Gravitational instantons with 𝑆1 symmetry
-
Steffen Aksteiner
,
Lars Andersson
,
Mattias Dahl
,
Gustav Nilsson
and
Walter Simon
Abstract
Uniqueness results for asymptotically locally flat and asymptotically flat
1 Introduction
A complete Ricci-flat four-manifold with at least quadratic curvature decay is called a gravitational instanton.
Apart from their intrinsic interest as geometric objects, the study of gravitational instantons is motivated by ideas from Yang–Mills theory and quantum gravity [29] (although the notion of gravitational instanton is sometimes introduced with additional assumptions, for example that the space is hyper-Kähler, we make no such a priori assumptions in this paper).
In this paper, we contribute to the classification of instantons by proving a set of uniqueness results for asymptotically locally flat (ALF)
Some early results related to the present work were presented by one of the authors in [54, 43].
In those papers, additional assumptions on the structure of the fixed point set were needed, as well as some non-trivial technical assumptions that are removed here.
The facts about the structure of the fixed point set that were used in the just mentioned papers are applications of the index theorems to instantons with
Recall that an ALF instanton has cubic volume growth and boundary at infinity 𝐿 of topology
The most well-known AF instanton is the Euclidean Kerr instanton, a 2-parameter family of instantons on
The Euclidean Kerr instanton is toric, that is, it admits an effective
The Chen–Teo instanton is a counterexample to one of the 1970s era uniqueness conjectures for gravitational instantons, the Euclidean Black Hole Uniqueness Conjecture [23, Conjecture 2], which stated that a non-flat AF instanton is in the Kerr family[2]. This shows that the Euclidean Black Hole Uniqueness Conjecture can hold only with additional assumptions, and moreover indicates that the classification problem for gravitational instantons merits renewed attention.
Examples of ALF-
For gravitational instantons with special geometry, some classification results are known.
There is a complete classification of hyper-Kähler instantons with total curvature in
The following theorem states the main results of this paper.
Let
If
If
If
Theorem 1.1 3 is not a rigidity result in the same sense as the first two points. However, it can be viewed as a step towards rigidity for the Chen–Teo family in view of the following conjecture.
Conjecture 1 ([1, Conjecture 1])
Let
An ALF non-Kähler Hermitian instanton admits a bounded Killing field, and is locally conformal to an extremal Kähler space. By analogy with the classification of compact Hermitian-Einstein spaces [38], it is plausible that a non-Kähler Hermitian instanton is actually toric, which in view of the classification result of [8] would imply the conjecture.
We further make the following progressively stronger conjectures, closely related to Conjecture 1.
The conclusion of Conjecture 1 holds for a non-Kähler ALF gravitational instanton which admits a non-vanishing Killing field with bounded norm.
In view of the results of Yau [58], the existence of one Killing field with bounded norm for a non-flat ALF instanton implies that one has either
Finally, we mention the following conjecture, analogous to those made in [23], and stronger than Conjectures 1 and 2.
An ALF gravitational instanton is Hermitian.
The problem raised by Conjecture 3 can be expected to be difficult, analogous to the question whether all compact Ricci-flat spaces have special holonomy [7, p. 19]. However, the ALF assumption provides structure not present in the compact case.
The Chen–Teo instanton was found using the Belinski–Zakharov soliton method, dressing a flat seed solution with four solitons and coalescing one pair of those. The computational methods employed by Chen and Teo did not enable them to explore solutions with more than four solitons. Thus, a priori, there could exist additional instantons which can be constructed along the same lines, but containing more solitons. This is an interesting direction of research.
In addition to the families of instantons mentioned above, some additional potential examples have been mentioned in the literature. These include the Euclidean signature version of the Kerr-NUT family of metrics, for example the so-called Kerr–Taub-bolt instanton; see [27, §III]. An argument showing that the Euclidean Kerr–Taub-bolt instanton must have line singularities was given in [14, §4.8]; see also [18, Appendix A].
We shall now indicate some of the background for the technique used in this paper. Recall that the Schwarzschild and Kerr instantons were constructed by Wick rotating known Lorentzian black hole metrics. The problem of classifying gravitational instantons has important parallels to the Black Hole Uniqueness Problem, and indeed, the divergence identity used in this paper arose out of attempts to prove the Black Hole Uniqueness Conjecture by generalizing Israel’s method [33, 44, 51] for the uniqueness problem from the static Lorentzian case to the stationary Lorentzian case.
In the case of a hypersurface orthogonal Killing vector field 𝜉, this method applies to spaces of both Riemannian and Lorentzian signature (with non-compact isometry group ℝ in the Lorentzian case). The key ingredients in the argument are divergence identities on the 3-dimensional orbit space, among which we select
which is closely related for example to [44, equations (2.12)–(2.13)].
Here
The above scheme can, with considerable effort, be generalized to the generic case when 𝜉 fails to be hypersurface-orthogonal.
Let
Restricting ourselves to the Riemannian case, we define real currents
Here
The 𝐺-signature theorem for an ALF-
Here we generalize these ideas to give restrictions on the fixed point set for ALF-
In the particular cases of instantons with topology
We end this introduction by pointing out that, in the Lorentzian case, the (anti-)self dual parts
2 ALF instantons
In this section, we introduce the notions of ALF four-manifold and ALF instanton and prove some results on the asymptotics of ALF spaces that will be needed in the proof of the main results.
We shall use the following conventions throughout the paper.
We use abstract index notation following [48].
Unless otherwise stated,
respectively.
We sometimes use index-free notation, and for example write
Definition 2.1 ([8, Definition 1.1])
We say that a complete Riemannian four-manifold
ℳ has an end diffeomorphic to
There is a triple
The transverse metric 𝛾 has constant curvature +1.
Let
(2.1)be a metric on
(2.2)where ℎ is a symmetric 2-tensor such that, for all non-negative integers 𝑘, there are constants
Definition 2.2 (ALF instanton)
Let
(1) In view of Definition 2.1,
(2) Definition 2.1 is compatible with the case when the orbits of 𝑇 do not close. This holds for generic AF models.
The following remark highlights some features of the geometry of AF instantons.
Remark 2.4 (AF instanton; see [1, Definition 2.1])
(1) Let
Introducing new Killing coordinates
we have that the identification
(2) If
then the action of
Lemma 2.5 ([8, Lemma 1.7])
Let
Let
Let 𝑔 be an ALF metric as in Definition 2.4.
As explained in [8, proof of Proposition 1.6], 𝜂 has the local form
on
(1) An ALF instanton with 𝐿 an
(2) An ALF instanton with 𝐿 an
An ALF instanton of topology
Definition 2.10 (ALF
S
1
instanton)
Let
The Euclidean Kerr instanton is AF with parameters
Recall that 𝐿 in Definition 2.1 has topology
Assuming that the orbits of 𝑇 are closed, the manifold 𝐿 in Definition 2.1 is a Seifert fibration, that is, a circle fibration over a base orbifold 𝑂.
Recall that, by Proposition B.2, if
The orbifold Euler characteristic is denoted by
If 𝑂 is the base of the Seifert fibration of a lens space, then either 𝑂 is orientable, in which case it has
For a manifold of type ALF-
where 𝐾 is the Gauss curvature.
In the present case, we have
3 The fixed point set
Let
Let 𝑃 be a nut.
Working in a suitable normal coordinate system, the skew-adjoint tensor
and thus has non-zero skew eigenvalues, the surface gravities
A nut with
The fixed point data of 𝑃 is the triple
A bolt 𝐵 is an orientable, totally geodesic two-surface in
and thus has skew eigenvalues
Lemma 3.1 (Generic period)
There is
In particular, the generic period for the
Proof
The exponential map based at any nut is surjective. The same statement holds for the normal exponential map at a bolt. It follows from this that all periods are mutually divisible. This proves the statement. ∎
3.1 Topological invariants
Let
see [30].
We also need to compute the signature of ℳ in terms of the fixed point set.
Since this computation involves a contribution from infinity, we will make the simplifying assumption that
The
From Novikov additivity of the signature and the 𝐺-signature theorem [4, Theorem 6.12] and [4, Proposition 6.18], we get
where the notation we use is related to the notation in [4, Proposition 6.18] by
Set
so
Further,
Note that Lemma 3.1 applies also to the bolt
Importantly, (3.4) holds as an algebraic identity with 𝑔 an independent variable; see [30, §5.8].
As remarked in the introduction, this fact plays an important role in [34]; see Section 3.2 below.
In particular, we can set
This argument to compute the signature also works for a general ALF manifold ℳ. Then the manifold 𝑁 bounding the Seifert fibered manifold 𝐿 can be constructed using equivariant plumbing as in [46, Chapter 2]. The signature and the fixed point data of 𝑁 can then be computed from the corresponding plumbing graph. Since we do not need this for the results of the present paper, we leave the details for future work.
3.2 Structure of the fixed point set
In [34], Jang has provided a complete analysis of the fixed point set for
allows us to rewrite equation (3.4) as the 𝐺-signature formula for a compact, oriented four-manifold
additional (fictitious) nuts. The algebraic facts regarding the 𝐺-signature formula used in [34] therefore apply directly in our situation.
Taking these facts into account, one may verify that the proofs of the following lemmas generalize to the case of ALF
Lemma 3.3 (Weight balance [34, Lemma 2.6])
Let
We shall make use of the following notation.
Let ℳ be a space with an
Lemma 3.4 (Companion nuts [34, Lemma 3.5])
Let
If
If
Lemma 3.5 (Nuts with highest weights [34, Lemma 3.6])
Let
If
If
Remark 3.6 (Examples of fixed point structures)
Let
For
for some mutually prime, positive integers
See [34, Theorem 7.2] for the case of
4 Geometry with a Killing field
In this section, we derive some useful facts for Ricci-flat four-manifolds with a non-vanishing Killing field.
The presentation here follows [25] but adds more detail.
Let
The scaling of
Define the following covariant expressions derived from
The next proposition states a number of identities for the fields that have just been introduced. The Lorentzian signature version is well known; see [42] and references therein. The proof is deferred to Appendix A.
The fields defined above have the following algebraic and differential properties:
(4.2)
Due to (4.2e) and Lemma 2.5, we can introduce the Ernst potentials
which, by (4.2f), (4.2b), solve the Ernst equations
With the twist 1-form
In view of (4.4) and (4.2e), we have that
and we also find
4.1 Charges
Following [25, §5], let
(4.5)
Hence integrating the currents over closed 3-surfaces enclosing but not intersecting nuts and bolts defines quasilocally conserved charges.
Next, we will compute the charges corresponding to
Let 𝐾 be a nut or a bolt in ℳ.
Let 𝑈 be an open set containing 𝐾 with smooth boundary 𝑆, so that neither 𝑈 nor 𝑆 intersects any other nut or bolt.
Let
We now discuss the geometry of bolts in more detail.
For a bolt 𝐵, we denote the volume form of its tangent bundle by
On 𝐵, we have
Define the curvature scalars
Since 𝐵 is totally geodesic, we have
and since ℳ is Ricci-flat,
For a nut 𝑃 with surface gravities
For a bolt 𝐵 with surface gravity 𝜅, we have
Proof
We will compute the integral (4.6) when
so
Here,
where the second term vanishes by the Bianchi identity.
Let
Near a nut 𝑃, we have
Let
With a linear change of coordinates
Together, we have
Next, working near a bolt 𝐵, introduce local coordinates
we compute the Taylor expansion of
We find that
Near the bolt 𝐵, we further have
so
On the surface 𝑆 around 𝐵, we have
Next,
which gives the result. ∎
5 Divergence identities
In this section, we derive a family of divergence identities (see (5.15) below) depending on a parameter 𝛽 taking real values.
An alternative derivation on the orbit space, following [53, 54], will be given in Appendix C.
Equation (5.15) with
After a suitable normalization of 𝜉, we may without loss of generality assume that the supremum of 𝜆 is 1.
Assume that the supremum of 𝜆 is 1. The following holds.
We have (5.1)
(5.1a)The twist potential 𝜔 can be normalized so that(5.1b)There are numbers
(5.2)If
Proof
Since
By Lemma B.1, we may normalize 𝜔 to have
By Proposition B.4, it follows that
where
With the normalization used in Lemma 5.1, we have
(5.4)
We also note the following equivalent conditions for
For any sign + or
Proof
If
On domains where
5.1 Characterizations of special geometry
The following result holds independently for the SD and ASD sides. Recall the Petrov classification of Weyl tensors on Riemannian 4-manifolds [35]; see also [49, 1].
Proof
We first show the equivalence of the conditions.
From (5.9) and
Suppose condition (1) holds. Then we have
which implies that
For
with
Proof
The gradient of (5.5c), using (4.2d) and (4.3), is equal to
By the Maxwell-type equations (4.2c) and the Bianchi identity applied to (5.5a), the contracted second derivative is given by
We notice that the last term can be absorbed by a lower order term involving (5.7), such that
Now (5.6) follows by expanding in 𝛽 as follows:
Here we used (5.11) and (5.9) for the second equality.
The third equality uses (5.5c) and
The quantities defined in Definition 5.3 are derived from the Weyl tensor in a manner analogous to the complex tensors used in characterizations of the Lorentzian Kerr geometry by Mars [41].
Identities analogous to (5.10) as well as wave equations for
For
Proof
For
where
which shows that (5.8) is a sum of squares for
Lemma 5.8 (Divergence identity)
Let
For
satisfy
Proof
To obtain (5.15), recall that the current
and (5.6). ∎
Let
so that
Let
where
In particular,
with equality only if
6 Integrating the divergence identity
In this section, we shall investigate the consequences of the divergence identity (5.17) presented in Lemma 5.10.
By Lemma 5.1,
and on the fixed point set
where
In this paper, we shall consider applications of identity (5.15) for
Near a nut 𝑃 with surface gravities
Near a bolt 𝐵 with surface gravity 𝜅, we have
Proof
From (4.2d), we have
Further,
Let
Proof
Since
and thus
Suppose
The set
The fact that
Proof
Taking the total anti-symmetrization (respectively the trace of (4.2c)) and using the first Bianchi identity (respectively the fact that
Fix small
For
If 𝜖 is sufficiently small, the sets
The sets
By construction,
Definition 6.6 (Length at infinity)
For
The remainder of this section is devoted to the proof of the following result.
Proposition 6.7 (Boundary terms)
Let
with equality if and only if ℳ is algebraically special.
Let
Proof
We note
and with
Using (5.5c), (5.16) and (5.13), we then have
The result follows. ∎
6.1 Contribution from the fixed point set
Lemma 6.9 (Boundary term at the nuts)
Let 𝑃 be a nut with surface gravities
Proof
By Lemma 6.2, Proposition 4.5 and Lemma 6.8, we have
and the result follows. ∎
Lemma 6.10 (Boundary term at the bolts)
Let 𝐵 be a bolt with surface gravity 𝜅.
Let
Proof
By Lemma 6.2, Proposition 4.5 and Lemma 6.8, we have
We use (4.8) and (4.7) to compute, on 𝐵,
Taking into account the fact that
In the last step here, the integral over the tube
Finally, the Gauss–Bonnet theorem tells us that
6.2 Contribution from infinity
Lemma 6.11 (Length at infinity)
Let
where the minimum is taken over all nuts and bolts in the fixed point set 𝐹, and all surface gravities.
The length at infinity
Proof
Assume for a contradiction there is an
Since
which is a contradiction. This proves the lemma. ∎
Lemma 6.12 (The boundary term at infinity)
Let
Proof
By (5.2), we have
as
We find that
We have
The manifold 𝐿 is a circle fibration over the orbifold 𝑂.
From Definition 2.1, we have that the transverse metric 𝛾 gives a metric on 𝑂 with constant curvature +1.
With respect to this fibration, the metric
We conclude that
6.3 The singular term
Lemma 6.13 (The singular term)
Proof
Proposition 6.3 tells us that 𝜆 is bounded away from zero near
By (6.4), we have, for sufficiently small
and the claim follows. ∎
The singular set
6.4 Proof of Proposition 6.7
Proposition 6.7 follows from Lemmas 5.10, 6.9, 6.10, 6.12, 6.13.
7 Proof of the main theorem
In this section, we will analyze the terms in the divergence identity derived in Proposition 6.7 and prove our main theorem. The first step is the following corollary to Proposition 6.7.
For a nut 𝑃 with fixed point data
Let
with equality if and only if
Proof
Since
Let 𝑃 be a nut.
If
If
Proof
Let 𝑃 have weights
We are now ready to apply inequality (7.1) to known spaces with ALF-
Let
Proof
In the present case, we have
In case there are no nuts,
We now assume
This means that equality holds in (7.2) and (7.1), for both signs.
Now consider the case
and then (7.2) is equivalent to
On the other hand, since the set
which means that equality must hold in view of Lemma 7.3.
Consider now the case where
so that (7.2) is equivalent to
From (3.5), we find that the set
Lemma 7.3 tells us that
In case
where the middle inequality uses again Lemma 7.3. Thus
which, once again by Lemma 7.3, means that
In either case, thus,
which finishes the proof. ∎
We next consider the case of Taub-bolt topology, which turns out to be rather similar to the Kerr case.
Let
Proof
From the assumption on the topology, we have
Finally, we consider the case of an
Let
Proof
Let
and thus the numbers of positively oriented and negatively oriented nuts are
Thus assume that there exists no nut with nut data
If
Henceforth, we can thus assume that
and since
Together with the possibilities of the choice of
Case:
When
and since the weight 2 occurs only once in this configuration, by Lemma 3.3, there must exist another nut, say
which is a contradiction, while if
and we conclude that
Finally, when
and as in the previous case, there is a nut
Case:
and the right-hand side is negative for
For the case
and since the weight 3 occurs an odd number of times, Lemma 3.3 tells us that there exists another nut, say
while if
which leads to a contradiction.
Case:
so
Case:
which is impossible.
Case:
which is a contradiction.
Case:
which again is a contradiction. ∎
7.1 Proof of Theorem 1.1
First assume that
8 Concluding remarks
In this paper, we have applied the divergence identity derived in Section 5 to the case of ALF
Funding source: Austrian Science Fund
Award Identifier / Grant number: 10.55776/P35078
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: W2431012
Funding statement: Walter Simon acknowledges funding by the Austrian Science Fund (FWF) [Grant DOI 10.55776/P35078]. The work of Lars Andersson was supported in part by NSFC under grant number W2431012.
A Proof of Proposition 4.3
First we recall the form of partly contracted volume forms,
By (A.2),
which is (4.2a), and similarly, (4.2b) follows via
To prove (4.2e), it is convenient to start with its dual,
The last step used (A.1) and dualizing the equation shows (4.2e). The divergence (4.2f) follows from
Finally, to prove (4.2g), apply a derivative to (4.2d),
In the second line, we used the Bianchi identity for the first term and a decomposition of contracted Weyl tensors, analogous to (A.2), for the last term.
The second term in the last line vanishes because it is SD and ASD in the index pair
B Asymptotics at infinity
In this appendix, we prove a couple of technical results on the asymptotic geometry of ALF
B.1 Asymptotics of Killing vector fields
We will prove that a bounded Killing field with closed orbits for a general ALF metric 𝑔 must be asymptotic to a multiple of the Killing field 𝑇 of the background metric
Let 𝑓 be a function on
Let
for some constant 𝑐 and 𝑇 has closed orbits in 𝐿. In particular,
Proof
Since
see [50, Proposition 8.1.3]. Using the Kato inequality, we find that
By Lemma B.1, there is a constant
We next work in local coordinates as discussed in Remark 2.7 on
for 𝑈 an open subset of
where
Since
for
Let the point
Then
Since 𝜉 is bounded as
Inserting
We next write
we see that
In the local coordinate system, we have found that
By continuity, the coefficients
By assumption, the orbits of 𝜉 are all closed.
Since 𝜉 is bounded, the orbits have uniformly bounded lengths.
If now
It remains to show that 𝑇 has closed orbits on 𝐿.
To see this, let
B.2 Asymptotics of solutions to Poisson equations.
Let
Let
Then there are constants
Proof
We have
Since 𝑇 is a Killing vector field for
or
By assumption,
for some 𝑓 satisfying
Let
Since
We may thus integrate from 𝑟 to ∞ to conclude that
or
Let
Then
If
so
and we conclude that the function
exists.
Taking the limit
If
so
and
Let 𝛿 be the smallest eigenvalue of 𝐷 which is larger than 3 and let 𝑃 be the spectral projection for 𝐷 corresponding to eigenvalues at least 𝛿.
Let
which we integrate from 1 to 𝑟 to get
or
Since
for
so
For
Since
and since
Together, we have
Finally,
for some
Next we consider the general case of
Let
Then there are constants
Proof
Since
we see that
C The quotient
C.1 Generalities
In this section, we consider the orbit space
Theorem C.1 ([56])
For a compact Lie group 𝐺 acting smoothly on
There exists a unique maximal orbit type.
The union
The 𝐺- action on ℳ restricts to
The quotient
In the present case, we have
We now recall [22] that, for an
tensor fields and differential forms on
tensor fields and forms on
In particular, we obtain a non-degenerate metric
We denote by
In local coordinates
where
In the coordinates introduced above, 1-forms
From (C.1), it follows that
which corresponds to the earlier observation that the current (4.5a) is conserved.
Excluding the half-flat case, we recall from Lemma 5.1 3 that
Rewriting [25, equation (5.18)], we recall that the field equations on
with respect to
The
As a consequence of (C.5), we also obtain
(1) As discussed in [25, Section 5], equations (C.5) and (C.6) are invariant under the (Geroch-)
(2) A special class of solutions is characterized by
C.2 The divergence identity on the quotient
We now recall the derivation of the family of divergence identities on the quotient
where
and 𝒞 denotes the trace-free part of a symmetric tensor.
The quantities
Furthermore,
We obtain the following identities [54].
On sets where
Proof
We first sketch the straightforward calculation in the case
where
(1) Identity (C.7) is equivalent to (5.15) for
(2) For
(3) When
where
(4) Still for
(5) In the general case,
C.3 The divergence identities on ℳ
We can now give an alternative proof of the divergence identity (5.15).
Proof
In suitable coordinates, both sides of (5.15) are analytic on
The family (C.7) suggests an attempt for direct integration on
Acknowledgements
We are grateful to Claude LeBrun for helpful remarks, and to Olivier Biquard for several helpful discussions and for sharing an early version of [8]. We thank the anonymous referees for valuable suggestions to improve the article.
References
[1] S. Aksteiner and L. Andersson, Gravitational instantons and special geometry, J. Differential Geom. 128 (2024), no. 3, 928–958. 10.4310/jdg/1729092451Search in Google Scholar
[2] S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces, Comm. Math. Phys. 299 (2010), no. 1, 89–127. 10.1007/s00220-010-1072-1Search in Google Scholar
[3] V. Apostolov, D. M. J. Calderbank and P. Gauduchon, Ambitoric geometry I: Einstein metrics and extremal ambikähler structures, J. reine angew. Math. 721 (2016), 109–147. 10.1515/crelle-2014-0060Search in Google Scholar
[4] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. 10.2307/1970717Search in Google Scholar
[5] C. Bär, On nodal sets for Dirac and Laplace operators, Comm. Math. Phys. 188 (1997), no. 3, 709–721. 10.1007/s002200050184Search in Google Scholar
[6] R. Beig and P. T. Chruściel, Killing vectors in asymptotically flat space-times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem, J. Math. Phys. 37 (1996), no. 4, 1939–1961. 10.1063/1.531497Search in Google Scholar
[7] A. L. Besse, Einstein manifolds, Class. Math., Springer, Berlin 2008. Search in Google Scholar
[8] O. Biquard and P. Gauduchon, On toric Hermitian ALF gravitational instantons, Comm. Math. Phys. 399 (2023), no. 1, 389–422. 10.1007/s00220-022-04562-zSearch in Google Scholar
[9] O. Biquard and V. Minerbe, A Kummer construction for gravitational instantons, Comm. Math. Phys. 308 (2011), no. 3, 773–794. 10.1007/s00220-011-1366-ySearch in Google Scholar
[10] R. Bott and L. W. Tu, Differential forms in algebraic topology, Grad. Texts in Math. 82, Springer, New York 1982. 10.1007/978-1-4757-3951-0Search in Google Scholar
[11] G. Chen and X. Chen, Gravitational instantons with faster than quadratic curvature decay (II), J. reine angew. Math. 756 (2019), 259–284. 10.1515/crelle-2017-0026Search in Google Scholar
[12] G. Chen and X. Chen, Gravitational instantons with faster than quadratic curvature decay. I, Acta Math. 227 (2021), no. 2, 263–307. 10.4310/ACTA.2021.v227.n2.a2Search in Google Scholar
[13] G. Chen and X. Chen, Gravitational instantons with faster than quadratic curvature decay (III), Math. Ann. 380 (2021), no. 1–2, 687–717. 10.1007/s00208-020-01984-9Search in Google Scholar
[14]
Y. Chen and E. Teo,
Rod-structure classification of gravitational instantons with
[15] Y. Chen and E. Teo, A new AF gravitational instanton, Phys. Lett. B 703 (2011), no. 3, 359–362. 10.1016/j.physletb.2011.07.076Search in Google Scholar
[16]
S. A. Cherkis and N. J. Hitchin,
Gravitational instantons of type
[17] P. T. Chruściel, On the invariant mass conjecture in general relativity, Comm. Math. Phys. 120 (1988), no. 2, 233–248. 10.1007/BF01217963Search in Google Scholar
[18] G. Clément and D. Gal’tsov, On the Smarr formulas for electrovac spacetimes with line singularities, Phys. Lett. B 802 (2020), Article ID 135270. 10.1016/j.physletb.2020.135270Search in Google Scholar
[19] A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compos. Math. 49 (1983), no. 3, 405–433. Search in Google Scholar
[20] E. Dobrowolski and T. Januszkiewicz, Arithmetic of an elliptic curve and circle actions on four-manifolds, Colloq. Math. 64 (1993), no. 1, 13–18. 10.4064/cm-64-1-13-18Search in Google Scholar
[21] H. Geiges and C. Lange, Seifert fibrations of lens spaces, Abh. Math. Semin. Univ. Hambg. 88 (2018), no. 1, 1–22. 10.1007/s12188-017-0188-zSearch in Google Scholar
[22] R. Geroch, A method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (1971), 918–924. 10.1063/1.1665681Search in Google Scholar
[23] G. W. Gibbons, Gravitational instantons: A survey, Mathematical problems in theoretical physics (Lausanne 1979), Lecture Notes in Phys. 116, Springer, Berlin (1980), 282–287. 10.1007/3-540-09964-6_335Search in Google Scholar
[24] G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977), no. 10, 2752–2756. 10.1103/PhysRevD.15.2752Search in Google Scholar
[25] G. W. Gibbons and S. W. Hawking, Classification of gravitational instanton symmetries, Comm. Math. Phys. 66 (1979), no. 3, 291–310. 10.1007/BF01197189Search in Google Scholar
[26] G. W. Gibbons and M. J. Perry, Black holes and thermal Green functions, Proc. Roy. Soc. Lond. Ser. A 358 (1978), no. 1695, 467–494. 10.1098/rspa.1978.0022Search in Google Scholar
[27] G. W. Gibbons and M. J. Perry, New gravitational instantons and their interactions, Phys. Rev. D (3) 22 (1980), no. 2, 313–321. 10.1103/PhysRevD.22.313Search in Google Scholar
[28] G. W. Gibbons, C. N. Pope and H. Römer, Index theorem boundary terms for gravitational instantons, Nuclear Phys. B 157 (1979), no. 3, 377–386. 10.1016/0550-3213(79)90109-3Search in Google Scholar
[29] S. W. Hawking, Gravitational instantons, Phys. Lett. A 60 (1977), no. 2, 81–83. 10.1016/0375-9601(77)90386-3Search in Google Scholar
[30] F. Hirzebruch, T. Berger and R. Jung, Manifolds and modular forms, Aspects of Math. E20, Friedrich Vieweg & Sohn, Braunschweig 1992. 10.1007/978-3-663-14045-0Search in Google Scholar
[31] C. J. Hunter, Action of instantons with a nut charge, Phys. Rev. D (3) 59 (1999), no. 2, Article ID 024009. 10.1103/PhysRevD.59.024009Search in Google Scholar
[32] A. D. Ionescu and S. Klainerman, On the uniqueness of smooth, stationary black holes in vacuum, Invent. Math. 175 (2009), no. 1, 35–102. 10.1007/s00222-008-0146-6Search in Google Scholar
[33] W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967), no. 5, 1776–1779. 10.1103/PhysRev.164.1776Search in Google Scholar
[34] D. Jang, Circle actions on oriented manifolds with discrete fixed point sets and classification in dimension 4, J. Geom. Phys. 133 (2018), 181–194. 10.1016/j.geomphys.2018.07.010Search in Google Scholar
[35] A. Karlhede, Classification of Euclidean metrics, Classical Quantum Gravity 3 (1986), no. 1, L1–L4. 10.1088/0264-9381/3/1/001Search in Google Scholar
[36] H. K. Kunduri and J. Lucietti, Existence and uniqueness of asymptotically flat toric gravitational instantons, Lett. Math. Phys. 111 (2021), no. 5, Paper No. 133. 10.1007/s11005-021-01475-1Search in Google Scholar
[37]
C. LeBrun,
Complete Ricci-flat Kähler metrics on
[38] C. LeBrun, On Einstein, Hermitian 4-manifolds, J. Differential Geom. 90 (2012), no. 2, 277–302. 10.4310/jdg/1335230848Search in Google Scholar
[39] P. Li, Circle action with prescribed number of fixed points, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 6, 1035–1042. 10.1007/s10114-015-3630-0Search in Google Scholar
[40] P. Li and K. Liu, Circle action and some vanishing results on manifolds, Internat. J. Math. 22 (2011), no. 11, 1603–1610. 10.1142/S0129167X11007380Search in Google Scholar
[41] M. Mars, A spacetime characterization of the Kerr metric, Classical Quantum Gravity 16 (1999), no. 7, 2507–2523. 10.1088/0264-9381/16/7/323Search in Google Scholar
[42] M. Mars, Uniqueness properties of the Kerr metric, Classical Quantum Gravity 17 (2000), no. 16, 3353–3373. 10.1088/0264-9381/17/16/317Search in Google Scholar
[43] M. Mars and W. Simon, A proof of uniqueness of the Taub-bolt instanton, J. Geom. Phys. 32 (1999), no. 2, 211–226. 10.1016/S0393-0440(99)00023-6Search in Google Scholar
[44] H. Müller zum Hagen, D. C. Robinson and H. J. Seifert, Black holes in static vacuum space-times, Gen. Relativity Gravitation 4 (1973), 53–78. 10.1007/BF00769760Search in Google Scholar
[45] M. Nozawa, An alternative to the Simon tensor, Classical Quantum Gravity 38 (2021), no. 15, Article ID 155001. 10.1088/1361-6382/ac0a87Search in Google Scholar
[46] P. Orlik, Seifert manifolds, Lecture Notes in Math. 291, Springer, Berlin 1972. 10.1007/BFb0060329Search in Google Scholar
[47] D. N. Page, Taub-NUT instanton with an horizon, Phys. Lett. B 78 (1978), no. 2–3, 249–251. 10.1016/0370-2693(78)90016-3Search in Google Scholar
[48] R. Penrose and W. Rindler, Spinors and space-time. Vol. 1, Cambridge Monogr. Math. Phys., Cambridge University, Cambridge 1987. 10.1017/CBO9780511524486Search in Google Scholar
[49] R. Penrose and W. Rindler, Spinors and space-time. Vol. 2, 2nd ed., Cambridge Monogr. Math. Phys., Cambridge University, Cambridge 1988. Search in Google Scholar
[50] P. Petersen, Riemannian geometry, 3rd ed., Grad. Texts in Math. 171, Springer, Cham 2016. 10.1007/978-3-319-26654-1Search in Google Scholar
[51] D. C. Robinson, A simple proof of the generalization of Israel’s theorem, Gen. Relativity Gravitation 8 (1977), no. 8, 695–698. 10.1007/BF00756322Search in Google Scholar
[52] P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983), no. 5, 401–487. 10.1112/blms/15.5.401Search in Google Scholar
[53] W. Simon, Characterizations of the Kerr metric, Gen. Relativity Gravitation 16 (1984), no. 5, 465–476. 10.1007/BF00762339Search in Google Scholar
[54] W. Simon, Nuts have no hair, Classical Quantum Gravity 12 (1995), no. 12, L125–L130. 10.1088/0264-9381/12/12/004Search in Google Scholar
[55] S. Sun and R. Zhang, Collapsing geometry of hyperkähler 4-manifolds and applications, Acta Math. 232 (2024), no. 2, 325–424. 10.4310/ACTA.2024.v232.n2.a2Search in Google Scholar
[56] T. tom Dieck, Transformation groups, De Gruyter Stud. Math. 8, Walter de Gruyter, Berlin 1987. Search in Google Scholar
[57] B. Weber, Energy and asymptotics of Ricci-flat 4-manifolds with a Killing field, Proc. Amer. Math. Soc. 147 (2019), no. 7, 3117–3130. 10.1090/proc/14014Search in Google Scholar
[58] S. T. Yau, Remarks on the group of isometries of a Riemannian manifold, Topology 16 (1977), no. 3, 239–247. 10.1016/0040-9383(77)90004-0Search in Google Scholar
© 2025 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- No compact split limit Ricci flow of type II from the blow-down
- Deriving Perelman’s entropy from Colding’s monotonic volume
- Gravitational instantons with 𝑆1 symmetry
- Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
- On the Rapoport--Zink space for GU(2, 4) over a ramified prime
- Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
- On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
- Anomaly flow: Shi-type estimates and long-time existence
Articles in the same Issue
- Frontmatter
- No compact split limit Ricci flow of type II from the blow-down
- Deriving Perelman’s entropy from Colding’s monotonic volume
- Gravitational instantons with 𝑆1 symmetry
- Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
- On the Rapoport--Zink space for GU(2, 4) over a ramified prime
- Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
- On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
- Anomaly flow: Shi-type estimates and long-time existence
