gamma\). Now the conformal metric \[\bar g_\Gamma:=\frac{1}{r^4}g_{\mathbb{C}^2/\Gamma}\] is also flat; indeed, under the isometry explicitly given by \[y\mapsto \frac{y}{|y|^2},\] \((\mathbb{C}^2\setminus\{0\})/\Gamma\) with the conformal metric \(\bar g_{\Gamma}\) can be isometrically identified with \((\mathbb{R}^4\setminus\{0\})/\Gamma\), but the complex structure becomes a non-standard one. In this way an \(\mathop{\mathrm{ALE}}\)-\(\Gamma\) model end is Hermitian.

5.2.2 \(\mathop{\mathrm{ALF}}\) models↩︎

Recall that the \(\mathop{\mathrm{ALF}}\)-\(A_k\) \((k\in \mathbb{Z})\) hyperkähler model end is given by the Gibbons-Hawking ansatz applied to \(\mathbb{R}^3\) with the harmonic function \[V=1+\frac{k+1}{2r}.\] It has an explicit expression as \[g=(1+\frac{k+1}{2r})(dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2))+(1+\frac{k+1}{2r})^{-1}(dt+\frac{(k+1)\cos\theta}{2}d\phi)^2, \quad r\in (0, \infty).\] Here \(0\leq\theta\leq\pi,0\leq\phi\leq2\pi\), and \(0\leq t\leq2\pi\).

If we reverse the orientation, then the metric \(g\) becomes Hermitian. Indeed, one can write down the associated complex structure explicitly as \[J:dr\mapsto(1+\frac{k+1}{2r})^{-1}(dt+\frac{(k+1)\cos\theta}{2}d\phi),\;d\theta\mapsto-\sin\theta d\phi,\] and the corresponding conformal extremal Kähler metric is given by \[g_K=(r+\frac{k+1}{2})^{-2}g.\] In this way an \(\mathop{\mathrm{ALF}}\)-\(A_{k}\) model end, with the reversed orientation, is Hermitian. A finite isometric quotient of this, which preserves the complex structure \(J\), is then an \(\mathop{\mathrm{ALF}}\) model end for Hermitian gravitational instantons.

Notice that the hyperkähler \(\mathop{\mathrm{ALF}}\)-\(D_k\) model ends, with the reversed orientation, are not Hermitian, since the \(\mathbb{Z}_2\) action does not preserve the complex structure \(J\). On the other hand, there are \(\mathop{\mathrm{ALF}}\) model ends which do not fall into the \(A_k\) family. As an example, one can consider the isometric quotient of the \(\mathop{\mathrm{ALF}}\)-\(A_k\) end by the action \(\phi\mapsto \phi+\pi,t\mapsto t+\pi\).

As a side remark, for the hyperkähler \(\mathop{\mathrm{ALF}}\)-\(A_k\) model end, there is another complex structure given by \[J':dr\mapsto-(1+\frac{k+1}{2r})^{-1}(dt+\frac{(k+1)\cos\theta}{2}d\phi),\;d\theta\mapsto-\sin\theta d\phi,\] such that the conformal metric \(r^{-2}g\) is scalar-flat Kähler.

5.2.3 \(\mathop{\mathrm{AF}}\) models↩︎

For \(\beta\in [0, 1)\), an \(\mathop{\mathrm{AF}}_\beta\) model end is the quotient of the form \[\mathbb{R}^4/\mathbb{Z}=(\mathbb{R}^3\times \mathbb{R}^1)/\mathbb{Z}=\mathbb{R}^1\times(\mathbb{R}^2\times\mathbb{R}^1)/\mathbb{Z},\] where the generator of \(\mathbb{Z}\) acts as a rotation by \(2\pi\beta\) on \(\mathbb{R}^2\) and a translation by \(1\) on \(\mathbb{R}^1\). When \(\beta=0\) this agrees with the \(\mathop{\mathrm{ALF}}\)-\(A_{-1}\) model. For \(\beta\neq 0\), the \(\mathop{\mathrm{AF_\beta}}\) model is locally hyperkähler, but not hyperkähler.

The \(\mathop{\mathrm{AF_\beta}}\) model ends exhibit interesting asymptotic behavior at infinity. When \(\beta = p/q\) for coprime integers \(p,q\), the space has an asymptotic cone given by \(\mathbb{C}_{1/q} \times \mathbb{R}\), where \(\mathbb{C}_{1/q}\) denotes the two-dimensional flat cone with cone angle \(2\pi/q\). When \(\beta \notin \mathbb{Q}\), the space instead has an asymptotic cone given by the two-dimensional flat half-plane \(\mathbb{H}\).

With respect to the decomposition \(\mathbb{R}^4 = \mathbb{R}^3 \times \mathbb{R}\), let \(\rho\) denote the radial coordinate on \(\mathbb{R}^3\) and \(t\) the coordinate on the \(\mathbb{R}\) factor. Let \(\mathbb{S}^2\) be the unit sphere in \(\mathbb{R}^3\). Then the flat metric on \(\mathbb{R}^4\) can be written as \[d\rho^2 + \rho^2 g_{\mathbb{S}^2} + dt^2.\] There exists a complex structure \(J\) such that \(J(d\rho) = -dt\), and whose restriction to \(\mathbb{S}^2\) agrees with the standard complex structure. With respect to \(J\), the conformally rescaled metric \(\rho^{-2} g_{\mathbb{R}^4}\) is Kähler. In particular, the \(\mathop{\mathrm{AF}}_\beta\) model end is Hermitian.

There are two non-standard asymptotic models that can be obtained from explicit solutions of 17 .

5.2.4 Special Kasner models↩︎

Let \(u = \log(-\varrho)\), and define on \((-\infty,-1) \times \mathbb{R}^3\) the special Kasner metric by \[\label{kasner} g_{\text{Kasner}} = -6\varrho \bigl(d\varrho^2 - \varrho (dx^2 + dy^2)\bigr) - \frac{1}{6\varrho} \, dt^2.\tag{19}\] A special Kasner model end is obtained by taking the quotient of this space by a lattice \(\Lambda \subset \mathbb{R}^3\). This metric is Hermitian with respect to both orientations.

This Type \(\mathrm{II}\) metric belongs to the Kasner family of Ricci-flat metrics, which are of the form \[\label{eqn:Kasner} dr^2 + r^{2p_1} d\theta_1^2 + r^{2p_2} d\theta_2^2 + r^{2p_3} d\theta_3^2,\tag{20}\] where \(p_1, p_2, p_3 \in \mathbb{R}\) satisfy \[p_1 + p_2 + p_3 = 1, \qquad p_1^2 + p_2^2 + p_3^2 = 1.\] The metric in 19 corresponds to the case \(p_1 = p_2 = \tfrac{2}{3}\) and \(p_3 = -\tfrac{1}{3}\).

It has quadratic volume growth and exactly quadratic curvature decay.

5.2.5 Anti-\(\mathop{\mathrm{ALH}}^*\) models↩︎

We take \(u = 0\) and define on \((-\infty,-1) \times \mathbb{R}^3\) the metric \[\label{Anti32ALH4232model} g_{\mathop{\mathrm{ALH}}^*} = -12\varrho \bigl(d\varrho^2 + dx^2 + dy^2\bigr) - \frac{1}{12\varrho}\bigl(dt - 12x\,dy\bigr)^2.\tag{21}\] This metric is isometric to the hyperkähler \(\mathop{\mathrm{ALH}}^*\) model defined earlier, but is Hermitian with respect to the opposite orientation. An anti-\(\mathop{\mathrm{ALH}}^*\) model is obtained by taking a quotient of \(g_{\mathop{\mathrm{ALH}}^*}\) by a cocompact lattice \(\Gamma\) in the Heisenberg group.

It has \(\frac{4}{3}\) volume growth rate and exactly quadratic curvature decay.

5.3 Examples↩︎

Some hyperkähler gravitational instantons can be viewed as Type \(\mathrm{II}\) metrics with respect to the reversed orientation. This includes the Calabi-Eguchi-Hanson metric on \(T^*S^2\) with reversed orientation, which is \(\mathop{\mathrm{ALE}}\), and the anti-Taub-NUT metric on \(\mathbb{R}^4\), which is \(\mathop{\mathrm{ALF}}\).

Page [107] discovered a “Taub-NUT instanton with horizon,” now commonly referred to as the Taub-Bolt metric. In this construction, the nut (i.e., the fixed point of the \(S^1\) action on \(\mathbb{R}^4\)) is replaced by a bolt, namely an \(S^2\). The underlying topology is \(\mathbb{CP}^2 \setminus \{\mathrm{pt}\}\). The metric is of cohomogeneity one and admits the explicit form \[ds^{2} = \frac{r^{2}-1}{r^{2}+1-\frac{5}{2}r}\,dr^{2} + \frac{r^{2}+1-\frac{5}{2}r}{r^{2}-1}\,(d\psi + 2\cos\theta\, d\phi)^{2} + (r^{2}-1)\,(d\theta^{2}+\sin^{2}\theta\, d\phi^{2}).\] The coordinates \((r,\theta)\) satisfy \(r \ge r_{0}\) and \(0 \le \theta \le \pi\), where \(r_{0}\) is the larger root of \(r^{2}+1-\frac{5}{2}r\). The coordinates \((\psi,\phi)\) satisfy the periodicity conditions \[(\psi,\phi)\sim(\psi+8\pi,\phi), \qquad (\psi,\phi)\sim(\psi+4\pi,\phi+2\pi).\] One can verify that the Taub-Bolt metric is of Type \(\mathrm{II}\) for both orientations. It is asymptotic to the Taub-NUT metric and is therefore \(\mathop{\mathrm{ALF}}\). With respect to the orientations \(dr \wedge d\theta \wedge d\phi \wedge d\psi\) and \(-dr \wedge d\theta \wedge d\phi \wedge d\psi\), the underlying complex surfaces can be identified with \(\mathcal{O}(-1)\) and \(\mathcal{O}(1)\), respectively. We refer to the former as the anti-Taub-Bolt gravitational instanton and the latter as the Taub-Bolt gravitational instanton.

The Kerr family of gravitational instantons arises from the Kerr solutions to the vacuum Einstein equations via a Wick rotation (i.e., replacing \(t \mapsto i t\)) [108]. Modulo scaling, this yields a one-parameter family of metrics on \(S^2 \times \mathbb{R}^2\), parametrized by \(\beta \in [0,1)\), each of which is asymptotic to the \(\mathop{\mathrm{AF}}_\beta\) model end. When \(\beta = 0\), one recovers the Schwarzschild metric.

An explicit formula is available (see, for example, [109]): \[g_a = \frac{\Delta}{\Sigma} (d\phi_1 + a \sin^2 \vartheta \, d\phi_2)^2 + \frac{\sin^2 \vartheta}{\Sigma} \bigl(a \, d\phi_1 - (s^2 - a^2) d\phi_2\bigr)^2 + \Sigma\left(\frac{1}{\Delta} ds^2 + d\vartheta^2\right), \label{eq:kerr95instanton}\tag{22}\] where \[\Sigma = s^2 - a^2 \cos^2 \vartheta, \qquad \Delta = s^2 - 2ms - a^2.\] Here \(a \in \mathbb{R}\), \(s \ge s_a\), and \(\vartheta \in [0,\pi]\), where \[s_a = m + \sqrt{m^2 + a^2}.\] We fix \(m = \tfrac{1}{2}\) to remove the scaling freedom.

One can verify that the Kerr family consists of Type \(\mathrm{II}\) metrics for both orientations.

A surprising discovery by Chen-Teo [110] is the existence of a one-parameter family (modulo scaling) of gravitational instantons on \((S^2 \times \mathbb{R}^2)\sharp \overline{\mathbb{CP}}^2\), parametrized by \(\beta \in (0,1)\). These metrics are constructed via the inverse scattering transform and admit an explicit expression in terms of rational functions. It was reconstructed using toric Kähler geometry by Biquard-Gauduchon [105], see the discussion in Section 5.4. There is also the study of Chen-Teo gravitational instanton from the twistor theory point of view [111].

It was shown by Aksteiner-Andersson [112] that the Chen-Teo metrics are of Type \(\mathrm{II}\) for exactly one choice of orientation. With this choice, they are referred to as the Chen-Teo gravitational instantons.

5.4 The results of Biquard-Gauduchon↩︎

One observes that all the examples above are toric, in the sense that there is a \(\mathbb{T}^2\)-action preserving both the metric and the complex structure. In [105], Biquard-Gauduchon provided a unified construction and interpretation of toric Type \(\mathrm{II}\) gravitational instantons with \(\mathop{\mathrm{ALF}}\) or \(\mathop{\mathrm{AF}}\) asymptotics. In this subsection, we briefly sketch their construction. Note that the terminology in [105] differs from ours: both \(\mathop{\mathrm{ALF}}\) and \(\mathop{\mathrm{AF}}\) asymptotics are referred to there as \(\mathop{\mathrm{ALF}}\).

As we have seen in Section 5.1, a Hermitian gravitational instanton \((M,g)\) already admits a Killing field \(\mathbb{K}\). Under an additional symmetry, by choosing the coordinate \(x+iy\) appropriately, one can further assume in the LeBrun-Tod ansatz 16 18 that the functions \(u\) and \(V\) are independent of the variable \(y\). As observed by Ward [113], a Bäcklund transformation then shows that local solutions to the reduced \(SU(\infty)\) Toda equation \[\label{eqn:reduced32Toda} (e^u)_{\varrho\varrho} + u_{xx} = 0\tag{23}\] correspond to axisymmetric harmonic functions \(U\) on \(\mathbb{R}^3\), i.e., \[\label{eqn:axisymmetric32harmonic} \frac{1}{\rho}(\rho U_\rho)_{\rho} + U_{zz} = 0,\tag{24}\] where \(\rho\) denotes the distance to the \(z\)-axis. The correspondence is given by \[\varrho = \tfrac{1}{2}\rho U_\rho, \qquad x = -\tfrac{1}{2}U_z, \qquad u = \log \rho^2.\] Conversely, any local solution \(u\) to 23 arises from such a construction. Note that 23 is nonlinear, whereas 24 is linear.

Therefore, in the toric setting, the LeBrun-Tod ansatz can be further reduced to axisymmetric harmonic functions. Recall that the ansatz 16 18 is obtained by performing Kähler reduction with respect to the Killing field \(\mathbb{K}\). To align with the notation of Biquard-Gauduchon [105], we instead consider the rescaled Killing field \[X := \sqrt{6k}\,\mathbb{K},\] for a nonzero constant \(k\). The only change in the formulas 16 18 is that 18 becomes \[V = \frac{1}{k}\bigl(-2\varrho + 6\varrho^2 u_{\varrho}\bigr).\]

By the above discussion, starting from an axisymmetric harmonic function \(U\), one can write the Ricci-flat metric \(g\) in the form \[\label{eq:harmonic32functioin32ansatz} g = e^{2\nu}(d\rho^2 + dz^2) + V\rho^2 dy^2 + V^{-1}(dt - F\,dy)^2,\tag{25}\] where \[\begin{align} e^{2\nu} &= \frac{1}{4} V\rho^2\bigl(U_{\rho z}^2 + U_{zz}^2\bigr),\\ V &= -\frac{1}{k}\left(\rho U_\rho + \frac{U_\rho^2 U_{zz}}{U_{\rho z}^2 + U_{zz}^2}\right),\\ F &= -\frac{1}{k}\left(-\frac{\rho U_\rho^2 U_{\rho z}}{U_{\rho z}^2 + U_{zz}^2} + \rho^2 U_z + 2H\right). \end{align}\] Here \(H\) is a harmonic conjugate of \(U\), and these expressions are valid on the domain where \(V > 0\) and \(U_{\rho z}^2 + U_{zz}^2 > 0\).

To understand the global geometry, one must impose appropriate behavior of the axisymmetric harmonic function \(U\) near the \(z\)-axis in \(\mathbb{R}^3\). This was analyzed by Biquard-Gauduchon [105] using toric Kähler geometry. More precisely, the associated Kähler metric \((M,g_K)\) is toric, and hence corresponds to a convex unbounded Delzant polytope \(\mathbf{\Delta} \subset \mathbb{R}^2\), together with a Kähler potential defined on its interior \(\mathbf{\Delta}^\circ\). The smoothness of the metric and the \(\mathop{\mathrm{ALF}}/\mathop{\mathrm{AF}}\) asymptotics are encoded in the behavior of the potential near \(\partial \mathbf{\Delta}\) and at infinity.

It was shown that \((\rho,z)\) defines a diffeomorphism from \(\mathbf{\Delta}\) onto the right half-plane \(\mathbb{H}\), which can be viewed as the quotient of \(\mathbb{R}^3\) by rotations about the \(z\)-axis. Thus, the axisymmetric harmonic function \(U\) is globally defined on \(\mathbb{H}^\circ\). By relating the above ansatz to the Kähler potential, it was proved in [105] that the behavior of \(U\) near the \(z\)-axis is given as follows:

Biquard-Gauduchon [105] then introduced the basic axisymmetric harmonic function \[U_0(\rho,z) = 2r - z \log \frac{r+z}{r-z},\] which corresponds geometrically to the flat metric and has asymptotic behavior near the \(z\)-axis determined by the piecewise linear function \[f(z) = |z|.\] The behavior of \(U\) at infinity can similarly be shown to satisfy \[U - U_0 = o(r).\] By superposing such basic solutions, it follows that for any convex piecewise linear function \[f(z) = A + \sum_{j=1}^n a_j |z - z_j|,\] with \(A > 0\), \(z_1 < \cdots < z_n\), and slope increasing from \(-1\) to \(1\) (so that each \(a_j \in (0,1)\) and \(\sum a_j = 1\)), one can associate a canonical axisymmetric harmonic function \[\label{eq:axisymmetric32harmonic32U} U = A \log \rho^2 + \sum_{j=1}^n a_j U_0(\rho, z - z_j).\tag{26}\] Using the ansatz 25 and choosing \(k = 2A\), which ensures that \(|X|_g^2 = V^{-1} \to 1\) at infinity, one obtains a Hermitian Ricci-flat metric defined on \(\mathbb{H}^\circ \times \mathbb{R}^2\), admitting an \(\mathbb{R}^2\)-translation symmetry. Taking the quotient of \(\mathbb{R}^2\) by a suitable lattice \(\Lambda\), the metric completion of \((\mathbb{H}^\circ \times (\mathbb{R}^2/\Lambda), g)\) yields a smooth \(4\)-manifold.

Each interval \((z_j, z_{j+1})\) along the \(z\)-axis is identified, via the diffeomorphism \((\rho,z)\), with an edge of the polytope \(\mathbf{\Delta}\). Let \(f_j'\) denote the slope of \(f\) on \((z_j, z_{j+1})\). By standard toric geometry, each edge of \(\mathbf{\Delta}\) determines a Killing field, which in our setting is a linear combination of \(\partial_y\) and \(\partial_t\) and vanishes along that edge. A direct computation shows that these vectors are explicitly given by \[\label{eq:BG32normailzed32rod32vector} \overline{\mathfrak v}_j = \begin{cases} \overline{\mathfrak{v}}_j = f_j'(\partial_{y} + F_j \partial_t), & \text{if } f_j' \neq 0, \\[0.5em] \overline{\mathfrak{v}}_j = \dfrac{1}{A} f_j^2 \partial_t, & \text{if } f_j' = 0. \end{cases}\tag{27}\] Here the constants \(F_j\) are determined by the following conditions:

• \[F_0 = -\frac{1}{A}\Bigl(A + \sum_{j=1}^n a_j z_j\Bigr)^2 + \frac{1}{A} \sum_{j=1}^n a_j z_j^2;\]

• if \(f_j' \neq 0\), then \[F_j - F_{j-1} = \frac{1}{A} f_j^2\left(\frac{1}{f_j'} - \frac{1}{f_{j-1}'}\right);\]

• if \(f_j' = 0\), then \[F_{j+1} - F_{j-1} = \frac{1}{A}\left(f_j^2\left(\frac{1}{f_{j+1}'} - \frac{1}{f_{j-1}'}\right) - 2(z_{j+1} - z_j) f_j\right).\]

In particular, these relations imply \[F_n = \frac{1}{A}\Bigl(A - \sum_{j=1}^n a_j z_j\Bigr)^2 - \frac{1}{A} \sum_{j=1}^n a_j z_j^2.\]

Locally, near each edge of \(\mathbf{\Delta}\), the Ricci-flat metric extends smoothly to a \(4\)-manifold if and only if the vector \(\overline{\mathfrak v}_j\) is primitive in the lattice \(\Lambda\), and any two adjacent vectors \(\overline{\mathfrak v}_j, \overline{\mathfrak v}_{j+1}\) generate \(\Lambda\).

Based on these strong constraints, [105] obtained a classification of toric Hermitian \(\mathop{\mathrm{ALF}}/\mathop{\mathrm{AF}}\) gravitational instantons:

Moreover, [105] gives an explicit description of all these metrics in terms of their associated toric polytopes. From left to right, the corresponding (unbounded) polytopes are those of the anti-Taub-NUT, Kerr, Chen-Teo, anti-Taub-Bolt, and Taub-Bolt gravitational instantons.

There is also related work by [114], which provides an alternative proof of the above toric classification under weaker assumptions at infinity.

There is a striking contrast between this result and the situation for Hermitian Einstein metrics with positive Einstein constant. For a Type \(\mathrm{II}\) Einstein metric with positive Einstein constant, up to a finite cover, the metric is either Kähler or Hermitian. It was proved by LeBrun [115] that all Hermitian positive Einstein \(4\)-manifolds are, up to scaling, given by the Page metric on \(\mathbb{CP}^2 \sharp \overline{\mathbb{CP}}^2\) [116], and the Chen-LeBrun-Weber metric on \(\mathbb{CP}^2 \sharp 2\overline{\mathbb{CP}}^2\) [117]. An observation of LeBrun [14] further shows that all such metrics are toric.

Motivated by this comparison with the positive Einstein case, it was conjectured in [112] that all \(\mathop{\mathrm{ALF}}\) and \(\mathop{\mathrm{AF}}\) Hermitian gravitational instantons are likewise toric. In particular, this would imply that the list in Theorem 12 exhausts all \(\mathop{\mathrm{ALF}}\) and \(\mathop{\mathrm{AF}}\) Hermitian gravitational instantons. This conjecture was proved in [118]; see also the discussion in Section 5.6.

5.5 Classification Results A: Asymptotic Structure↩︎

As in the hyperkähler case, we now have a complete classification of the asymptotic ends of Hermitian gravitational instantons.

In the remainder of this subsection, we sketch a proof of the above theorem. In the non-collapsed case, \((M,g)\) must be \(\mathop{\mathrm{ALE}}\) by [2], so we henceforth assume that \((M,g)\) is collapsed at infinity. Recall that there is a canonical Killing field \(\mathbb{K}\).

Claim. \(\mathbb{K}\) has no zeroes outside a compact set.

To see this, we first note that the moment map \(\xi := \mathop{\mathrm{Scal}}_{g_K}\) is a Morse-Bott function on \(M\), whose critical submanifolds have indices \(0\), \(2\), or \(4\). Since \(\mathop{\mathrm{Scal}}_{g_K}\) is nowhere vanishing, it follows from 11 and the maximum principle that \(\xi\) cannot admit a local minimum. Moreover, it has exactly one maximal critical submanifold, while all other critical submanifolds consist of isolated points of index \(2\).

It therefore suffices to show that the index \(2\) critical points remain in a compact set. Suppose, for contradiction, that this is not the case. Then there exists a sequence of index \(2\) critical points \(q_j\) tending to infinity. Let \(a_j > 0\) and \(-b_j < 0\) denote the associated weights of \(\mathbb{K}\) at \(q_j\). Fix a base point \(p\), and consider the rescaled spaces \((M, g_j, p) := (M, r_j^{-2} g, p)\), where \(r_j = \mathop{\mathrm{dist}}_g(p, q_j)\).

For sufficiently small \(\delta\), the balls \(B_j := B(q_j, \delta) \subset (M, g_j)\) have bounded curvature and non-collapsed universal covers \((\widetilde{B}_j, \widetilde{q}_j)\). Passing to a subsequence, we may assume that \((B_j, q_j)\) converges to a limit ball \(B_\infty := B(q, \delta)\), and that the universal covers \((\widetilde{B}_j, \widetilde{q}_j)\) converge to a limit \((\widetilde{B}_\infty, \widetilde{q})\). The actions of the deck transformation groups \(G_j\) on \(\widetilde{B}_j\) converge to an isometric action of a Lie group \(G_\infty\) of positive rank on \(\widetilde{B}_\infty\), and \[B_\infty = \widetilde{B}_\infty / G_\infty.\]

Since \(\mathbb{K}\) vanishes at \(q_j\), after suitable normalization we may assume that it converges to a nonzero Killing field \(\mathbb{K}_\infty\) on \(\widetilde{B}_\infty\) that vanishes at \(\widetilde{q}\). It then follows that \(\mathbb{K}_\infty\) must vanish along the orbit \(G_\infty \cdot \widetilde{q}\). Consequently, we must have either \(a_j / b_j \to 0\) or \(a_j / b_j \to \infty\). In either case, the \(\mathbb{R}\)-action generated by \(\mathbb{K}\) limits to an effective isometric \(\mathbb{R}^2\)-action on \(\widetilde{B}_\infty\) that fixes the orbit \(G_\infty \cdot \widetilde{q}\), which is impossible.

We now divide into 2 cases

• \(\mathbb{K}\) induces an \(S^1\)-action;

• \(\mathbb{K}\) induces a higher rank \(\mathbb{T}^k\)-action.

5.5.1 \(\mathbb{K}\) induces an \(S^1\)-action↩︎

In this case, we may perform the Kähler reduction. For simplicity, we assume that the \(S^1\)-action is free; in general, one must work in the orbifold setting (see [118]). Then each level set \(\{\xi = \epsilon\}\) is an \(S^1\)-bundle over a compact Riemann surface \(\Sigma\).

In 16 , the metric \[\underline g := e^u (dx^2 + dy^2)\] can be viewed globally as a family of metrics on \(\Sigma\) parametrized by \(\varrho\). Note that at infinity \(\xi = \mathop{\mathrm{Scal}}_{g_K} \to 0\), and hence \(\varrho \to \infty\).

Integrating the \(SU(\infty)\) Toda equation \[(e^u)_{\varrho\varrho} + u_{xx} + u_{yy} = 0\] over \(\Sigma\), and applying the Gauss-Bonnet formula, we obtain \[\label{eq:e94u32volume32of32sigma} \int_{\Sigma} e^u \, dx\,dy = 2\pi \chi(\Sigma)\varrho^2 + a\varrho + b,\tag{28}\] for some constants \(a\) and \(b\). Using 18 , one also finds \[\label{eq:Ve94u32volume32of32sigma} \int_{\Sigma} V e^u \, dx\,dy = -6a\varrho^2 - 12b\varrho.\tag{29}\] Suppose that the period of the vector field \(\mathbb{K}\) is \(\mathfrak{p}\). Then we obtain the volume formula \[\label{eq:volume32of32rho32annuli} \mathop{\mathrm{Vol}}(D_0 \le \varrho \le D_1) = -2\mathfrak{p} a (D_1^3 - D_0^3) - 6\mathfrak{p} b (D_1^2 - D_0^2).\tag{30}\] Since \(\varrho > 0\), it follows from 28 –30 that \(\chi(\Sigma) > 0\), and hence \(\Sigma\) is conformally equivalent to \(S^2\).

We now consider an asymptotic cone \((\mathcal{C}, O)\), defined as the pointed Gromov-Hausdorff limit of the rescaled spaces \[(M_j, g_j, p) := (M, r_j^{-2} g, p),\] for a sequence \(r_j \to \infty\). As before, for any \(q \in \mathcal{C} \setminus \{O\}\), a ball \(B_\infty\) around \(q\) is of the form \[B_\infty = \widetilde{B}_\infty / G_\infty,\] where \((\widetilde{B}_\infty, \tilde{g}_\infty)\) arises as the limit of the local universal covers. Our goal is to pass the conformally Kähler structure to the limit \(\widetilde{B}_\infty\). This is more delicate than in the hyperkähler case, since Type \(\mathrm{II}\) metrics may degenerate to Type \(\mathrm{I}\) metrics in the limit.

On \(M_j\), we define \[\xi_j := (\xi(q_j))^{-1}\xi, \qquad \varrho_j := \xi_j^{-1}.\] By the scalar curvature formula under conformal change, we obtain \[\label{eq:scaled32down32lambda95j} 6\Delta_{g_j}\xi_j + r_j^2 \xi(q_j)^3 \xi_j^4 = 0.\tag{31}\] Noting that \(\xi(q_j) \le C r_j^{-2/3}\), the Cheng-Yau gradient estimate yields a Harnack inequality for \(\xi_j\). In particular, after appropriate normalization and passing to a subsequence, there exists a smooth positive limit function \(\xi_\infty\) on \(\widetilde{B}_\infty\) such that the conformal metric \(\xi_\infty^2 \widetilde{g}_\infty\) is Kähler.

Moreover, \(\xi_\infty\) descends to \(B_\infty\) and extends to a globally defined function on \(\mathcal{C} \setminus \{O\}\). From volume comparison and the formula 30 , it follows that \(\xi_\infty\) is non-constant, with \(\xi_\infty \to 0\) at \(O\) and \(\xi_\infty \to \infty\) at infinity.

After pulling back to each local cover \(\widetilde{B}_j\), we may assume that the vector fields \(\mathbb{K}_j := -J \nabla_{g_j} \xi_j^{-1}\) and functions \(V_j := |\mathbb{K}_j|_{g_j}^{-2}\) converge, respectively, to \[\mathbb{K}_\infty := -J \nabla_{g_\infty} \xi_\infty^{-1}, \qquad V_\infty := |\mathbb{K}_\infty|_{g_\infty}^{-2}\] on \(\widetilde{B}_\infty\). We may rewrite 16 as \[\label{eqn5-30}g_j=V_j(d\varrho_j^2+e^{u_j}(dx^2+dy^2))+r_j^{-4}\xi^{-2}(q_j)V_j^{-1}\eta^2,\tag{32}\] where \[V_j=\xi^{-3}(q_j)r_j^{-2}(-12\varrho_j+6\varrho_j^2\partial_{\varrho_j}u_j)\] and \[u_j=u+\log\xi^2(q_j).\] We may also assume that \(u_j\) converges to a smooth limit \(u_\infty\). Denote \[h_{\Sigma,j}:=e^{u_j}(dx^2+dy^2)\] as the metric on \(\Sigma\) determined by \(\varrho_j=1\). Using the above local convergence, one sees that for sufficiently large \(j\), the metrics \(h_{\Sigma,j}\) have uniformly bounded curvature. Furthermore, since \(\chi(\Sigma) > 0\), the sequence \((\Sigma, h_{\Sigma,j})\) cannot collapse everywhere. Without loss of generality, we may choose \(q_j\) such that its image \(q_{\Sigma,j} \in (\Sigma, h_{\Sigma,j})\) has injectivity radius bounded from below. It then follows that, after passing to a subsequence, the balls \(B_\epsilon(q_{_{\Sigma,j}})\subset (\Sigma,h_{\Sigma,j})\) converges smoothly to a non-collapsed limit \((B_{\epsilon}(q_{\Sigma,\infty}),h_{\Sigma,\infty})\).

After passing to a subsequence, we may assume that one of the following occurs:

• \(\xi_j^3 r_j^2\) converges to a positive constant. Geometrically, this means that the Ricci-flat metric \((\widetilde{B}_\infty, \widetilde{g}_\infty)\) is of Type \(\mathop{\mathrm{II}}\).

• \(\xi_j^3 r_j^2\) converges to zero. Geometrically, this means that \((\widetilde{B}_\infty, \widetilde{g}_\infty)\) is of Type \(\mathop{\mathrm{I}}\).

However, one can show that the first situation cannot occur, by considering the volume of the metric annuli \(A_{1/2,2}\) in \((M,g_j,p)\) and using the choice of \(q_j\), 30 , and 32 .

In the second case, the Kähler metric \((\widetilde{B}_\infty,\lambda^{2/3}_\infty\widetilde{g}_\infty)\) has vanishing scalar curvature, and we have \[-12\varrho_{\infty}+6\varrho_\infty^2\partial_{\varrho_\infty}u_\infty\equiv0.\] It then follows that \(h_{\Sigma,\infty}\) has constant curvature one. This implies that \(h_{\Sigma_j}\) has bounded diameter and converges to the round metric on \(S^2\). The asymptotic cone \(\mathcal{C}\) is given by the metric \[g_\infty=V_\infty(d\varrho_\infty^2+\varrho_\infty^2g_{S^2}).\] on \(\mathbb{R}^3\).

A further argument based on LeBrun’s ansatz for scalar-flat Kähler metrics ([103]) shows that \(V_\infty\) is harmonic with respect to the flat metric \[d\varrho_\infty^2 + \varrho_\infty^2 g_{S^2}.\] It then follows from 28 –29 , together with the non-maximal volume growth assumption, that \(V_\infty\) must be constant. This implies that \(\mathcal{C}\) is itself isometric to \(\mathbb{R}^3\).

In general, when the \(S^1\)-action is not necessarily free, a similar argument shows that \(\mathcal{C}\) must be a quotient of the form \(\mathbb{R}^3 / \mathbb{Z}_k\), where \(\mathbb{Z}_k\) acts on an \(\mathbb{R}^2\)-factor. In particular, the asymptotic cone is unique.

From this, together with a PDE analysis of 17 –18 , one obtains an asymptotic expansion for \(u\) and \(V\). This implies that \((M, g)\) is up to scaling either \(\mathop{\mathrm{ALF}}\) or \(\mathop{\mathrm{AF}}_\beta\) for some \(\beta \in \mathbb{Q}\).

5.5.2 \(\mathbb{K}\) induces a higher rank \(\mathbb{T}^k\)-action↩︎

If \(k \geq 3\), then one can explicitly write down solutions to the LeBrun-Tod ansatz and verify that such solutions cannot arise when \(M\) is complete. Thus, we may assume \(k = 2\). In this case, the unique maximal submanifold of \(\xi = \mathop{\mathrm{Scal}}_{g_K}\) must be a fixed point. In particular, this implies that \(M\) is simply connected.

Since the \(T^2\)-action has a fixed point, it follows that there exists a global moment map \[(x_1, x_2) : M \to \mathbb{R}^2.\] In this setting, although a global Kähler reduction with respect to the vector field \(\mathbb{K}\) is not available, the local analysis still applies and allows one to study the structure of an asymptotic cone \((\mathcal{C}, O)\). As before, away from \(O\), the space \(\mathcal{C}\) is locally given as a quotient of the form \((\widetilde{B}_\infty, \widetilde{g}_\infty)/G_\infty\). One can show that \(\widetilde{g}_\infty\) must be of Type \(\mathop{\mathrm{I}}\). Moreover, \(\mathcal{C} \setminus \{O\}\) is a two-dimensional manifold with boundary, equipped with a Riemannian metric \(g_\infty\), such that the conformally related metric \(V_\infty^{-1} g_\infty\) is flat.

Using arguments similar to those in [119], one concludes that the flat space is a half-plane \(\mathbb{H}\). Since \(V_\infty\) is harmonic, it must be constant. Therefore, \(\mathcal{C}\) itself is the flat half-plane \(\mathbb{H}\), and the local limit \((\widetilde{B}_\infty, \widetilde{g}_\infty)\) is also flat.

The above discussion implies that \((M,g)\) is asymptotically flat, i.e., \(|\mathop{\mathrm{Riem}}| = o(r^{-2})\). By the resolution of a conjecture of Petrunin-Tuschmann in [119], it follows that the end of \(M\) cannot be simply connected, and hence must be a finite quotient of \(S^2 \times S^1 \times [1,\infty)\). Let \(K \subset M\) be a compact set, and consider the universal cover \(\widetilde{M}\) of \(M \setminus K\). One can then perform a global Kähler reduction on \(\widetilde{M}\), which leads to the conclusion that \((M,g)\) is \(\mathop{\mathrm{AF}}_\beta\) for some \(\beta \notin \mathbb{Q}\).

5.6 Classification Results B: Global Structure↩︎

The following result provides a complete classification of Hermitian gravitational instantons in the collapsed case, and a partial classification in the non-collapsed case.

In the remainder of this subsection, we sketch a proof of the above theorem. We begin with the \(\mathop{\mathrm{ALF}}\) and \(\mathop{\mathrm{AF}}\) cases. The strategy is to show that such a gravitational instanton must be toric, and then to appeal to Theorem 12.

Let \((g_K, J)\) denote the conformally Kähler metric. If the Killing field \(\mathbb{K}\) generates a \(\mathbb{T}^2\)-action, then the metric is already toric. Thus, we may assume that \(\mathbb{K}\) generates only an \(S^1\)-action. In this case, \((M,g)\) is either \(\mathop{\mathrm{ALF}}\) or \(\mathop{\mathrm{AF}}_\beta\) with \(\beta \in \mathbb{Q}\). By the asymptotic classification (Theorem 13), the Kähler metric \(g_K\) exhibits cusp behavior at infinity, in the sense that locally it is asymptotic to a Poincaré cusp bundle over a round \(\mathbb{S}^2\).

There are two main steps:

1. Show that \((M, J)\) admits an effective holomorphic \((\mathbb{C}^*)^2\)-action extending the \(S^1\)-action generated by \(\mathbb{K}\);

2. Prove a Calabi-type theorem, producing a compact torus \(\mathbb{T}^2 \subset (\mathbb{C}^*)^2\) that preserves \(g_K\).

Step (1) follows from a compactification result. Denote by \(h\) the Hermitian metric on \(K_M^{-1}\) induced by \(g_K\). An observation of LeBrun [14] implies that the Hermitian metric \(\mathop{\mathrm{Scal}}_{g_K}^{-2} h\) on \(K_M^{-1}\) has positive curvature.

One can naturally compactify \((M, J)\) to a two-dimensional complex orbifold \(\overline{M}\) by adding a point to each \(\mathbb{C}^*\)-orbit at infinity. The complement \(\overline{M} \setminus M\) is a divisor \(D = \mathbb{CP}^1\). The \(\mathbb{C}^*\)-action induced by \(\mathbb{K}\) extends to \(\overline{M}\) and fixes \(D\).

By studying the positive curvature \(2\)-form of the Hermitian metric \(\mathop{\mathrm{Scal}}_{g_K}^{-2} h\), one shows that the line bundle \(-(K_{\overline{M}} + D)\) is positive on \(\overline{M}\). Hence \((\overline{M}, D)\) is a log del Pezzo surface. By performing suitable blow-ups, one can further show that \(\overline{M}\) is birational to a ruled surface over \(\mathbb{CP}^1\), and hence is rational. Taking a minimal resolution of \(\overline{M}\) with the lifted holomorphic \(\mathbb{C}^*\)-action, and then passing to a minimal model \(M'\), we obtain a surface that still admits a holomorphic \(\mathbb{C}^*\)-action. It follows that \(M'\) must be either \(\mathbb{CP}^2\) or a Hirzebruch surface. Under these strong constraints, one concludes that \(\overline{M}\) admits a holomorphic toric \((\mathbb{C}^*)^2\)-action containing the \(S^1\)-action generated by \(\mathbb{K}\).

For Step (2), recall that for a compact extremal Kähler metric, Calabi [3] proved that the isometry group contains a maximal compact subgroup of the identity component of the holomorphic automorphism group. In our setting, one can extend this result using the fact that \(g_K\) has Poincaré-type behavior at infinity.

Finally, we consider the \(\mathop{\mathrm{ALE}}\) case. In contrast to the \(\mathop{\mathrm{ALF}}\) and \(\mathop{\mathrm{AF}}\) cases, the Kähler metric \(g_K\) is incomplete on \(M\), and its completion \(\overline{M}\) is obtained by adding an orbifold point at infinity with orbifold group \(\Gamma \subset U(2)\). The compactified surface \(\overline{M}\) is still a log del Pezzo surface, but in this case one cannot conclude that \(\overline{M}\) is toric. The metric \(g_K\) extends to an orbifold Kähler metric \(\overline{g}_K\) on \(\overline{M}\), which is Bach-flat. Its scalar curvature \(s_{\overline{g}_K}\) vanishes at the orbifold point and is positive elsewhere.

In general, for \(\Gamma \subset U(2)\), there exist infinitely many log del Pezzo surfaces with a single orbifold point admitting a holomorphic \(\mathbb{C}^*\)-action fixing that point. However, under the assumption \(\Gamma \subset SU(2)\), there are only finitely many such orbifolds. One can compute the minimum value of \(\mathop{\mathrm{Scal}}_{g_K}\) in terms of the Kähler class and the holomorphic extremal vector field, as essentially carried out by LeBrun-Simanca [122].

In our setting, one can explicitly verify that the minimum of \(\mathop{\mathrm{Scal}}_{g_K}\) is nonzero, except in the case of the orbifold obtained from \(\mathbb{P}(\mathcal{O}(2)\oplus \mathcal{O})\) by contracting the \((-2)\)-curve, with the \(\mathbb{C}^*\)-action given by rotation along the fibers. This exceptional case corresponds to the anti-Calabi-Eguchi-Hanson gravitational instanton.

6 Type \(\mathop{\mathrm{III}}\) gravitational instantons↩︎

The study of Type \(\mathop{\mathrm{III}}\) gravitational instantons is largely an open area. On the one hand, there are not many known topological or geometric constraints; on the other hand, it is also challenging to solve the PDE system for general Einstein metrics. To date there are only two classes of examples of Type \(\mathop{\mathrm{III}}\) (for both orientations) gravitational instantons are known, by [123] and [106]. In this section we will explain these results.

6.1 Dimension reduction↩︎

Let \((M,g)\) be an oriented 4-dimensional Riemannian manifold. Suppose \(X\) is a Killing vector field on \(M\) and denote by \(\eta=X^{\flat}\) its dual 1-form. Then one can check that \[d^*(\eta\wedge d\eta)=2(\mathop{\mathrm{Ricci}}_g.\eta) \wedge \eta.\] In particular, if \(g\) is Ricci-flat, then locally one can find an twist potential \(\mathcal{E}\), which is unique up to a constant addition, such that \[d\mathcal{E}=*(\eta\wedge d\eta).\] The twist 1-form \(d\mathcal{E}\) measures the non-integrability of the orthogonal distribution to \(X\). Clearly \[\mathcal{L}_{X}\mathcal{E}=\langle \eta, d\mathcal{E}\rangle=*(\eta\wedge *d\mathcal{E})=0.\] The twist was first defined in the context of general relativity for a stationary vacuum spacetime, i.e, a Lorentzian Einstein metric with a time-like Killing field.

Now suppose we have two commuting Killing fields \(X_1, X_2\), with dual 1-forms \(\eta_1, \eta_2\) respectively. Then we get two twist potentials \(\mathcal{E}_1, \mathcal{E}_2\) respectively. Denote \[\Theta_i:=*(\eta_1\wedge\eta_2\wedge d\eta_i),\;i=1, 2.\] Then \[d\Theta_1=d\langle\eta_2, d\mathcal{E}_1\rangle=d\mathcal{L}_{X_2}\mathcal{E}_1=\mathcal{L}_{X_2} \eta_1=0,\] and similarly, \(d\Theta_2=0\). So \(\Theta_1\) and \(\Theta_2\) are both constants. They are zero precisely when the distribution orthogonal to the span of \(X_1, X_2\) is integrable. This is the case if \(X_1\) and \(X_2\) have a common zero.

The above discussion motivates one to work with Riemannian metrics \(g\) on \(B^2\times \mathbb{T}^2\) which are \(\mathbb{T}^2\)-invariant. Assuming the orthogonal distribution is integrable, they are locally of the form \[g=\underline g+\rho \sum_{1\leq\alpha, \beta\leq 2}\Phi_{\alpha\beta}d\phi_\alpha d\phi_\beta\] where \(\underline g\) is a metric on \(B\), \(\rho>0\) is a function on \(B\), and \(\Phi\) is a smooth map from \(B\) into \(\mathrm{SL}(2;\mathbb{R})/\mathop{\mathrm{SO}}(2)\); the latter is identified with the space of positive definite symmetric 2 by 2 matrices with determinant 1. We also normalize the area form such that \[\int_{\mathbb{T}^2} d\phi_1d\phi_2=1.\] To derive the dimension reduction of the Ricci-flat equation in this setting, we recall that a Ricci-flat metric is a critical point of the Einstein-Hilbert functional \[\mathop{\mathrm{EH}}(g)=\int_{M} \mathop{\mathrm{Scal}}_g \mathop{\mathrm{dvol}}_g.\] A straightforward computation shows that \[\mathop{\mathrm{Scal}}_g=\mathop{\mathrm{Scal}}_{\underline g}-2\rho^{-1}\Delta_{\underline g}\rho+\frac{1}{2}\rho^{-2}|d\rho|^2_{\underline g}-\frac{1}{4}\mathop{\mathrm{Tr}}(\Phi^{-1}d\Phi )_{\underline g}^2.\] So we may write \[\mathop{\mathrm{EH}}(g)=\int_B (\mathop{\mathrm{Scal}}_{\underline g}\rho-2\Delta_{\underline g}\rho+\frac{1}{2}\rho^{-1}|d\rho|^2_{\underline g}-\frac{1}{4}\rho\mathop{\mathrm{Tr}}(\Phi^{-1}d\Phi )_{\underline g}^2)\mathop{\mathrm{dvol}}_{\underline g}.\] To derive the Euler-Lagrange equation of \(\mathop{\mathrm{EH}}\), we work at a critical point of \(\mathop{\mathrm{EH}}\) and consider compactly supported variations of \(\rho, \Phi\) and \(\underline{g}\) separately. First, if we conformally vary \(\underline g\) to \((1+u)\underline g\), then noticing that the last three integrals are conformally invariant, so \[\delta\mathop{\mathrm{EH}}=\int_B -\frac{1}{2}\rho \Delta_{\underline g}u \mathop{\mathrm{dvol}}_{\underline g}=-\frac{1}{2}\int_B u\Delta_{\underline g} \rho \mathop{\mathrm{dvol}}_{\underline g} .\] This implies that \(\rho\) is a harmonic function on \(B\). Let \(z\) be its harmonic dual, i.e., \(dz=*d\rho\). Generically we may assume \(\rho+\sqrt{-1}z\) gives local conformal coordinates for \(\underline g\).

If we consider the variation of \(\Phi\), then we get \[\label{eqn:axisymmetric} (\rho\Phi^{-1}\Phi_\rho)_\rho+(\rho\Phi^{-1}\Phi_z)_z=0.\tag{33}\] This is the same as saying that \(\Phi\) defines an axisymmetric harmonic map in 3 dimensions, if we use cylindrical coordinates \((\rho, z, \theta)\).

If we vary the conformal class of \(\underline g\), i.e., vary \(\underline g\) by \(h\) with \(\mathop{\mathrm{tr}}_{\underline g}h=0\), then we have \[\label{eqn6-4}\nabla^2\rho+\frac{1}{2}\rho^{-1}d\rho\otimes d\rho -\frac{\rho}{4}\mathop{\mathrm{Tr}}(d\Phi\Phi^{-1})^2=c \underline g\tag{34}\] for some constant \(c\). Finally, if we vary \(\rho\), we get \[\label{eqn6-5} \mathop{\mathrm{Scal}}_{\underline g}=\Delta_{\underline g}\log\rho+\frac{1}{2}\rho^{-2}|d\rho|^2_{\underline g}+\frac{1}{4}\mathop{\mathrm{Tr}}(\Phi^{-1}d\Phi)_{\underline g}^2.\tag{35}\] One can see that 34 and 35 combine to one equation \[\mathop{\mathrm{Ricci}}_{\underline g}=\nabla^2\log\rho+\frac{1}{4}\mathop{\mathrm{Tr}}(\Phi^{-1}d\Phi+\rho^{-1}d\rho \mathop{\mathrm{Id}})^2_{\underline{g}}.\] If we write the metric \[\underline g=e^{2\nu}(d\rho^2+dz^2),\] then this is equivalent to the system \[\begin{align} \label{e:eqn24633} \left\{\begin{aligned} &\partial_z\nu=\frac{1}{4\rho}\mathop{\mathrm{Tr}}(\mathcal{U}\mathcal{V}),\\ &\partial_\rho\nu=-\frac{1}{2\rho}+\frac{1}{8\rho}\mathop{\mathrm{Tr}}(\mathcal{U}^2-\mathcal{V}^2), \end{aligned}\right. \end{align}\tag{36}\] where we denote \[\mathcal{U}=\mathop{\mathrm{Id}}+\rho\Phi^{-1}\Phi_\rho,\] and \[\mathcal{V}=\rho\Phi^{-1}\Phi_z.\] The compatibility between the two equations in 36 follows from 33 . It also implies that \(\nu\) is determined up to constant by \(\Phi\).

The upshot is that in the above setting, a Ricci-flat metric locally reduces to an augmented harmonic map, which consists of an axisymmetric harmonic map \(\Phi\) together with a choice of the augmentation \(\nu\). Such a formalism is well-studied in the general relativity literature, see [124]–[126].

6.2 Rod structures↩︎

We are interested in toric gravitational instantons, namely, gravitational instantons with an effective action of a torus \(\mathbb{T}^2\). The topology is encoded in a rod structure, which plays the role of a Delzant polytope in toric Kähler geometry. For a rigorous and precise definition of a rod structure and related terminologies we refer to [106]. For the expository purpose in this paper, we give a less technical and slightly different definition from [106].

Given a rod structure \(\mathfrak R\), we can build an oriented smooth toric four manifold (without specifying a Riemannian metric) as follows. In the \((\rho, z)\) plane we denote \[\mathbb{H}:=\{\rho\geq 0\}.\] Denote by \(\mathbb{H}^\circ\) the interior of \(\mathbb{H}\) and we identify \(\partial\mathbb{H}\) with \(\mathbb{R}\) so the turning points and the rods are naturally subsets of \(\partial\mathbb{H}\). Consider the product \[M^\circ:=\mathbb{H}^\circ\times \mathbb{T}^2,\] where \(\mathbb{T}^2:=\mathbb{R}^2/\Lambda\). We endow \(M^\circ\) with the canonical orientation induced by the 4-form \(d\rho\wedge dz\wedge d\phi_1\wedge d\phi_2\), where \(\phi_1, \phi_2\) are the natural coordinates on \(\mathbb{R}^2\). It admits a natural translation action of the 2-torus \(\mathbb{T}^2\). The open manifold \(M^\circ\) can be partially compactified into a smooth toric manifold \(M\), by adding points with stabilizers over \(\partial\mathbb{H}\): over each rod \(\mathfrak I_j\) we add a copy of \(\mathbb{S}^1\), which is the quotient of \(\mathbb{T}^2\) by the subgroup generated by the rod vector \(\mathfrak v_j\), and at each turning point \(z_j\) we add a point whose stabilizer is the whole \(\mathbb{T}^2\).

Changing the orientation of this \(4\)-manifold corresponds to replacing \(\mathfrak R\) by its opposite \(\overline{\mathfrak R}\). The opposite rod structure has the same turning points as \(\mathfrak R\), but the rod vectors \(\mathfrak v_j\) are replaced by \(R_1\mathfrak v_j\), and the asymptotic Type \(\sharp\) is replaced by \(\overline{\sharp}\). Here \(R_1\) denotes the reflection in \(\mathbb{R}^2\) that swaps the two coordinate axes, and the Type transforms according to \[\overline{\mathop{\mathrm{AE}}}=\mathop{\mathrm{AE}}, \qquad \overline{\mathop{\mathrm{ALF}}^{\pm}}=\mathop{\mathrm{ALF}}^{\mp}, \qquad \overline{\mathop{\mathrm{AF_\beta}}}=\mathop{\mathrm{AF}}_{-\beta}.\]

Given a rod structure \(\mathfrak R\), we may associate to it a nonnegative integer degree \(d(\mathfrak R)\ge 0\). Starting from the line \([\mathfrak v_0]\), for each \(j=1,\ldots,n\) we rotate the line \([\mathfrak v_{j-1}]\) counterclockwise by an angle less than \(\pi\) to obtain the line \([\mathfrak v_j]\). If \(\mathfrak v_n=\mathfrak v_0\), this procedure produces a loop in \(\mathbb{R}\mathbb{P}^1\). If \(\mathfrak v_n\neq \mathfrak v_0\), we perform one additional counterclockwise rotation from \([\mathfrak v_n]\) to \([\mathfrak v_0]\) to close the loop. We then define \(d(\mathfrak R)\) to be the winding number of this loop, with a shift by \(-1\) when \(\mathfrak R\) is of Type \(\mathop{\mathrm{AE}}\) in the \(\mathop{\mathrm{ALF}}^+\) case. This shift is not essential and is introduced only to maintain uniform notation. In general, we have \(d(\overline{\mathfrak R})\neq d(\mathfrak R)\). One can see that for a rod structure \(\mathfrak R\) with asymptotic Type \(\mathop{\mathrm{AE}}/\mathop{\mathrm{ALF}}/\mathop{\mathrm{AF_\beta}}\), if \(\mathfrak R\) has more than 2 turning points, then \(d(\mathfrak R)>1\).

6.3 Model examples↩︎

Examples of toric gravitational instantons can be obtained by applying the Gibbons-Hawking ansatz to an axisymmetric harmonic function on \(\mathbb{R}^3\). The associated axisymmetric harmonic maps also serve as local and asymptotic models for solving general axisymmetric harmonic map equation; see Section 6.4.

Denote the flat metric on \(\mathbb{R}^3\) by \(d\rho^2+dz^2+\rho^2d\phi_2^2\). For an axisymmetric harmonic function \(V\), we can locally write the Gibbons-Hawking ansatz as \[\label{eq:Type32I32metric} g=V(d\rho^2+dz^2+\rho^2d\phi_2^2)+V^{-1}(d\phi_1+Fd\phi_2)^2\tag{37}\] with \[dF=\rho V_\rho dz-\rho V_zd\rho.\] Then the associated axisymmetric harmonic map \(\Phi\) is given by \[\label{eqn:GH32harmonic32map} \Phi=\rho^{-1}\left(\begin{matrix} V^{-1} & V^{-1}F \\ V^{-1}F & \rho^2 V+V^{-1}F^2 \end{matrix}\right).\tag{38}\] The reversed orientation can be obtained by swapping the \(\phi_1\) and \(\phi_2\) coordinates, which means that the associated axisymmetric harmonic map is given by \[\label{eqn:GH32harmonic32map32with32reversed32orientation} \Phi=\rho^{-1}\left(\begin{matrix} \rho^2 V+V^{-1}F^2 & V^{-1}F \\ V^{-1}F & V^{-1} \end{matrix}\right).\tag{39}\]

As more specific examples, the flat \(\mathbb{R}^4\) can be obtained by applying 38 to \[V=\frac{1}{2r}, \;\; F=\frac{z}{2r}.\] and the rod structure has exactly one turning point \(z_0=0\), and with rod vectors \[\mathfrak v_0=(\frac{1}{2}, 1)^{\top}, \;\;\mathfrak v_1=(-\frac{1}{2}, 1)^{\top}.\] For the Taub-NUT metric we have the same rod structure but with \[V=1+\frac{1}{2r}, \;\; F=\frac{z}{2r}.\]

If we now let \(V=1\) and \(F=0\) we get the flat product \(\mathbb{R}^3\times \mathbb{S}^1\), and the associated harmonic map is \[\label{e:standard32harmonic32map32AF}\Phi^{\mathop{\mathrm{AF}}_0}=\begin{pmatrix} \rho^{-1} & 0 \\ 0 & \rho \end{pmatrix}.\tag{40}\] The rod structure has no turning points and the rod vector is \(\mathfrak v_0=(0, 1)^{\top}\). For \(\beta\in \mathbb{R}\) we may deform \(\Phi^{\mathop{\mathrm{AF}}_0}\) to \[\Phi^{\mathop{\mathrm{AF_\beta}}}=\frac{1}{\rho}\left(\begin{matrix}\rho^2&\beta\rho^2\\\beta\rho^2&\beta^2\rho^2+1\end{matrix}\right).\] This gives rise to the quotient \(M_\beta:=(\mathbb{R}^3\times \mathbb{R})/\langle q_\beta\rangle\), where \(q_\beta\) acts as a rotation by angle \(2\pi\beta\) on \(\mathbb{R}^3\) and a translation by distance \(1\) on \(\mathbb{R}^1\).

6.4 Tame harmonic maps↩︎

In order to construct toric gravitational instantons using axisymmetric harmonic maps, we need to prescribe the asymptotics of harmonic maps near the rods and at infinity. This leads to the notion of tameness for axisymmetric harmonic maps defined on \(\mathbb{H}^\circ\). We denote \[r=\sqrt{\rho^2+z^2}.\] For a non-zero \(v\in \mathbb{R}^2\), there is a model harmonic map \(\Phi_v\), which is given by \[\Phi_{v}:=P_v^{\top}\left(\begin{matrix} \rho&0\\0&\rho^{-1} \end{matrix}\right) P_v,\] where \(P_v\) is an element in \(SO(2)\) which rotates \(v\) to the direction of \([(1, 0)^{\top}]\). For \(\beta\in \mathbb{R}\) we also define \[\Phi^{\mathop{\mathrm{AF_\beta}}}_{v}=\rho^{-1}P_v^{\top}\left(\begin{matrix}\rho^2&\beta\rho^2\\\beta\rho^2&\beta^2\rho^2+1\end{matrix}\right)P_v.\] When \([v]=[(1, 0)^{\top}]\), this is the harmonic map associated to the flat space \(M_\beta\).

Similarly, for two linearly independent vectors \(v_1, v_2\in \mathbb{R}^2\), there are 3 model axisymmetric harmonic maps \(\Phi^{\mathop{\mathrm{AE}}}_{v_1, v_2}, \Phi^{\mathop{\mathrm{ALF^+}}}_{v_1, v_2}, \Phi^{\mathop{\mathrm{ALF^{--}}}}_{v_1, v_2}\). When \([v_1]=[(\frac{\alpha}{2}, 1)^{\top}]\) and \([v_2]=[(-\frac{\alpha}{2}, 1)^{\top}]\) for \(\alpha>0\), we have \[\label{e:turning32point32model} \Phi^{\mathop{\mathrm{AE}}}_{v_1, v_2}:=\frac{1}{\rho}\left(\begin{matrix} \frac{2r}{\alpha} & z\\ z & \frac{\alpha r}{2} \end{matrix}\right),\tag{41}\] and \(\Phi^{\mathop{\mathrm{ALF^+}}}\) and \(\Phi^{\mathop{\mathrm{ALF^{--}}}}\) are given by the expression in 38 and 39 respectively, with \(V=1+\frac{\alpha}{2r}\) and \(F=\frac{\alpha z}{2r}\).

When \(\alpha=1\), these are the harmonic maps associated to the flat \(\mathbb{R}^4\), the Taub-NUT metric and the anti-Taub-NUT metric respectively. For more general \(v_1, v_2\), there is a similar definition by reducing to the above case using a rotation in \(SO(2)\).

We have a general existence result, which is essentially due to [127], [124]. The key reason is due to the fact that the source space \(\mathbb{R}^3\) is flat and the target space \(SL(2;\mathbb{R})/SO(2)\) is non-positively curved.

6.5 From tame harmonic maps to conical gravitational instantons↩︎

Given a harmonic map \(\Phi\) that is strongly tamed by a rod structure \(\mathfrak R\) of asymptotic Type \(\sharp\), we may, by the discussion in Section 6.1, choose an augmentation \(\nu : \mathbb{H}^\circ \to \mathbb{R}\), which is unique up to an additive constant. As explained in Section 6.1, there is a smooth oriented toric \(4\)-manifold \(M\) associated with \(\mathfrak R\). The augmented harmonic map \((\Phi,\nu)\) then determines a toric Ricci-flat metric \(g\) on the open subset \(M^\circ\).

The proof of Theorem 19 involves three main ingredients. We fix a choice of \(\nu\).

First, there is a local regularity theorem along the interior of each rod \(\mathfrak I_j\). This relies on analytic estimates for axisymmetric harmonic maps developed in [128]–[130]. These estimates were originally proved in the context of general relativity to handle singular behavior near rods, and in our setting they imply the following: for each \(j = 0, \cdots, n\), there exists a constant \(c_j\) such that the metric \(g_j\) associated to the augmented harmonic map \((\Phi, \nu + c_j)\) extends as a Lipschitz metric across the \(2\)-sphere \(S_j \subset M\) corresponding to the rod \(\mathfrak I_j\).

Using the ellipticity of the Einstein equation together with harmonic coordinates, one can further show that \(g_j\) is in fact smooth across \(S_j\). In particular, the original metric \(g\) may have cone singularities along \(S_j\), with cone angle \[\vartheta_j = 2\pi e^{c_j}.\] Intrinsically, this can be expressed as \[\vartheta_j = 2\pi \lim_{\rho \to 0} \frac{1}{e^\nu \rho} \sqrt{\rho \, \Phi(\mathfrak v_j, \mathfrak v_j)}.\]

Note that these cone singularities are analytically mild: they arise from an incorrect normalization of the rod vector \(\mathfrak v_j\). If one rescales \(\mathfrak v_j\) to \(e^{-c_j}\mathfrak v_j\), then the metric becomes smooth. In particular, the curvature of \(g\) remains locally bounded.

Secondly, there is a singularity removal theorem at each turning point \(z_j\). To analyze this, consider the two adjacent rods \(\mathfrak I_{j-1}\) and \(\mathfrak I_j\). The above discussion shows that \(g\) has cone angle \(\vartheta_{j-1}\) along \(\mathfrak I_{j-1}\) and cone angle \(\vartheta_j\) along \(\mathfrak I_j\). In a neighborhood of \(z_j\), we may replace the lattice \(\Lambda\) by a new lattice \(\Lambda'\), generated by \(2\pi \vartheta_{j-1}^{-1}\mathfrak v_{j-1}\) and \(2\pi \vartheta_j^{-1}\mathfrak v_j\). With respect to \(\Lambda'\), the metric \(g\) becomes smooth along the corresponding \(2\)-spheres \(S_{j-1}\) and \(S_j\). One can then show that \(g\) also extends smoothly across the point associated with the turning point \(z_j\). This again relies on local elliptic PDE analysis and the removable singularity theorem for Einstein metrics due to Bando-Kasue-Nakajima [2].

Finally, there is a unique choice of \(\nu\) such that \(\vartheta_0 = 2\pi\). By the definition of asymptotic models and the normalization conditions on \(\mathfrak v_0\) and \(\mathfrak v_n\) in the definition of a rod structure, it follows that \(\vartheta_n = 2\pi\) as well. Hence, \(g\) has no cone singularities outside a compact set.

There is also a rigidity result for globally tame harmonic maps. Note that there is a natural action of \(SL(2;\mathbb{R})\) on the set of rod structures, given by simultaneously rotating all rod vectors.

Finally, the following theorem characterizes the Type of \(g\) in terms of the degree of the rod structure.

In both the Type \(\mathrm{I}\) and Type \(\mathrm{II}\) cases, toric gravitational instantons (possibly with conical singularities) reduce to axisymmetric harmonic functions via the Gibbons-Hawking and LeBrun-Tod ansatz. In particular, they can be classified and admit explicit expressions. In the Type \(\mathop{\mathrm{I}}\) case this is easy to write down, see Section 6.3; in the Type \(\mathrm{II}\) case, this was carried out in [105], see Section 5.4.

6.6 Cone angles and rod lengths↩︎

Given the above discussion, it becomes clear that, in order to construct toric gravitational instantons, the central task is to choose a rod structure \(\mathfrak R\) such that all cone angles \(\vartheta_j\) are equal to \(2\pi\). A natural idea is to achieve this by varying the rod lengths. At first glance, this is a reasonable strategy, since the number of parameters matches the number of constraints, and it is not difficult to see that the cone angles \(\vartheta_j\) depend continuously on the rod lengths. However, in general, we do not have an explicit description of the associated axisymmetric harmonic maps, and hence of the cone angles, making the situation more subtle.

First, it may happen that the cone angles \(\vartheta_j\) do not vary at all when the rod lengths are changed. As a simple example, consider the double cover of the Calabi-Eguchi-Hanson metric branched along the totally geodesic \(2\)-sphere. This yields a gravitational instanton with cone angle \(4\pi\) along the \(2\)-sphere. It is associated with a rod structure of asymptotic Type \(\mathop{\mathrm{AE}}\) with three rods, whose rod vectors are \((1,0)^{\top}\), \((1,1)^{\top}\), and \((0,1)^{\top}\). In this case, there is only one finite rod, and varying its length merely rescales the metric. The cone angle remains equal to \(4\pi\) and hence does not change.

More generally, one has the following (see [106], Remark 7.8):

Secondly, there are cases in which the cone angles vary in a favorable way. As an example, consider the Kerr metrics, which form an explicit family of smooth \(\mathop{\mathrm{AF}}_\beta\) gravitational instantons \(g_\beta\) on \(\mathbb{R}^2 \times S^2\) for \(\beta \in [0,1)\). The associated rod structure consists of three rods with directions \((1,0)^\top\), \((0,1)^\top\), and \((1,0)^\top\), where the vector \((0,1)^\top\) generates the screw rotation at infinity in the \(\mathop{\mathrm{AF}}_\beta\) model.

When \(\beta = 0\), one recovers the Schwarzschild metric, while as \(\beta \to 1\), an explicit analysis shows that the length \(l_\beta\) of the finite rod tends to infinity. Now fix \(\beta \in (0,1)\). For any \(\gamma \in (0,1)\), one can modify the smooth metric \(g_\gamma\) to produce a conical \(\mathop{\mathrm{AF}}_\beta\) gravitational instanton with varying rod length as follows.

First, replace the lattice \(\Lambda\) by the lattice generated by \((1,0)^\top\) and \((0,\beta\gamma^{-1})^\top\). This produces a toric Ricci-flat metric with cone angle \[\vartheta_{\beta,\gamma} = 2\pi \beta \gamma^{-1}\] along a \(2\)-sphere. Next, rescale the metric by a factor \(\beta^{-1}\gamma\). The resulting metric \(g_{\beta,\gamma}\) has \(\mathop{\mathrm{AF}}_\beta\) asymptotics, with associated rod length \[l_{\beta,\gamma} = l_\gamma \, \beta^{-1} \gamma.\] As \(l_{\beta,\gamma} \to \infty\) (since \(\beta\) is fixed and \(l_\gamma\to\infty\) happens when \(\gamma\to1\), \(l_{\beta,\gamma}\to\infty\) happens precisely when \(\gamma \to 1\)), the cone angle \(\vartheta_{\beta,\gamma}\) tends to \(2\pi\beta < 2\pi\), while as \(l_{\beta,\gamma} \to 0\) (i.e., \(\gamma \to 0\)), the cone angle \(\vartheta_{\beta,\gamma}\) diverges to infinity. By continuity of the cone angle, there exists a critical value \(\gamma_0\) such that \(\vartheta_{\beta,\gamma_0} = 2\pi\), and the corresponding metric \(g_{\beta,\gamma_0}\) is smooth. Of course, we already know this occurs when \(\gamma = \beta\), but this discussion illustrates that, in favorable situations, one can tune the cone angles to \(2\pi\) by varying the rod length.

More generally, in the Type \(\mathrm{II}\) case, the results of Biquard–Gauduchon [105] provide explicit relations between cone angles and rod lengths, which in particular imply the uniqueness of the critical rod length (up to scaling).

6.7 The gravitational instantons in [106]↩︎

Fix \(\beta\in (0, 1)\) and an integer \(n\geq 1\). For \(\boldsymbol{l}=(l_1, \cdots, l_n)\in \mathbb{R}^n_+\), we consider the rod structure \(\mathfrak{R}_{\boldsymbol{l}}\) with

• turning points \(\{z_j\}_{j=1}^{n+1}\), such that \(z_1=0\) and \(z_{j+1}-z_{j}=l_j\) for \(j=1, \cdots, n\);

• lattice \(\Lambda=\mathbb{Z}^2\subset \mathbb{R}^2\);

• rod vectors \(\mathfrak{v}_0=\mathfrak{v}_{n+1}=(1,0)^{\top}\) and for \(j\neq0,n+1\), \(\mathfrak{v}_{j}=(0,1)^{\top}\) when \(j\) is odd, and \(\mathfrak{v}_{j}=(-1,1)^{\top}\) when \(j\) is even;

• asymptotic Type \(\mathop{\mathrm{AF_\beta}}\)

We let \((\Phi_{\boldsymbol{l}}, \nu_{\boldsymbol{l}})\) be the augmented harmonic map which is strongly tamed by \(\mathfrak R_{\boldsymbol{l}}\), and let \(g_{\boldsymbol{l}}\) be the associated conical \(\mathop{\mathrm{AF_\beta}}\) gravitational instanton on \(X_n\). Here \(X_n\) is the oriented 4-manifold diffeomorphic to \[\label{eqn:definition32of32Xn}X_n:=\begin{cases}(S^2\times \mathbb{R}^2)\# k {\mathbb{CP}^2}\# k \overline{\mathbb{CP}^2}, & \text{if } n=2k+1; \\ (S^2\times \mathbb{R}^2)\# k{\mathbb{CP}^2} \# (k+1)\overline{\mathbb{CP}^2}, & \text{if } n=2k+2. \end{cases}\tag{42}\] They are simply-connected and non-spin. In particular, it is clear that they do not admit Type \(\mathop{\mathrm{I}}\) gravitational instantons.

We give a brief outline of the main ideas in the proof here. Denote by \(\vartheta_j(\boldsymbol{l})\) the cone angle along the 2-sphere associated to \(\mathfrak I_j\). By construction, we know that \(\vartheta_0=\vartheta_{n+1}=2\pi\). We define the angle map \[\mathcal{A}:\mathbb{R}^n_+\to\mathbb{R}^n_+;(l_1,\ldots,l_n)\mapsto (\vartheta_1(\boldsymbol{l}),\ldots,\vartheta_n(\boldsymbol{l})).\] It suffices to show that the image of \(\mathcal{A}\) contains the point \[{\boldsymbol{\vartheta}}_0:=(2\pi,\ldots,2\pi).\] Denote \[\mathcal{A}_{0}:\mathbb{R}^n_+\to\mathbb{R}^n_+; (l_1,\ldots,l_n)\mapsto(e^{-l_1}(2\pi+1),\ldots,e^{-l_n}(2\pi+1)).\] Theorem 24 follows from the following result and a simple degree argument.

This is in turn a consequence of the following two statements

• There exists an \(N>0\) such that if \(l_j>N\) for some \(j\), then \((l_1, \cdots, l_n)\in \mathcal{S}\).

• There exits an \(\epsilon>0\) such that if \(l_j<2N\) for all \(j\) and \(l_j<\epsilon\) for some \(j\), then \((l_1, \cdots, l_n)\in \mathcal{S}.\)

As mentioned above, the main difficulty lies in the lack of explicit control over the cone angles in general. Nevertheless, for the purposes of the above statements, it suffices to obtain limiting information on the cone angles when the rod structure degenerates, i.e., when one of the rod lengths tends to zero or infinity. This is achieved via a bubbling analysis of harmonic maps, carried out in [106], by comparison with a family of background maps constructed by gluing suitably rescaled model maps through an induction on scale.

It is worth noting that, although the rod structures under consideration are of asymptotic Type \(\mathop{\mathrm{AF}}_\beta\), the bubbling limits may involve rod structures of asymptotic Type \(\mathop{\mathrm{AE}}\), corresponding to non-collapsed limits of the associated toric gravitational instantons. On the other hand, during the bubbling process, the lattice \(\Lambda\) appearing in the definition of the rod structure may degenerate. As a result, the limiting harmonic maps need not correspond to genuine rod structures, and hence may fail to give rise to toric gravitational instantons. This phenomenon reflects the presence of collapsing in the degeneration.

While the bubbling analysis does not provide a complete description of the cone angles, it yields two key estimates that are sufficient to establish the two statements above.

First, suppose that the maximal rod length, say \(l_j\), tends to infinity. By rescaling the rod structure by \(l_j^{-1}\), one obtains a degeneration of the lattice \(\Lambda\). For the rod structures under consideration, one can show that the corresponding cone angle \(\vartheta_j\) converges to either \(2\pi\beta\) or \(2\pi(1-\beta)\), depending on the parity of \(j\). In particular, it is always strictly less than \(2\pi\). This also explains the requirement that \(\beta \in (0,1)\) in our construction. The Kerr metrics discussed in Section 6.6 provide a concrete example of this phenomenon.

Second, if a bubble limit has a rod structure \(\mathfrak R'\) of asymptotic Type \(\mathop{\mathrm{AE}}\), then one obtains an angle comparison estimate: if the rod structure has \(k\) turning points, then \[\vartheta_1 \ge \vartheta_k, \qquad \vartheta_{k-1} \ge \vartheta_0.\] In particular, if \(\vartheta_k = 2\pi\), then \(\vartheta_1 \ge 2\pi\). This comparison ultimately follows from the Bishop–Gromov volume comparison principle.

6.8 The gravitational instantons of Khuri-Weinstein-Yamada [131]↩︎

In [131], another class of toric Type \(\mathop{\mathrm{III}}\) gravitational instantons was constructed earlier via an explicit ansatz, which in turn depends on a simplification of the axisymmetric harmonic map formalism. There is a very similar construction in the Lorenztian setting [132]. There is an associated generalized rod structure consisting of infinitely many turning points arranged in a periodic configuration. Equivalently, one may view this as a rod structure in which \(\mathbb{R}\) is replaced by \(S^1\).

The construction is based on the observation that when a map \(\Phi: \mathbb{H}^\circ\rightarrow SL(2;\mathbb{R})/SO(2)\) is diagonalizable, i.e., of the form \[\Phi = \begin{pmatrix} e^{u_1} & 0 \\ 0 & e^{u_2} \end{pmatrix},\] then \(\Phi\) is an axisymmetric harmonic map if and only if \(u_1\) and \(u_2\) are axisymmetric harmonic functions. Correspondingly the image of \(\Phi\) is a geodesic in \(SL(2; \mathbb{R})/SO(2)\). In the special case when \(u_1 = -\log \rho\) and \(u_2 = \log \rho\), this yields the flat product \(\mathbb{R}^3 \times S^1\).

To construct nontrivial toric gravitational instantons, [123] considers axisymmetric harmonic functions that are periodic in the \(z\)-direction. Equivalently, one works with axisymmetric harmonic functions on the product \(Q = S^1 \times \mathbb{R}^2\). More precisely, divide \(S^1\) into two segments of equal length, denoted by \(I_1\) and \(I_2\). These can be naturally viewed as subsets of \(Q\). One can then construct Green’s functions \(u_\alpha\) (\(\alpha = 1,2\)) on \(Q\) such that \(u_\alpha\) is asymptotic to \(-\log \rho\) near points in the interior \(I_\alpha^\circ\) and also at infinity. Using \(u_1\) and \(u_2\) to define \(\Phi\), one can then solve for \(\nu\) according to 36 , which is uniquely determined up to an additive constant.

This yields a toric Ricci-flat metric on \[M^\circ = (\mathbb{H}^\circ / \mathbb{Z}) \times (\mathbb{R}^2 / \Lambda),\] where the lattice \(\Lambda\) is generated by \((1,0)^\top\) and \((0,1)^\top\). As in the previous discussion, one obtains a toric gravitational instanton \((M,g)\) with, in general, cone singularities along two 2-spheres. The cone angles along the two rods are given by \[2\pi \lim_{\rho\rightarrow 0}e^{-\log \rho - \nu + \frac{1}{2}u_i}.\] By adjusting \(u_i\) by suitable additive constants, one can arrange for the metric to extend smoothly.

The underlying manifold \(M\) is diffeomorphic to \(S^4\setminus T^2\). It has fundamental group isomorphic to \(\mathbb{Z}\). The universal cover \(\widetilde{M}\) is an infinite connected sum of \(S^2\times S^2\), which admits a complete Ricci-flat metric but it has infinite \(b_2\) and infinite curvature energy. By taking finite covers of \(M\) we can also get a gravitational instanton on the connected sum of an arbitrary number of \(S^2\times S^2\)’s.

The metric \(g\) also has an interesting asymptotic, namely, it is asymptotic to the special Kasner metric \[dr^2+r^{-\frac{2}{3}}dx_1^2+r^{\frac{4}{3}}dx_2^2+r^{\frac{4}{3}}dx_3^2.\] It has quadratic volume growth and exactly quadratic curvature decay.

7 Open questions↩︎

In this section, we list some open questions related to gravitational instantons.

7.1 Toric gravitational instantons↩︎

Following the construction of toric \(\mathop{\mathrm{AF_\beta}}\) gravitational instantons in Theorem 24, it is a natural problem to develop a systematic understanding of general toric gravitational instantons.

In particular, for other types of asymptotic behavior, one may ask:

It is well known that toric gravitational instantons give rise to integrable systems, and one may hope to apply the inverse scattering methods to generate explicit solutions. This approach has already proved successful in the discovery of the Chen-Teo gravitational instantons. Note that the new gravitational instantons in Theorem 24 were constructed by a topological degree argument. This leads to:

A first step in this direction would be to recover the gravitational instantons constructed in Theorem 24 by means of integrable systems, especially for those rod structures with low degree. There is the very recent work [133] that provides an explicit construction for the \(\mathop{\mathrm{AF_\beta}}\) gravitational instantons constructed in [106] on \(X_3\).

7.2 Asymptotic classification↩︎

In the non-collapsed setting, a gravitational instanton is known to be \(\mathop{\mathrm{ALE}}\) by [2], so the main difficulty lies in the collapsed setting. As explained in this article, the asymptotic classification in the collapsed case has already been completed in the hyperkähler and Hermitian settings. The Type \(\mathop{\mathrm{II}}\) case can always be reduced to the Hermitian case, up to passing to a double cover. The Type \(\mathop{\mathrm{I}}\) case can be reduced to the hyperkähler case, up to passing to a finite cover, provided the fundamental group is finite. This naturally leads to the following question:

Observe that the known examples of Type \(\mathop{\mathrm{III}}\) gravitational instantons, discussed in Section 6, are all asymptotic to Type \(\mathop{\mathrm{I}}\) or Type \(\mathop{\mathrm{II}}\) model ends (up to orientation reversal). There are many Ricci-flat ends with symmetry which are Type \(\mathop{\mathrm{III}}\) (for both orientations), for example, the general Kasner family of ends given in 20 . One may ask

In general, one of the main difficulties in addressing Problem 4 is the lack of a geometric understanding of asymptotic cones.

There is also a topological finiteness question

All of these questions are known to have affirmative answers for general complete Riemannian manifolds whose curvature decays faster than quadratically, for example when \(|\mathop{\mathrm{Riem}}_g| = O(r^{-2-\epsilon})\) for some \(\epsilon > 0\); see [65], [134], [135]. If, in addition, the end is assumed to be simply connected, then the asymptotic cone must be flat: it is \(\mathbb{R}^m\) when \(m = \dim M \neq 4\), and it is either \(\mathbb{R}^4\) or \(\mathbb{R}^3\) when \(m = 4\). This follows from the work of Petrunin-Tuschmann [135], together with the proof of the Petrunin-Tuschmann conjecture in [119], which excludes the two dimensional half-plane \(\mathbb{H}\) when \(m=4\). There is also a partial result toward Problem 4 in [61], obtained under an additional asymptotic holonomy assumption and curvature decay.

The diversity of asymptotic models of gravitational instantons also generates various analytic questions, for example, Fredholm theory for natural elliptic operators and Hodge theory on gravitational instantons. Especially interesting is the case where the asymptotics is not standard, for example, in the \(\mathop{\mathrm{ALG}}^*, \mathop{\mathrm{ALH}}^*, \mathop{\mathrm{AF_\beta}}(\beta\notin\mathbb{Q})\) cases.

This will likely involve the techniques of microlocal analysis. See, for example, [136], [137] and references therein.

7.3 Rigidity with fixed asymptotics↩︎

Given a prescribed asymptotic structure at infinity, there are several longstanding rigidity questions. In the \(\mathop{\mathrm{ALE}}\) case, one has the conjecture of Bando-Kasue-Nakajima:

There are some partial results on this question, see for example [120], [138], [139]. To understand \(\mathop{\mathrm{ALE}}\) gravitational instantons of Type \(\mathop{\mathrm{II}}\) requires an extension of Theorem 15 in the case when \(\Gamma\) is not in \(SU(2)\).

In the \(\mathop{\mathrm{AF}}\) case, there is a Riemannian black hole uniqueness conjecture, which has been discussed by many authors; see, for example, Gibbons [140], Lapedes [141], Yau ([71], Problem 116), and Aksteiner-Andersson-Dahl-Nilsson-Simon [142]. In general relativity, the black hole uniqueness conjecture asserts that the only asymptotically flat, stationary vacuum Einstein spacetime is the Kerr spacetime. The naive Riemannian analogue-namely, the uniqueness of Kerr metrics among \(\mathop{\mathrm{AF}}\) gravitational instantons, was disproved by the Chen-Teo metrics, and more generally by the metrics constructed in Theorem 24. A modified version of the Riemannian black hole uniqueness conjecture were proposed in [106].

Evidence for this is provided by the uniqueness result of Mars-Simon [143] (with a mild extension in [106]), which implies that the above conjecture holds in the presence of an isometric semi-free \(S^1\)-action compatible with the \(\mathop{\mathrm{AF}}_0\) asymptotics. The proof relies on a conformal argument due to Bunting-Masood-ul-Alam [144], which reduces the problem to an application of the positive mass theorem.

There is a natural extension of the above conjecture to the \(\mathop{\mathrm{ALF}}\) case. Again, supporting evidence is available under the assumption of an \(S^1\)-symmetry.

7.4 Topological obstructions and rigidity↩︎

Besides the general topological constraints arising from nonnegative Ricci curvature, an obstruction specific to gravitational instantons is given by the Hitchin-Thorpe inequality. Recall that for a compact oriented Einstein 4-manifold \((M, g)\), we have \[2\chi(M) + 3\tau(M) \geq 0.\] where \(\tau(M)\) is the signature of \(M\), and the equality holds if and only if \(g\) is Ricci-flat and Type \(\mathop{\mathrm{I}}\). This follows from a combination of the Gauss-Bonnet-Chern theorem and the signature formula \[\tau(M)=\frac{1}{12\pi^2}\int_M(|\mathop{\mathrm{W}}^+|^2-|\mathop{\mathrm{W}}^-|^2)\mathop{\mathrm{dvol}}_g.\] For gravitational instantons with given asymptotics, an expected general form of the Hitchin-Thorpe inequality is \[2\bigl(\chi(M) - \delta(g)\bigr) + 3\bigl(\tau(M) - \eta(g)\bigr)\geq0.\] Here the correction term \(\delta(g)\) to \(\chi(M)\) is given in Lemma 1, while the correction term \(\eta(g)\) to \(\tau(M)\) involves the limiting eta invariant; see, for example, [138], [139], [145], [146]. In particular, for the standard \(\mathop{\mathrm{ALE}}\), \(\mathop{\mathrm{ALF}}\), \(\mathop{\mathrm{ALG}}\), and \(\mathop{\mathrm{ALH}}\) asymptotics, these correction terms can be computed explicitly.

As a consequence, if \(X\) admits a hyperkähler gravitational instanton with \(\mathop{\mathrm{AL\sharp}}\) (\(\sharp = E, F, G, H\)) asymptotics, then any gravitational instanton on \(X\) with the same \(\mathop{\mathrm{AL\sharp}}\) asymptotics must be hyperkähler, up to orientation reversal. This also yields certain nonexistence results under prescribed asymptotics.

Motivated by this, it is a natural to ask similar rigidity questions.

There are some partial results on these questions assuming an \(S^1\)-symmetry; see [142].

For manifolds with simple topology, one expects a general rigidity

This is known to hold when the curvature decays faster than quadratic rate, by [119], [146].

This is related to Question 5. The only known examples of non-flat gravitational instantons with \(b_1>0\) are the ones in [123].

Recall that for compact Einstein 4-manifolds, there are obstructions arising from smooth topology, for example through the Seiberg-Witten theory [147]–[152]. One may ask whether analogous phenomena occur in the noncompact setting.

7.5 Mass and renormalized volume↩︎

Since a gravitational instanton necessarily has vanishing scalar curvature, results concerning nonnegative scalar curvature impose constraints on gravitational instantons as well. The classical positive mass theorem applies to asymptotically Euclidean (\(\mathop{\mathrm{AE}}\), i.e., \(\mathop{\mathrm{ALE}}\) with trivial \(\Gamma\)) manifolds \((M,g)\) with nonnegative scalar curvature. Since gravitational instantons exhibit a variety of interesting asymptotic geometries, it is natural to seek a notion of mass that can be defined and computed in terms of the asymptotic structure.

There has already been a lot of progress in this direction; see, for example, [138], [153]–[161]. It is particularly interesting to understand the condition of non-negative Ricci curvature, as opposed to non-negative scalar curvature.

In the classical setting, zero mass is achieved only by flat metrics. For the asymptotics of gravitational instantons, typically flat models cannot be realized, and thus one expects the existence of a definite mass gap. This is analogous to the Penrose-type inequality in general relativity. In the Riemannian setting, the area of the horizon can be replaced by the area of certain minimal submanifolds.

There are related results, see, for example [156], [160].

In a different direction, recall that for \(\mathop{\mathrm{ALE}}\) gravitational instantons, Biquard–Hein [162] introduced a notion of renormalized volume \(\mathcal{V}\), using a canonical foliation of the end by constant-mean-curvature hypersurfaces (see also [163] for a different description). This renormalized volume is a global geometric invariant of an \(\mathop{\mathrm{ALE}}\) gravitational instanton. It is similar to the notion of renormalized volume for Poincaré-Einstein manifolds.

7.6 Stability↩︎

Gravitational instantons are critical points of the Einstein–Hilbert functional. For reasons arising from physics and from Ricci flow, it is important to investigate their stability.

Note that, a priori, there may be several different notions of stability. There are many results on this problem; see, for example, [164]–[171]. In particular, it is shown by Biquard-Ozuch [168] that Hermitian gravitational instantons are all linearly unstable. A natural question is whether the Type \(\mathop{\mathrm{III}}\) gravitational instantons discussed in Section 6 are stable.

7.7 Gauge theory↩︎

There are deep connections between gravitational instantons and gauge theory, which we did not discuss in detail in this survey. A general problem is

For hyperkähler gravitational instantons, there are explicit conjectures connecting to moduli spaces in gauge theory. On the one hand, there is a modularity conjecture ([172]) which predicts that any hyperkähler gravitational instanton with quadratic volume growth arises as some Hitchin moduli spaces (of Higgs bundles), see [59], [61], [173], [174] for progress in this direction. On the other hand, hyperkähler gravitational instantons are also expected to arise as suitable monopole moduli spaces, see [50], [56], [57], [175]–[179] for conjectures and progress.

It is also a natural question to study gauge theory over gravitational instantons.

For the flat \(\mathbb{R}^4\) this is the famous ADHM construction. For \(\mathop{\mathrm{ALE}}\) and \(\mathop{\mathrm{ALF}}\) hyperkähler gravitational instantons, there are extensive studies on this topic, see, for example [180]–[184] and the references therein.

7.8 Submanifold geometry↩︎

Dually, it is natural to investigate special submanifolds, for example, complete or compact minimal submanifolds, submanifolds of constant mean curvature, etc, in a gravitational instanton.

There is a long line of research on understanding stable minimal surfaces in Ricci-flat manifolds. For example, this is related to a question of Yau ([185]), see, for example, [186]–[188].

7.9 Moduli and bubbling↩︎

Even though hyperkähler gravitational instantons have been classified, it remains to understand the moduli compactification of these spaces, similar to studying the case of compact hyperkähler metrics on the K3 manifolds in [23]. See [189], [190] for progress in this direction.

Due to noncompactness, taking limits of gravitational instantons would inevitably involve choosing base points and scales. These will be important in understand general bubbling analysis for singularity formations. On the other hand, as is discussed in [23], it would be an interesting question to understand the multi-scale moduli of hyperkähler metrics, either compact, or non-compact, encoding the bubbling information, and relate them to period domains (in the spirit of [87]).

7.10 Relation with compact Einstein metrics↩︎

As explained in Section 4.7, hyperkähler gravitational instantons can arise from singularity formation of Ricci-flat metrics on the K3 manifold. More generally, Type \(\mathop{\mathrm{I}}\) gravitational instantons may also come out of singularity formation of general compact Kähler-Einstein metrics in complex dimension 2. In the non-collapsing situation, which is automatic when the Ricci curvature is positive, this is well-understood, see, for example, [191]–[193]. In the negatively curved case, collapsing may occur and one may see non-\(\mathop{\mathrm{ALE}}\) gravitational instantons, see, for example, [194], [195].

This would give extra geometric information on the KSBA moduli space for algebraic surfaces of general type.

In the more general Riemannian setting, a natural question is

Note that currently we do not have many examples of compact Einstein metrics in 4 dimensions. In particular, all known examples with non-negative Einstein constants are of either Type \(\mathop{\mathrm{I}}\) or Type \(\mathop{\mathrm{II}}\) for a choice of orientation. Note that the notion of Type can be defined for general Einstein metrics. Naturally we can ask

In the Ricci-flat case, this is related to a longstanding question of Page:

Following a proposal of Gibbons-Pope and Page, there are extensive results towards understanding a potential variant of the Kummer construction, where one reverses the orientation of some of the Calabi-Eguchi-Hanson spaces, to construct a Ricci-flat manifold not diffeomorphic to the K3 manifold, see [196], [197]. These results are all negative, suggesting the difficulty of Question 29.

7.11 Infinite energy↩︎

In general, for a complete oriented Ricci-flat 4-manifold, the second Betti number \(b_2\) could be infinity. There are two classes of known examples, both given by explicit constructions.

The first class of examples is given by Anderson-Kronheimer-LeBrun [72], which applies the Gibbons-Hawking ansatz to a positive harmonic function on \(\mathbb{R}^3\) with a suitably chosen infinite set of singularities. In particular, these examples are hyperkähler and they are often referred to as Type \(A_\infty\). Their geometry is much more complicated than gravitational instantons. In particular, it is known [198] that the asymptotic cones may not be unique and may not be metric cones.

The second class of examples is given by the universal covers of the gravitational instantons of [123], as discussed in Section 6.8.

A general conjecture is

This is related to

It is also unknown whether an exotic \(\mathbb{R}^4\) can admit a complete Riemannian metric with non-negative Ricci curvature.

References↩︎