About a Family of ALF Instantons with Conical Singularities
We apply the techniques developed in our previous article to describe some interesting families of ALF gravitational instantons with conical singularities. In particular, we completely understand the 5-dimensional family of Chen-Teo metrics and prove that only 4-dimensional subfamilies can be smooth...
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| description | We apply the techniques developed in our previous article to describe some interesting families of ALF gravitational instantons with conical singularities. In particular, we completely understand the 5-dimensional family of Chen-Teo metrics and prove that only 4-dimensional subfamilies can be smoothly compactified so that the metric has conical singularities.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 079, 19 pages
About a Family of ALF Instantons
with Conical Singularities
Olivier BIQUARD a and Paul GAUDUCHON b
a) Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France
E-mail: olivier.biquard@sorbonne-universite.fr
b) École Polytechnique, CNRS, CMLS, F-91120 Palaiseau, France
E-mail: paul.gauduchon@polytechnique.edu
Received June 21, 2023, in final form October 10, 2023; Published online October 20, 2023
https://doi.org/10.3842/SIGMA.2023.079
Abstract. We apply the techniques developed in our previous article to describe some
interesting families of ALF gravitational instantons with conical singularities. In particular,
we completely understand the 5-dimensional family of Chen–Teo metrics and prove that
only 4-dimensional subfamilies can be smoothly compactified so that the metric has conical
singularities.
Key words: gravitational instantons; toric geometry; conformally Kähler metrics
2020 Mathematics Subject Classification: 53C25; 53C55
Dedicated to Jean-Pierre Bourguignon
on the occasion of his 75th birthday,
with our admiration and gratitude.
1 Introduction
In a previous paper [3], the authors of the present paper have provided a complete classification,
as well as an effective mode of construction, of so-called toric Hermitian ALF gravitational
instantons. These are four-dimensional, complete, non-compact oriented Ricci-flat Riemannian
(positive definite, smooth) manifolds, which are toric, i.e., admits an effective metric action
of the torus T2 = S1 × S1, are conformally Kähler – but non-Kähler – and, at infinity, are
diffeomorphic to the product R× L, where L is locally a S1-bundle over the sphere S2; the AF
case is when L = S2 × S1.
This class of gravitational instantons includes the Riemannian versions, obtained by Wick
rotations, of well-known Lorentzian space-times, namely (i) the Schwarzschild space, (ii) the
family of Kerr spaces, (iii) the self-dual Taub-NUT space, equipped with the opposite orienta-
tion to the one induced by its hyperkähler structure, and (iv) the Taub-bolt space, discovered
in 1978 by Don Page [15]. Apart from the Taub-NUT space, these spaces share the feature of
being of type D+D−, meaning that their self-dual and anti-self-dual Weyl tensors, W+ and W−
respectively, are both degenerate and non-vanishing, hence giving rise to an ambikähler struc-
ture, as defined in [2], in their conformal cases, cf. Section 2.2 below, see also [9] for the Kerr
spaces (in this terminology, the self-dual Taub-NUT space is of type D+O−, meaning that W+
is degenerate and non-vanishing, while W− ≡ 0).
It has been conjectured for a long time that no other 4-manifold could be admitted in the
family of toric Hermitian ALF gravitational instantons. In 2011 however, a new 1-parameter
This paper is a contribution to the Special Issue on Differential Geometry Inspired by Mathemati-
cal Physics in honor of Jean-Pierre Bourguignon for his 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Bourguignon.html
mailto:olivier.biquard@sorbonne-universite.fr
mailto:paul.gauduchon@polytechnique.edu
https://doi.org/10.3842/SIGMA.2023.079
https://www.emis.de/journals/SIGMA/Bourguignon.html
2 O. Biquard and P. Gauduchon
family of toric ALF – in fact AF – gravitational instantons was discovered by Yu Chen and
Edward Teo, [6], and these instantons were shown to be Hermitian by Steffen Aksteiner and
Lars Andersson [1]. In contrast with the former examples, the Chen–Teo instantons are not
derived from a Lorentzian space by a Wick rotation, and don’t give rise anymore to an ambitoric
structure, but nevertheless still admit a toric Kähler structure in their conformal class. Moreover,
in view of the ALF condition, this toric Kähler structure, as well as the toric Kähler structure
associated to any toric Hermitian ALF gravitational instanton, can be chosen to be locally close
at infinity to the product of the standard sphere S2, of curvature 1, with a hyperbolic cusp, of
curvature −1.
In [3], it has been shown that, together with the above mentioned gravitational instantons,
issuing from classical Lorentzian spaces, the Chen–Teo instanton provides the missing puzzle-
piece needed for a full classification of the toric Hermitian ALF gravitational instantons. In
this approach, the conformal Kähler structure plays a crucial role and, together with a recent
ansatz due to Paul Tod, allows for a simple construction of these instantons, as explained in
the next section. In particular, it provides a new, simple definition of the Chen–Teo instantons.
As a matter of fact, this approach first provides a general description of these metrics on the
open set where the toric action is free. The actual construction of instantons then amounts
to smoothly compactifying the manifold together with the metric along the invariant divisors,
encoded by the edges of the moment polytope. This phase of the construction actually requires
strong restrictions, and eventually ends up with the above mentioned complete classification.
A larger class is obtained by smoothly compactifying the manifold as above but by allowing
conical singularities of the metric along some of them. Metrics of this kind will be called regular.
Examples of such metrics can be found inside the well-known family of so-called Kerr–Taub-NUT
metrics introduced by Gibbons–Perry in [12], cf. [3, Section 7.2.1].
In 2015, Chen–Teo introduced a new 5-parameter family of toric ALF Ricci-flat metrics – in
fact, a 4-parameter family, if we ignore the scaling, as we shall do in the sequel – including the
above mentioned 1-parameter sub-family of Chen–Teo instantons.
Apart from the latter, none of them is smooth – a consequence of the classification in [3],
see also Section 3.2 below – but some of them are regular, as defined above. The main goal
of this paper is to provide an alternative description of the Chen–Teo family, to detect inside
this family the regular elements and the topology of their boundary at infinity, cf. Theorem 3.3
below, and to describe some distinguished regular sub-families and their limits.
2 Toric Hermitian ALF gravitational instantons: a quick review
In this section, we recall some main features of toric Hermitian ALF gravitational instantons,
taken from [3].
2.1 General presentation
It turns out, cf. [3, Corollary 5.2], that each of them is completely encoded by a positive,
continuous, convex, piecewise affine function f on R of the form
f(z) = A+
r∑
i=1
ai|z − zi|, (2.1)
for some positive integer r, where z denotes the standard parameter of R, A is a positive real
number, the coefficients ai are positive real numbers, with
∑r
i=1 ai = 1, and the zi, i = 1, . . . , r,
called angular or turning points, denote the points of discontinuity on R of the slope of f . For
convenience, we put: z0 := −∞ and zr+1 := +∞, cf. Figure 1 for the piecewise affine function of
About a Family of ALF Instantons with Conical Singularities 3
the Chen–Teo instanton. On each open interval (zi, zi+1), i = 0, . . . , r, the slope of f is constant,
denoted by f ′
i . It is required that
f ′
0 = −1 < f ′
1 < · · · < fr−1 < f ′
r = 1.
The coefficients ai are related to the slopes f ′
i by
ai =
1
2
(f ′
i − f ′
i−1), i = 1, . . . , r.
According to the Tod ansatz, cf. [17] and also [3, Section 3], the geometry of a toric Hermitian
Ricci-flat metric is determined by a harmonic, axisymmetric (real) function U = U(ρ, z), defined
on the Euclidean space R3, with the following notation: if u1, u2, u3 denote the standard
coordinates of R3, the pair (ρ, z) – the so-called Weyl–Papapetrou coordinates – are defined by
ρ :=
(
u21 + u22
) 1
2 , z = u3; U being axisymmetric means that it is invariant by the S1-action:
eiθ · u = (cos θu1 + sin θu2,− sin θu1 + cos θu2, u3), hence is a function of ρ, z, and the condition
of being harmonic is then expressed by Uzz + Uρρ +
1
ρUρ = 0, where, as usual, Uρ, Uz, Uρ,z etc.
denote the partial derivatives with respect to ρ and z. For any such generating function U , the
corresponding metric is then given, in the Harmark form [5, 13], by
g =
(dt− Fdx3)
⊗2
V
+ V ρ2dx3 ⊗ dx3 + e2ν(dρ⊗ dρ+ dz ⊗ dz), (2.2)
on the open set, where ρ ̸= 0, where t, x3 are angular coordinates, and V , F , e2ν are functions
of ρ, z, defined by
V = −1
k
(
ρUρ +
U2
ρUzz
U2
ρz + U2
zz
)
, e2ν =
1
4
V ρ2
(
U2
ρz + U2
zz
)
, (2.3)
F = −1
k
(
−
ρU2
ρUρz
U2
ρz + U2
zz
+ ρ2Uz + 2H
)
, (2.4)
where H, the conjugate function of U , is defined, up to an additive constant, by
Hz = ρUρ, Hρ = −ρUz, (2.5)
cf. Section 2.2 for the significance of the constant k.
The functions H and F are both defined up to an additional constant. Indeed, in the
expression (2.2) of the metric, the 1-form η = dt − Fdx3 is well defined, but the pair (t, F ) is
subject to the transform (t, F ) 7→ (t + cx3, F + c), for any constant c, by which η, the vector
field ∂t and x3 remain unchanged, while the vector field ∂x3 becomes ∂x3 − c∂t. In particular,
the vector field ∂x3 + F∂t remains unchanged.
In the current ALF case, it was shown in [3, Section 5] that the generating function U of any
toric Hermitian ALF gravitational instanton is defined on the whole space R3, except on the
z-axis ρ = 0, that, near the z-axis, U is close to f(z) log ρ2, while, at infinity, it is asymptotic to
the harmonic axisymmetric function U0 defined by
U0(ρ, z) = 2
(
ρ2 + z2
) 1
2 − z log
(
ρ2 + z2
) 1
2 + z(
ρ2 + z2
) 1
2 − z
.
It follows that the generating function U of any toric Hermitian ALF gravitational instanton is
actually entirely determined by the above piecewise affine function f(z), via the formula
U(ρ, z) = A log ρ2 +
r∑
i=1
aiU0(ρ, z − zi). (2.6)
4 O. Biquard and P. Gauduchon
By setting
di :=
(
ρ2 + (z − zi)
2
) 1
2 ,
and by noticing that the constant k in (2.3)–(2.4) is equal to 2A, cf. below, we get the following
expressions of U , its first and second derivatives, and H:
U(ρ, z) = A log ρ2 + 2
r∑
i=1
aidi −
r∑
i=1
ai(z − zi) log
(di + z − zi)
(di − z + zi)
, (2.7)
Uρ =
2
ρ
(
A+
r∑
i=1
aidi
)
, Uz = −
r∑
i=1
ai log
(di + z − zi)
(di − z + zi)
, (2.8)
Uρρ = − 2
ρ2
(
A+
r∑
i=1
aidi
)
+ 2
r∑
i=1
ai
di
, Uρz =
2
ρ
r∑
i=1
ai
(z − zi)
di
, Uzz = −2
r∑
i=1
ai
di
,
and
H(ρ, z) = 2Az +
r∑
i=1
ai(z − zi)di +
1
2
ρ2
r∑
i=1
ai log
(di + z − zi)
(di − z + zi)
,
up to constant. We then get
V =
1
A
(
A+
r∑
i=1
aidi
)( (∑r
i=1
ai
di
)(
A+
∑r
i=1 aidi
)(∑r
i=1
ai(z−zi)
di
)2
+
(∑r
i=1
aiρ
di
)2 − 1
)
, (2.9)
e2ν =
1
A
(
A+
r∑
i=1
aidi
)( r∑
i=1
ai
di
(
A+
r∑
i=1
aidi
)
−
(( r∑
i=1
ai(z − zi)
di
)2
+
( r∑
i=1
aiρ
di
)2))
, (2.10)
and
F =
1
A
((
A+
∑r
i=1 aidi
)2(∑r
i=1
ai(z−zi)
di
)(∑r
i=1
ai(z−zi)
di
)2
+
(∑r
i=1
aiρ
di
)2 − 2Az −
r∑
i=1
ai(z − zi)di
)
. (2.11)
It is easy to show that( r∑
i=1
ai(z − zi)
di
)2
+
( r∑
i=1
aiρ
di
)2
≤ 1,
r∑
i=1
ai
di
r∑
i=1
aidi ≥ 1.
It then readily follows that
V ≥ 1 +A
r∑
i=1
ai
di
,
and that V tends to 1 at infinity.
On the z-axis ρ = 0, for any z where f ′(z) ̸= 0, we infer
V (0, z) =
1
A
f(z)
(f ′(z))2
(
f(z)
r∑
i=1
ai
|z − zi|
− (f ′(z))2
)
, (2.12)
e2ν(0, z) = (f ′(z))2V (0, z),
About a Family of ALF Instantons with Conical Singularities 5
F (0, z) =
1
A
(
(f(z))2
f ′(z)
− 2Az −
r∑
i=1
ai(z − zi)|z − zi|
)
=
1
A
(
(f(z))2
f ′(z)
−H(0, z)
)
. (2.13)
From (2.13), we infer that on any interval (zi, zi+1), i = 0, . . . , r, F (0, z) is constant, say equal
to Fi. If f
′
i ̸= 0 and f ′
i−1 ̸= 0, since H is continuous on the z-axis, we then have
Fi − Fi−1 =
1
A
f2
i
(
1
f ′
i
− 1
f ′
i−1
)
;
if, however, f ′
i = 0, then f ′
i−1 ̸= 0 and f ′
i+1 ̸= 0 and we then get
Fi+1 − Fi−1 =
1
A
(
f2
i
(
1
f ′
i+1
− 1
f ′
i−1
)
− 2(zi+1 − zi)fi
)
,
cf. [3, Propsition 7]. From (2.13) again, we get
F0 = − 1
A
(
A+
r∑
i=1
aizi
)2
+
1
A
r∑
i=1
aiz
2
i ,
Fr =
1
A
(
A−
r∑
i=1
aizi
)2
− 1
A
r∑
i=1
aiz
2
i ,
up to an additional constant, hence
Fr − F0 =
2
A
(
A2 +
( r∑
i=1
aizi
)2
−
r∑
i=1
aiz
2
i
)
. (2.14)
It follows that
A =
1
4
(
Fr − F0 +
(
(Fr − F0)
1
2 + 16
r∑
i=1
aiz
2
i − 16
( r∑
i=1
aizi
)2) 1
2
)
.
In particular, the metric is AF, i.e., satisfies Fr − F0 = 0, if and only if
A2 =
r∑
i=1
aiz
2
i −
( r∑
i=1
aizi
)2
.
Remark 2.1. As explained above, to any convex, piecewise affine function f(z) as defined
in (2.1) is associated a generating function U(ρ, z), defined by (2.6), hence by (2.7); conversely,
it follows from (2.8) that f(z) is determined by U(ρ, z) via the formula f(z) = 1
2(ρUρ)(0, z). The
corresponding Ricci-flat metric g is then expressed by (2.2), where the functions V (ρ, z), F (ρ, z),
e2ν(ρ, z) are given by (2.3)–(2.4), hence by (2.9)–(2.11). For any real constants α > 0, β, f(z)
may be replaced by the function f̃(z) := 1
αf(αz+ β) = A
a +
∑r
i=1 ai|z− z̃i|, with z̃i =
zi−β
α , and
the corresponding generating function is then replaced by
Ũ(ρ, z) =
1
a
U(αρ, αz + β) =
A
a
log ρ2 +
r∑
i=1
aiU0(ρ, z − z̃i).
The corresponding Ricci-flat metric is then g̃(ρ, z, t, x3) :=
1
α2 g(αρ, αz+β, αt, x3), meaning that g̃
is homothetic to g by a factor 1/α2, via the change of variables (ρ, z, t, x3) 7→ (αρ, αz+β, αt, x3).
Also notice that, by denoting f̃i := f̃(z̃i) and fi := f(zi), we have f̃i = fi/α, i = 1, . . . , r.
6 O. Biquard and P. Gauduchon
2.2 The Kähler environment
By definition, a toric Hermitian ALF gravitational instanton, say (M, g), admits a Kähler met-
ric, gK , in the conformal class of g, which is actually toric as well, meaning that the torus
action is Hamiltonian, i.e., admits a moment map The fact that g is conformally Kähler implies
that the self-dual Weyl tensor W+ of g, regarded as a (symmetric, trace-less) operator on the
self-dual part of Λ2M , is degenerate, meaning that W+ has a simple, non-vanishing, simple
eigenvalue, say λ, and a repeated eigenvalue −λ
2 . It is also required that λ is not constant and
everywhere non-vanishing. According to [8], the Kähler metric gK is then equal to λ2/3g or
a constant multiple. In view of the ALF condition, it is convenient to set: gK =
(
k−1λ
)2/3
g,
where the constant k is chosen in such a way that gK is asymptotic at infinity to the product
of the standard sphere of radius 1 and the Poincaré cusp of sectional curvature −1 (more detail
in [3, Section 2]).
The conformal factor
(
k−1λ
) 2
3 is then equal to x21, where x1 denotes the moment of the
Hamiltonian Killing vector field ∂t, and the scalar curvature, ScalgK , of the Kähler metric gK is
then equal to 6k
2
3λ
1
3 = 6kx1, and tends to 0 at infinity. In particular, gK = x21g is extremal,
even Bach-flat, since it is conformal to an Einstein metric. The constant k is actually the same
as the constant k appearing in (2.3)–(2.4), and turns out to be equal to 2A [3, equation (89)].
In terms of the generating function U , the Kähler form, ωK , and the volume form, vgK , of gK
have the following expression:
ωK =
2
U2
ρ
(1
ρ
(Uzzdρ− Uρzdz) ∧ (dt− Fdx3)− V (Uρzdρ+ Uzzdz) ∧ dx3
)
,
and
vgK =
1
2
ωK ∧ ωK =
4
ρU4
ρ
(
U2
ρz + U2
zz
)
dt ∧ dx3 ∧ dz ∧ dρ. (2.15)
From (2.15), we can infer that the volume of (M, gK) is finite, and the image of the moment
map is a convex, pre-compact polytope in the Lie algebra t of the torus T2, cf. Figure 2, which
is the picture, taken from [3, Section 8], of the moment polytope of the Chen–Teo instan-
ton. Notice that, apart from the dashed edge E∞, representing the boundary at infinity, each
edge Ei, i = 0, 1, . . . , r is associated to the interval (zi, zi+1) on the z-axis ρ = 0.
The moment with respect of ωK of the Killing vector fields ∂t and ∂x3 – which, in general,
don’t form a basis of Λ – are denoted by x1 and µ respectively, with
x1 =
2
Hz
, µ = − 1
A
zHz + ρHρ − 2H
Hz
,
where, we recall, H is defined by (2.5) [3, Proposition 6.1]. Notice however that in general ∂t
and ∂x3 , regarded as elements of the Lie algebra t of the torus T2 don’t form a basis of the
lattice Λ in t induced by T2. In restriction to the boundary ρ = 0, the moments are functions
of z, with the following expressions on each interval (zi, zi+1), where f ′
i ̸= 0,
x1 =
1
f(z)
, µ = − Fi
f(z)
+
1
A
(
f(z)
f ′(z)
− z
)
.
The expression (2.2) of the metric g holds on the open set, M0, where ρ ̸= 0, i.e., where the
torus action is free. In the toric Kähler setting, the boundary ρ = 0 of this open set is formed
of (r − 1) compact invariant divisors, isomorphic to 2-spheres, and of two divisors isomorphic
to punctured spheres, corresponding to a point at infinity for each of them, encoded by the
r + 1 edges of the moment polytope. To each edge of the moment polytope, itself encoded by
About a Family of ALF Instantons with Conical Singularities 7
an interval (zi, zi=1), i = 0, . . . , r, is associated a Killing vector field, vi, regarded as an element
of t, actually a primitive element of Λ: vi is then the generator of a S1-action of period 2π, and
vanishes on the corresponding invariant divisor. It follows from (2.12) that vi has the following
form:
vi = f ′
i(∂x3 + Fi∂t) if f ′
i ̸= 0, vi =
1
A
f2
i ∂t if f ′
i = 0. (2.16)
More generally, if the metric admits a conical singularity along the invariant divisor Ei, of
angle 2παi, then
vi = αif
′
i(∂x3 + Fi∂t) if f ′
i ̸= 0, vi =
1
A
αif
2
i ∂t if f ′
i = 0. (2.17)
The conditions that (M0, g) will smoothly extend to the boundary, possibly with conical
singularities of g on the invariant divisors, is that each pair vi, vi+1 be a basis of the lattice Λ,
i.e., that each pair be related to the next one by an element of the group GL(2,Z) of 2 × 2
matrices with integer coefficients and determinant equal to ±1, i.e.,(
vi−1
vi
)
=
(
ℓi −ϵi
1 0
)(
vi
vi+1
)
,
hence
ℓivi = vi−1 + ϵivi+1, i = 1, . . . , r − 1,
where the ℓi are integers and ϵ = ±1, cf. [3, Section 8].
As already mentioned, these conditions turn out to be quite restrictive, in particular impose
that r cannot exceed 3. For each value 1, 2 or 3 of r, the only toric Hermitian ALF gravitational
instantons are then as follows, cf. Theorems A and 8.2 in [3]:
� r = 1: The self-dual Taub-NUT instanton, i.e., the Euclidean self-dual Taub-NUT on R4,
with the orientation opposite to the one induced by its hyperkähler structure. Its piecewise
affine function is f(z) = 2n+|z| and its generating function is U(ρ, z) = 2n log ρ2+U0(ρ, z).
� r = 2:
(i) The Taub-bolt instanton, discovered by D. Page in 1978, whose piecewise affine func-
tion is f(z) = 3b+ 1
2 |z + b|+ 1
2 |z − b|, b = 3
4 |n|.
(ii) The Euclidean Kerr metrics, discovered by R. Kerr in 1963, with
f(z) = m+
1
2
(
1− a
b
)
|z+ b|+ 1
2
(
1 +
a
b
)
|z− b|, 0 < |a| < b =
(
m2 + a2
) 1
2 .
(iii) The Euclidean Schwarzschild metric, discovered by K. Schwarzschild in 1918, with
f(z) = m+ 1
2 |z+m|+ 1
2 |z−m|, which can be viewed as a particular case of Euclidean
Kerr metric, with a = 0 and b = m.
� r = 3: The 1-parameter family of Chen–Teo instantons, discovered by Yu Chen and
Edward Teo in 2011, cf. [6], with
f(z) =
1
2
(
1− p
3
2 − q
3
2 + q|z + q
1
2 − q|+ |z|+ p|z − p
1
2 + p|
)
, (2.18)
0 < p < 1, 0 < q < 1, p+ q = 1, (2.19)
f1 = pq
1
2 , f2 = pq, f3 = p
1
2 q. (2.20)
In contrast with the previous cases, the Chen–Teo instantons are not the Euclidean form
of Lorentzian spaces, and their anti-self-dual Weyl tensor W− is not degenerate, as shown
in [1].
8 O. Biquard and P. Gauduchon
z1 = q − q
1
2
f1 = pq
1
2
z3 = p
1
2 − p
f3 = p
1
2 q
f2 = pq
f ′
2 = q
f ′
0 = −1
f ′
1 = −p
z2 = 0
f ′
3 = 1
Figure 1. The piecewise affine function of the Chen–Teo metric.
(0, 0)
V4 = (−p, 1)V3 =
(
− p
2A(1− p
3
2 − q
3
2 − 2p
1
2 q), 1
)
V0 = (p,−1) V1 =
( p
2A(1− p
3
2 + q
3
2 ),−1
)
V2 =
( p
2A(1− p
3
2 + q
3
2 ), 1
2A(p
3
2 − q
3
2 − p+ q))
)
E0
E1
E2
E3
E∞
Figure 2. The Chen–Teo moment polytope in the x, y-plane, with respect to the Z-basis v1 = −p(∂x3 +
F1∂t), with F0 = 0, where x = −p(y + F1x1), y = µ+ 1
2A
(
p
3
2 − q
3
2 − p+ q
)
, and 2A = 1− p
3
2 − q
3
2 . The
slope of the edge between V2 and V3 is −1 for any value of the parameter p.
More generally, we shall consider smooth completions of the metric g given by (2.2) admitting
conical singularities along the invariant divisors, as described by Theorem 7.5 in [3, Section 7].
As shown in [3, Section 7], this can be done for the whole family of Kerr–Taub–NUT family,
introduced by G.W. Gibbons and M.J. Perry in 1980, which includes the instantons mentioned
above when r = 2.
This also concerns the Chen–Teo 4-parameter family introduced in [7], which we shall explore
in the next section.
2.3 The self-dual Eguchi–Hanson metric
The Eguchi–Hanson metric was first discovered by Tohru Eguchi and Andrew J. Hanson in [10],
and by Eugenio Calabi in [4]; it is also a member of the Gibbons–Hawking family of hyperkähler
metrics [11]. Like the Taub-NUT metric quoted above, also a member of the Gibbons–Hawking
family, the Eguchi–Hanson metric is of type O+D− with respect to the orientation determined by
the hyperkähler structure, meaning that W+ ≡ 0, while W− is degenerate, but non-zero. With
respect to the opposite orientation it is then of type D+O− and will then be called the self-dual
Eguchi–Hanson metric. This can be written in Harmark form (2.2), with ρ =
(
r2 − b2
) 1
2 sin θ,
z = r cos θ, V = r
r2−b2
, F = − cos θ, e2ν = r
r2−b2 cos2 θ
. It can be shown that the simple eigenvalue
About a Family of ALF Instantons with Conical Singularities 9
of W+ is λ+ = 2b2
r3
and the conformal Kähler metric is then conveniently chosen to be gK = 1
r2
g,
whose Kähler class is then ωK = −dr
r2
∧ (dt+cos θdx3)− 1
r sin θdθ ∧ dx3, so that the moment x1
of ∂t be equal to 1
r and k = 2b2. Unlike the self-dual Taub-NUT metric, the self-dual Eguchi–
Hanson metric is ALE, not ALF, but its generating function, UEH, is nevertheless of the same
type (2.6) as the generating functions of the gravitational instantons considered in this note,
namely
UEH(ρ, z) =
1
2
U0(ρ, z + b) +
1
2
U0(ρ, z − b)
= d1 −
1
2
(z + b) log
d1 + z + b
d1 − z − b
+ d2 −
1
2
(z − b) log
d2 + z − b
d2 − z + b
,
with d1 =
(
ρ2 + (z + b)2
) 1
2 and d2 =
(
ρ2 + (z − b)2
) 1
2 , and the corresponding piecewise affine
function is then
fEH(z) =
1
2
|z + b|+ 1
2
|z − b|.
It may be observed that the positive constant A appearing in the general expression (2.1) is here
equal to 0 and that the identity k = 2A is here no longer valid, showing again that the self-dual
Eguchi–Hanson metric does not belong to the family of gravitational instantons considered in
this paper. It may however be viewed as a limit, as already observed by Don Page in [15],
cf. also [3, Section 7.2]. In the current setting, this can be viewed by considering the following
one-parameter family of metrics encoded by their piecewise affine functions of the form
fA(z) = A+
1
2
|z + b|+ 1
2
|z − b|,
normalized by the condition A+ b = 1, cf. Remark 2.1, with A, b ≥ 0; notice that f1 = f2 = 1,
and that most metrics in this family have conical singularities along invariant divisors, of angles
2παi, i = 1, 2, 3. The vector fields attached to the corresponding polytopes, cf. (2.16)–(2.17),
are then v0 = −α0(∂x3 + F0∂t), v1 = α1
1
A∂t, v2 = α2(∂x3 + F2∂t), and the regularity condition
is then: ( v0v1 ) =
(
ℓ −ϵ
1 0
)
( v1v2 ), for some integer ℓ and ϵ = ±1; we then have α0 = ϵα2, hence ϵ = 1,
α0 = α2, and we can actually assume α0 = α2 = 1, and ℓα = A(F2 − F0), by setting α1 = α,
hence, by (2.14), ℓα = 2
(
A2 − b2
)
= 2(A− b) = 2(2A− 1), where we can assume that ℓ is equal
to 1, 0 or −1. When A runs in the open interval (0, 1), the corresponding metric is smooth in the
following three cases:
(
A = 3
4 , b =
1
4 , ℓ = 1
)
,
(
A = 1
2 , b =
1
2 , ℓ = 0
)
and
(
A = 1
4 , b =
3
4 , ℓ = −1
)
,
corresponding to the “positive” Taub-bolt metric, the Schwarzschild metric and the “negative”
Taub-bolt metric respectively.1
When A ∈
(
0, 12
)
we can take ℓ = −1, the topology is that of the negative Taub-bolt metric
(the total space of O(1)), the angle 4π(1 − 2A) goes from 0 when A → 1
2 to 4π when A → 0,
hence the metric tends to the pull-back, from O(2) to O(1), of the self-dual Eguchi–Hanson
metric, with a conical singularity of angle 4π. When A ∈
(
1
2 , 1
)
, we have ℓ = 1, the topology
is that of the positive Taub-bolt metric (the total space of O(−1)), again the angle 4π(2A− 1)
goes from 0 when A → 1
2 to 4π when A → 1. The limit for A = 1 is the Taub–Nut metric on R4
which is generated by the function f1(z) = 1 + |z|.
There is a symmetry around A = 1
2 : the metrics for A = 1
2 ± a are the same with the
orientation reversed, up to scale. So it may seem curious that the limits for A = 0 and A = 1
1The “positive” and the “negative” Taub-bolt metrics are actually the same metric on the same manifold,
namely the complex projective plane CP2 with a deleted point, with however opposite orientations, hence two
different conformal Kähler structures: the “positive” Taub-bolt has the natural orientation of the tautological
line bundle O(−1) over CP1, the “negative” one the natural orientation of the dual line bundle O(1). Similarly,
the hyperkähler Eguchi–Hanson metric lives on the oriented manifold O(−2), while the self-dual Eguchi–Hanson
lives on the dual line bundle O(2).
10 O. Biquard and P. Gauduchon
are the selfdual Eguchi–Hanson metric and the Taub-NUT metric. This contradiction is solved
by understanding that these are limits at different scales: the Taub-NUT metric is obtained
when A → 1 by shrinking the 2-sphere to a point, and by rescaling there is a bubble which
is O(−1) with the 2-cover of the Eguchi–Hanson metric. This is precisely what we see on the
other side A → 0, with the opposite orientation.
Finally notice the change of topology and of orientation at
(
A = 1
2 , ℓ = 0, b = 1
2
)
, encoding
the (Riemannian) Schwarzschild metric, which lives on the product S2 × R2, with its natural
two orientations.
3 The Chen–Teo family
The Chen–Teo 4-parameter family is actually relevant to the general treatment of the preced-
ing section, i.e., is included and probably coincides with the family of toric Hermitian ALF
gravitational instantons, with r = 3, when the z-axis admits 3 angular points, z1 < z2 < z3.
The convex piecewise affine function f as then the following general form:
f(z) = A+
1
2
(1− p)|z − z1|+
1
2
(p+ q)|z − z2|+
1
2
(1− q)|z − z3|, (3.1)
where
−1 < −p < q < 1,
are the slopes of f , on the open intervals (−∞, z1), (z1, z2), (z2, z3), (z3,∞) respectively. The
pair (p, q) then belongs to the open domain of R2 defined by
−1 < p < 1, −1 < q < 1, p+ q > 0.
We denote f1 := f(z1), f2 := f(z2), f3 := f(z3), and, in addition to p, q, we introduce two
positive parameters a, b by
a := f2
1 /f
2
2 , b := f2
3 /f
2
2 .
Alternatively,
√
a− 1 = p
(z2 − z1)
f2
,
√
b− 1 = q
(z3 − z2)
f2
. (3.2)
Then a > 1 if p > 0, a < 1 if p < 0 and a = 1 if p = 0; similarly, b > 0 if q > 0, b < 1 if q < 0
and b = 1 if q = 0, and
lim
p→0
(a− 1)
p
=
2(z2 − z1)
f2
, lim
q→0
(b− 1)
q
=
2(z3 − z2)
f2
. (3.3)
Notice that the parameters a, b, as well as the parameters p, q, are insensitive to the transform
described in Remark 2.1.
From (3.1), we get f2 = A+ f2
2
( (√a−1)
p (1− p) + (
√
b−1)
q (1− q)
)
, hence
A = f2
(
p+ q −
√
aq(1− p)−
√
bp(1− q)
)
2pq
=
1
2
(
f2
(√
a+
√
b
)
− (z3 − z1)
)
.
Remark 3.1. For further use, it will be convenient to “normalize” the convex piecewise affine
function f(z), via the transform described in Remark 2.1, in order that z2 = 0 and f2 = 1, hence
f1 =
√
a, f3 =
√
b. The convex piecewise affine function f(z) is then given by (3.1), with
A =
√
a+
√
b
2
− 1
2
(√
a− 1
p
+
√
b− 1
q
)
, z1 =
1−
√
a
p
, z2 = 0, z3 =
√
b− 1
q
.
About a Family of ALF Instantons with Conical Singularities 11
3.1 Regularity
As in the introduction, we call a metric regular if on some smooth compactification it has at
worst conical singularities. In order to test the regularity of these metrics, we introduce the
angles 2πα0, 2πα1, 2πα2, 2πα3, attached to each divisor, where the αi are all positive, and we
consider the corresponding Killing vector fields, when p ̸= 0, q ̸= 0,
v0 = −α0
(
∂
∂x3
+ F0
∂
∂t
)
, v1 = −pα1
(
∂
∂x3
+ F1
∂
∂t
)
,
v2 = qα2
(
∂
∂x3
+ F2
∂
∂t
)
, v3 = α3
(
∂
∂x3
+ F3
∂
∂t
)
, (3.4)
where, for i = 0, 1, 2, 3, Fi denotes the (constant) value of F in the interval (zi, zi+1) on the
axis ρ = 0, cf. [3, Lemma 7.1]. The regularity conditions are then(
v0
v1
)
=
(
ℓ1 −ϵ1
1 0
)(
v1
v2
)
,
(
v1
v2
)
=
(
ℓ2 −ϵ2
1 0
)(
v2
v3
)
,
where ℓ1, ℓ2 are integers, and ϵ1, ϵ2 are equal to ±1, hence
ℓ1v1 = v0 + ϵ1v2, ℓ2v2 = v1 + ϵ2v3,
or else, in view of (3.4),
ℓ1pα1 + ϵ1qα2 = α0, (3.5)
pα1 + ℓ2qα2 = ϵ2α3, (3.6)
ℓ1pα1F1 + ϵ1qα2F2 = α0F0, (3.7)
pα1F1 + ℓ2qα2F2 = ϵ2α3F3. (3.8)
In view of (3.5), in (3.7) the Fi may be replaced by Fi+c for any constant c, and likewise in (3.8)
in view of (3.6). Also recall, cf. [3, Proposition 7.3], that the Fi are related by
F1 − F0 = −f2
1
A
(1− p)
p
= −f2
2
A
a(1− p)
p
,
F2 − F1 =
f2
2
A
(p+ q)
pq
,
F3 − F2 = −f2
3
A
(1− q)
q
= −f2
2
A
b(1− q)
q
. (3.9)
In particular,
A(F3 − F0) =
f2
2
pq
(
p+ q − aq(1− p)− bp(1− q)
)
. (3.10)
As observed above, in view of (3.5)–(3.6), (3.7)–(3.8) can be rewritten as
ℓ1p(F1 − F0)α1 + ϵ1q(F2 − F0)α2 = 0, (3.11)
ϵ2p(F1 − F3)α1 + ϵ2ℓ2q(F2 − F3)α2 = 0. (3.12)
Since α1 and α2 are both positive, it follows that
(F2 − F0)(F1 − F3) = ϵ1ℓ1ℓ2(F1 − F0)(F2 − F3),
12 O. Biquard and P. Gauduchon
hence
(F2 − F0)(F1 − F3)
(F1 − F0)(F2 − F3)
= ϵ1ℓ1ℓ2,
or, equivalently,
n :=
(F3 − F0)(F1 − F2)
(F1 − F0)(F2 − F3)
= ϵ1ℓ1ℓ2 − 1. (3.13)
In view of (3.9)–(3.10), n, defined by (3.13), has the following expression:
n =
(p+ q)(p+ q − aq(1− p)− bp(1− q))
aq(1− p)bp(1− q)
, (3.14)
and will be called the normalized total NUT-charge, cf. [7, Section III.B]. By (3.13), n is then
an integer, whenever the metric is regular. Notice that n = 0 if and only if F3 − F0 = 0, i.e., if
and only if the metric is AF.
Remark 3.2. Notice that (3.14) can be rewritten as
n =
(p+ q)
ab(1− p)(1− q)
(
a+ b− (a− 1)
p
− (b− 1)
q
)
.
It follows from (3.14) and (3.2) that n is well defined at p = 0 or q = 0 and that the quantity
a+ b− (a−1)
p − (b−1)
q has the sign of n. In particular, a regular metric is AF if and only if the
parameters p, q, a, b are related by a+ b− (a−1)
p − (b−1)
q = 0.
From (3.5)–(3.8), we infer
α1 =
1
ℓ1p
(F0 − F2)
(F1 − F2)
α0 =
1
ϵ2p
(F3 − F2)
(F1 − F2)
α3, (3.15)
α2 =
1
ϵ1q
(F0 − F1)
(F2 − F1)
α0 =
1
ϵ2ℓ2q
(F3 − F1)
(F2 − F1)
α3, (3.16)
α3 =
ϵ2
ℓ1
(F0 − F2)
(F3 − F2)
α0 =
ϵ2ℓ2
ϵ1
(F0 − F1)
(F3 − F1)
α0. (3.17)
In view of (3.9), we then get
α1 = ϵ2
b(1− q)
(p+ q)
α3, α2 = ϵ1
a(1− p)
(p+ q)
α0, (3.18)
from which we infer
ϵ1 = ϵ2 = 1. (3.19)
It then follows that
n = ℓ1ℓ2 − 1, v2 = ℓ1v1 − v0, v3 = ℓ2v2 − v1 = nv1 − ℓ2v0, (3.20)
so that v0 ∧ v3 = nv0 ∧ v1, hence
n = det (v0, v3),
since the pair (v0, v1) is a basis of the lattice Λ. From (3.15)–(3.17) and (3.9), we easily infer
that the integers ℓ1, ℓ2 can be rewritten as
ℓ1 =
(p+ q − aq(1− p))
bp(1− q)
α0
α3
=
(
1 +
aq(1− p)
p+ q
n
)
α0
α3
, (3.21)
About a Family of ALF Instantons with Conical Singularities 13
ℓ2 =
(p+ q − bp(1− q))
aq(1− p)
α3
α0
=
(
1 +
bp(1− q)
p+ q
n
)
α3
α0
. (3.22)
From (3.18) and (3.19), the conical parameters α1, α2 are given by
α1 =
b(1− q)
p+ q
α3, α2 =
a(1− p)
p+ q
α0, (3.23)
while the relations (3.5), (3.6), (3.11) and (3.12) are expressed by
ℓ1pα1 + qα2 = α0, (3.24)
pα1 + ℓ2qα2 = α3, (3.25)
−ℓ1ap(1− p)α1 + (p+ q − aq(1− p))α2 = 0, (3.26)
(p+ q − bp(1− q))α1 − ℓ2bq(1− q)α2 = 0. (3.27)
By using the expressions of α1, α2 given by (3.23), it is easily checked that these relations are
all satisfied.
So far, we assumed that pq ̸= 0. In view of (3.3), the cases when p = 0, q > 0 or q = 0, p > 0
are then obtained by continuity. When p tends to 0, then q > 0, since p+ q > 0, and
ℓ1 = (1 + n)
α0
α3
, ℓ2 =
α3
α0
, α1 =
b(1− q)
q
α3, α2 =
1
q
α0,
and
n =
(1 + q)
b(1− q)
− q
b(1− q)
lim
p→0
(a− 1)
p
− 1.
Similarly, when q tends to 0, then p > 0 and
ℓ1 =
α0
α3
, ℓ2 = (1 + n)
α3
α0
, α1 =
1
p
α3, α2 =
a(1− p)
p
α0
and
n =
(1 + p)
a(1− p)
− p
a(1− p)
lim
q→0
(b− 1)
q
− 1.
Recall that a metric of the Chen–Teo 4-parameter family is said to be regular if it can be
smoothly compactified along the invariant divisors, Di, encoded by the edges Ei of the momen-
tum polytope, i = 0, . . . , r, with a suitable choice of conical singularities of angles 2παi along
each Di. In view of the above, this happens if and only if we can choose α0, α1, α2, α3, all pos-
itive, satisfying the conditions (3.24)–(3.27), hence, equivalently, the conditions (3.21)–(3.23),
in fact (3.21)–(3.22) only, since α1 and α2 are then be defined by (3.23). We first observe that,
in this case, it follows from (3.23) that α1 and α2 are completely determined by α0 and α3,
as p + q > 0; moreover, only the quotient α0/α3 is relevant, so that we can arrange that, say,
α0 = 1. This being understood, we can formulate the following statement:
Theorem 3.3. Let (M, g) be an element of the 4-parameter Chen–Teo family, of parameter p,
q, a, b; let n be the total NUT-charge of g:
(1) (M, g) is regular (that is has a smooth compactification on which it has at worst conical
singularities) if and only if n is an integer.
(2) If (M, g) is regular, then the boundary at infinity is diffeomorphic to L, where L is
14 O. Biquard and P. Gauduchon
(i) a lens space of type ℓ/n, where ℓ is a factor of n+ 1, if n ̸= −1;
(ii) the sphere S3, if n = −1;
(iii) S1 × S2, if n = 0.
Proof. (i) We already know that n is an integer whenever g is regular. For the converse, in
view of the above, we simply have to show that if n is an integer there always exist α0, α3
positive, in fact only α3 > 0 if we choose α0 = 1, satisfying the conditions (3.21)–(3.22), where
ℓ1, ℓ2 is some pair of integers such that n = ℓ1ℓ2 − 1. From (3.14), we infer that(
1 +
aq(1− p)
p+ q
n
)(
1 +
bp(1− q)
p+ q
n
)
= ℓ1ℓ2 = n+ 1, (3.28)(
1 +
aq(1− p)
p+ q
n
)
=
p+ q − aq(1− p)
bp(1− q)
, (3.29)
and (
1 +
bp(1− q)
p+ q
n
)
=
p+ q − bp(1− q)
aq(1− p)
. (3.30)
If n ≥ 0, hence ℓ1ℓ2 > 0, it follows from (3.28) that either
(
1+ aq(1−p)
p+q n
)
and
(
1+ bp(1−q)
p+q n
)
are
both positive or both negative; the second case is in fact excluded, due to the constraints on the
parameters p, q, a, b: indeed, since n ≥ 0, if p > 0, q > 0, then
(
1+ aq(1−p)
p+q n
)
and
(
1+ bp(1−q)
p+q n
)
are clearly both positive, and this is still the case if p ≥ 0, q < 0, because of (3.29), or if p < 0,
q ≥ 0, because of (3.30). Thus, ℓ1, ℓ2 are both positive, and α3 is then defined by (3.21)–(3.22).
If n < −1, hence ℓ1ℓ2 < 0, then either
(
1 + aq(1−p)
p+q n
)
> 0 and
(
1 + bp(1−q)
p+q n
)
< 0 or vice
versa. In the former case, we can choose ℓ1 > 0, ℓ2 < 0, in the latter case, chose instead ℓ1 < 0,
ℓ2 > 0, and, in both cases, define α3 by α3 =
p+q−aq(1−p)
ℓ1bp(1−q) = ℓ2aq(1−p)
p+q−bp(1−q) .
If n = −1, then ℓ1ℓ2 = 0, so that either ℓ1 = 0 or ℓ2 = 0 or both. In the former case, we can
define α3 =
ℓ2(p+q)
p+q−bp(1−q) , where ℓ2 may be any integer of the sign of p+q−bp(1−q), and likewise
if ℓ2 = 0. The most interesting case is when ℓ1 = ℓ2 = 0, i.e., when p+q = aq(1−p) = bp(1−q);
then, we can choose α3 = α0 = 1, a = p+q
q(1−p) , b =
p+q
p(1−q) , α1 =
1
p , α2 =
1
q .
(ii) If (M, g) is regular, we know by (3.20) that v3 = nv1−ℓ2v0. According to Proposition 4.1
in [3], the function z + iρ identifies the interior of the moment polytope P , equipped with the
complex structure induced by g, with the Poincaré upper half-plane. At infinity, the topology
of (M, g) is then R × L, where L is obtained, from the product [0, 1] × T2, by identifying the
circle {0} × S1 with the circle {1} × S1 via the rotation 2iπ ℓ2
n , where {0} × S1 encodes the
orbit of v0, around E0, and {1} × S1 the orbit of v3 = nv1 − ℓ2v0, around E3, cf. [14, 16] and
Figure 3. ■
3.2 The case when the metric is smooth
If α0 = α1 = α2 = α3 = 1, i.e., if (M, g) is a gravitational instanton, the system (3.24)–(3.27)
becomes
ℓ1p+ q = 1, p+ ℓ2q = 1, (3.31)
p+ q − a(1− p) = 0, p+ q − b(1− q) = 0. (3.32)
By (3.31), p+ q = 1− (ℓ1 − 1)p = 1− (ℓ2 − 1)q. From (3.32), we then infer
a− 1
p
=
2− ℓ1
1− p
,
b− 1
q
=
2− ℓ2
1− q
.
About a Family of ALF Instantons with Conical Singularities 15
R ≃ ξ → ∞
Dv0 Dv3
ℓ/n
−R R
Figure 3. Lens space at infinity, obtained “by attaching two solid tori S1 × D2 together by a diffeo-
morphism S1 × ∂D2 → S1 × ∂D2 sending a meridian {x} × ∂D2 to a circle of slope ℓ/n”, cf. [14]. The
disk Dv0 , resp. Dv3 , is formed by the orbits of the Killing vector field v0, resp. v3. The red half-circle is
the hyperbolic geodesic in the Poincaré upper half-plane relating −R to R on the real axis, and R tends
to infinity.
It follows that ℓ1 < 2 and ℓ2 < 2. From (3.31), we infer that ℓ1p = 1−q > 0 and ℓ2q = 1−p > 0.
If p > 0 and q > 0, hence ℓ1 = ℓ2 = 1, we thus get n = 0, p + q = 1, a = 1
q , b = 1
p , which
characterizes the Chen–Teo instanton, cf. (2.18)–(2.20). If p > 0 and q < 0, then ℓ1 > 0, hence
ℓ1 = 1, so that p + q = 1 and q = ℓ2q, which is impossible, since ℓ2q > 0. Similarly, we cannot
have p < 0 and q > 0. We thus recover the fact, already established in [3, Section 7], that
the only toric Hermitian ALF gravitational instantons with 3 angular points are the Chen–Teo
gravitational instantons.
3.3 Some particular cases in the general ALF case
If α0 = α3 = 1, then, by (3.24)–(3.25), we get
ℓ1pα1 + qα2 = 1, pα1 + ℓ2qα2 = 1,
while, by (3.23), we have
a =
(p+ q)α2
1− p
, b =
(p+ q)α1
1− q
.
From (3.21)–(3.22), we also infer
ℓ1 =
1− qα2
pα1
= 1 + qα2n, ℓ2 =
1− pα1
qα2
= 1 + pα1n,
n =
ℓ1 − 1
qα2
=
ℓ2 − 1
pα1
=
1− pα1 − qα2
pqα1α2
,
and
a− 1
p
=
1− ℓ1α1 − α2
1− p
,
b− 1
q
=
1 + α1 − ℓ2α2
1− q
.
Particular case 3.4. We first consider the case when, in addition to α0 = α3 = 1, we suppose
that α2 = 1, and we then put: α1 = α (similar developments can be done, by simply swapping p
and q if we suppose instead that α1 = 1 and α2 = α). We then have
ℓ1pα+ q = 1, pα+ ℓ2q = 1, (3.33)
16 O. Biquard and P. Gauduchon
and
a =
p+ q
1− p
, b =
(p+ q)α
1− q
,
ℓ1 = 1 + qn =
1− q
pα
, ℓ2 = 1 + pαn =
1− pα
q
,
n =
ℓ1 − 1
q
=
ℓ2 − 1
pα
=
1− pα− q
pqα
,
and
a− 1
p
=
2− ℓ1α
1− p
,
b− 1
q
=
1 + α− ℓ2
1− q
.
Interesting 1-parameter families are obtained by taking q = 0, from which we infer: p > 0,
hence 1
2 < p < 1 – since we have then a > 1; from (3.33) we also get pα = 1, hence 1 < α < 2,
and ℓ1 = 1, hence n = ℓ2 − 1; we also infer: a = p
1−p , b = 1, a−1
p = 2−α
1−p and b−1
q = 1 + α − ℓ2.
For any n = ℓ2 − 1, we thus get a 1-parameter family of regular metrics parametrised either
by p ∈ (12 , 1) or, equivalently, by the angle 2πα ∈ (2π, 4π).
When p tends to 1, i.e., when α tends to 1, for any ℓ2, a tends to +∞, b = 1 and
√
b−1
q
tends to 2−ℓ2
2 ; in view of Remark 3.1, the metric then tends to the metric encoded by the affine
piecewise function
fα=1(z) = 1− (2− ℓ2)
4
+
1
2
∣∣∣z + (2− ℓ2)
4
∣∣∣+ 1
2
∣∣∣z − (2− ℓ2)
4
∣∣∣.
When p tends to 1
2 , i.e., when α tends to 2, for any ℓ2, a = 1, implying that z1 = z2
and
√
a−1
p = 0, and
√
b−1
q tends to 3−ℓ2
2 . In view of Remark 3.1 again, the metric then tends to
the metric encoded by the piecewise affine function
fα=2(z) = 1− (3− ℓ2)
4
+
1
2
∣∣∣z + (3− ℓ2)
4
∣∣∣+ 1
2
∣∣∣z − (3− ℓ2)
4
∣∣∣.
This limit when the angle goes to 4π corresponds to the process described in [3, Section 9] where
the S2 with the conical singularity disappears at the limit 4π with a bubble which should be
the 2-cover of the self-dual Eguchi–Hanson metric (see the family of Section 2.3 for an example
of this phenomenon).
By successively considering the particular cases when ℓ2 = 2, n = 1, ℓ2 = 0, n = −1, ℓ2 = −1,
n = −2 and the AF case ℓ2 = 1, n = 0, we thus get the following 1-parameter families.
(i) ℓ2 = 2, n = 1: fα=1(z) = 1 + |z|, which encodes the self-dual Taub-NUT gravitational
instanton, and fα=2(z) = 3
4 + 1
2 |z + 1
4 | +
1
2 |z − 1
4 |, which encodes the positive Taub-bolt
metric.
(ii) ℓ2 = 0, n = −1: fα=1(z) = 1
2 + 1
2 |z + 1
2 | +
1
2 |z − 1
2 |, which encodes the Schwarzschild
gravitational instanton, and fα=2(z) = 1
4 +
1
2 |z+
3
4 |+
1
2 |z−
3
4 |, which encodes the negative
Taub-bolt metric.
(iii) ℓ2 = −1, n = −2: fα=1(z) = 1
4 +
1
2 |z+
3
4 |+
1
2 |z−
3
4 |, which encodes the negative Taub-bolt
gravitational instanton, and fα=2(z) = 1
2 |z + 1| + 1
2 |z − 1|, which encodes the self-dual
Eguchi–Hanson metric.
(iv) ℓ2 = 1, n = 0 (this is an AF case): fα=1(z) = 3
4 + 1
2 |z +
1
4 |+
1
2 |z −
1
4 |, which encodes the
positive Taub-bolt gravitational instanton, and fα=2(z) = 1
2 + 1
2 |z +
1
2 |+
1
2 |z −
1
2 |, which
encodes the Schwarzschild metric.
About a Family of ALF Instantons with Conical Singularities 17
3.4 The AF case
As mentioned above, the normalized total NUT-charge n is equal to zero if and only if the metric
is AF. We then have ℓ1ℓ2 = 1, while it follows from (3.21)–(3.22) that ℓ1 = ℓ−1
2 = α0
α3
. Since ℓ1
and ℓ2 are both positive integers, we eventually infer that
ℓ1 = ℓ2 = 1,
so that
α0 = α3.
The conditions (3.24)–(3.27) then become
pα1 + qα2 = α0 = α3,
−ap(1− p)α1 + (p+ q − aq(1− p))α2 = 0,
(p+ q − bp(1− q))α1 − bq(1− q)α2 = 0.
Without loss of generality, we can suppose that α0 = α3 = 1. We thus get
pα1 + qα2 = 1,
as well as
a =
(p+ q)
1− p
α2, b =
(p+ q)
1− q
α1.
Particular case 3.5. An interesting case is when α1 = α2 =: α > 0. This happens if and
only if
a =
1
1− p
, b =
1
1− q
,
and then
α =
1
p+ q
.
If p + q = 1, i.e., if α tends to 1, we thus recover the Chen–Teo instanton. If, however, p + q
tends to 2, i.e., if both p and q tend to 1, then α tends to 1
2 , while the normalised piecewise
affine function, cf. Remark 3.1, tends to f(z) = 1 + |z|; we thus obtain a quotient by Z/2Z of
the self-dual Taub-NUT space. Finally, if p + q tends to 0, i.e., p and q both tend to 0 and α
then tends to +∞, then a and b both tend to 1,
√
a−1
p =
√
b−1
q = 1
2 , and the piecewise affine
function tends to fEH(z) = 1
2 |z + 1
2 | +
1
2 |z − 1
2 |, which encodes the self-dual Eguchi–Hanson
metric, cf. Section 2.3.
Particular case 3.6. An interesting case is with α0 = α2 = α3 = 1, and we then put: α1 =: α.
We thus get
a =
p+ q
1− p
, b =
p+ q
p
and α =
1− q
p
.
Interesting 1-parameter families are obtained by fixing the parameter q (this actually amounts
to fixing the asymptotic behavior of the metric). Then 1−q
2 < p < 1 (the first inequality coming
from a > 1). We get a family of AF examples which are smooth except for one conical singularity
along a S2 with angle 2πα ∈ (2π(1 − q), 4π); q being fixed, this family is parametrised either
18 O. Biquard and P. Gauduchon
by p, or by α, or, better by τ := α − 1, so that −q < τ < 1. In view of Remark 2.1, for each
value of τ it is easy to check that the corresponding metric is encoded by the following piecewise
affine function
f q,τ (z) =
(1 + qτ)
1
2
2q(1− q)
(
(1 + qτ)
1
2 − q(q + τ)
1
2 − (1− q)
3
2
)
+
(q + τ)
2(1 + τ)
∣∣∣∣z + (1− τ2)
(q + τ)
1
2
(
(1 + qτ)
1
2 + (q + τ)
1
2
)∣∣∣∣+ (1 + qτ)
2(1 + τ)
|z|
+
(1− q)
2
∣∣∣∣z − (1 + τ)
(1− q)
1
2
(
(1 + qτ)
1
2 + (1− q)
1
2
)∣∣∣∣.
When τ = 0, i.e., α = 1 and p = 1− q, the corresponding metric is smooth: it is the Chen–Teo
gravitational instanton of parameter q, p = 1− q, whose piecewise affine function is
f τ=0(z) =
1
2pq
(
1− p
3
2 − q
3
2
)
+
q
2
∣∣∣∣z + 1
q
1
2 (1 + q
1
2 )
∣∣∣∣+ 1
2
|z|+ p
2
∣∣∣∣z − 1
p
1
2 (1 + p
1
2 )
∣∣∣∣.
If τ = −q, i.e., α = 1− q, we get
f τ=−q(z) =
(1 + q)
1
2
2q
(
(1 + q)
1
2 − (1− q)
)
+
(1− q)
2
|z|+ (1− q)
2
∣∣∣∣z − 1
1 + (1 + q)
1
2
∣∣∣∣.
If τ = 1, i.e., α = 2, we get
f τ=1(z) =
(1 + q)
1
2
2q
(
(1 + q)
1
2 − (1− q)
1
2
)
+
(1 + q)
2
|z|
+
(1− q)
2
∣∣∣∣z − 2
(1− q)
1
2
(
(1 + q)
1
2 + (1− q)
1
2
)∣∣∣∣.
The limit for the angle 4π is again obtained by blowing down the S2 to a point and is a Kerr
metric. The limit for the angle 2π(1− q) is a Kerr–Taub-bolt metric with a conical singularity:
it changes the topology at infinity because there is a bubble at infinity. The special case q = 0
was already studied in Section 3.3, case (iv).
Acknowledgements
We thank the referees for their careful reading of the article.
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1 Introduction
2 Toric Hermitian ALF gravitational instantons: a quick review
2.1 General presentation
2.2 The Kähler environment
2.3 The self-dual Eguchi–Hanson metric
3 The Chen–Teo family
3.1 Regularity
3.2 The case when the metric is smooth
3.3 Some particular cases in the general ALF case
3.4 The AF case
References
|
| id | nasplib_isofts_kiev_ua-123456789-212005 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T07:02:01Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Biquard, Olivier Gauduchon, Paul 2026-01-22T09:14:47Z 2023 About a Family of ALF Instantons with Conical Singularities. Olivier Biquard and Paul Gauduchon. SIGMA 19 (2023), 079, 19 pages 1815-0659 2020 Mathematics Subject Classification: 53C25; 53C55 arXiv:2306.11110 https://nasplib.isofts.kiev.ua/handle/123456789/212005 https://doi.org/10.3842/SIGMA.2023.079 We apply the techniques developed in our previous article to describe some interesting families of ALF gravitational instantons with conical singularities. In particular, we completely understand the 5-dimensional family of Chen-Teo metrics and prove that only 4-dimensional subfamilies can be smoothly compactified so that the metric has conical singularities. We thank the referees for their careful reading of the article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications About a Family of ALF Instantons with Conical Singularities Article published earlier |
| spellingShingle | About a Family of ALF Instantons with Conical Singularities Biquard, Olivier Gauduchon, Paul |
| title | About a Family of ALF Instantons with Conical Singularities |
| title_full | About a Family of ALF Instantons with Conical Singularities |
| title_fullStr | About a Family of ALF Instantons with Conical Singularities |
| title_full_unstemmed | About a Family of ALF Instantons with Conical Singularities |
| title_short | About a Family of ALF Instantons with Conical Singularities |
| title_sort | about a family of alf instantons with conical singularities |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212005 |
| work_keys_str_mv | AT biquardolivier aboutafamilyofalfinstantonswithconicalsingularities AT gauduchonpaul aboutafamilyofalfinstantonswithconicalsingularities |