Families of degenerating Poincar\'e-Einstein metrics on $\mathbb{R}^4$

We provide the first example of continuous families of Poincar\'e-Einstein metrics developing cusps on the trivial topology $\mathbb{R}^4$. We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Pleba\'nski-Demia\'nski. We additionally indicate how to construct similar examples on more complicated topologies.


Introduction
An Einstein metric satisfies for some real number Λ: Ric(g) = Λg.
(1) This is a central equation in Geometry and in several instances of Physics, especially in dimension 4. A Poincaré-Einstein metric is a noncompact Einstein metric with a specific asymptotic behavior giving rise to a conformal boundary metric at infinity, the simplest example being the Poincaré model for hyperbolic space whose conformal infinity is the round sphere.Poincaré-Einstein metrics were first notably used to construct a number of conformal invariants of the boundary geometry; see [FG85,FG12].More recently, they have also played an important role in the physics literature in relationship with AdS/CFT correspondence; see [Wit98,Biq05].From several perspectives, dimension 4 is a threshold dimension in topology and geometry.In this dimension, there are three ways for compact Einstein or Poincaré-Einstein metrics on a given manifold to degenerate: orbifold singularity formation, collapsing and cusp formation.
Orbifold formation has been widely studied and is now reasonably understood.Numerous examples of curves of such degenerations have been produced in the Kähler and Poincaré-Einstein settings, see [LS94,Biq13,Biq16].All such degenerations have moreover been reconstructed by gluing-perturbation [Ozu19a,Ozu19b].
Despite deep general results such as [CT06], the collapsing and cusp formation remain comparatively mysterious.The collapsing situation has received a lot of attention and many examples of curves of Einstein metrics collapsing have been produced on K3 surfaces, see for instance [HSVZ21,Fos19].The third situation of cusp formation has however never been observed except from "trivial" examples of (warped) products of degenerating surfaces and from sequences of metrics requiring infinitely many different topologies [And06,Bam12].More concretely, the following question was left open:

Question 0.1 ([And05]). "Another interesting open question is whether cusps can actually form within a given or fixed component of [the moduli space of Poincaré-Einstein metrics], on a fixed manifold M ."
A simple but not so appealing example showing that this exists is the so-called topological black hole metric.The metric is V (r) −1 dr 2 + V (r)dθ 2 + r 2 g N for V (r) := −1 + r 2 − 2m/r 2 with m large enough and g N the metric of a hyperbolic surface.Letting g N degenerate creates a cusp that extends to the conformal infinity.This naive example answers Anderson's question but, to the authors' knowledge, does not seem to have been mentioned before.This is still a 2-dimensional behavior and we provide many more interesting examples here.
Another intriguing question is whether cusp formation requires some topology -like orbifold degeneration requires nontrivial 2-homology.Anderson conjectured that it was the case: Question 0.2 ([And05])."It would also be very interesting to know if the possible formation of cusps is restricted by the topology of the ambient manifold M .[...] One might conjecture for instance that on the 4-ball cusp formation is not possible." We instead provide explicit examples of continuous families of smooth Poincaré-Einstein metrics on R 4 developing different kinds of cusps.We moreover find curves of metrics without any degeneration in the bulk but forming various conical, cusp or naked singularities in their conformal infinity.

Debever and Plebański-Demiański's local family of metrics
In this article, we study families of Poincaré-Einstein metrics exhibiting the above three types of degenerations focusing on the least understood case of cusp formation.These examples are surprisingly explicitly given in coordinates and are found in the families of Einstein metrics whose Lorentzian counterparts were discovered by Debever [Deb71] and which were given in more convenient coordinates by Plebański-Demiański [PD76].These metrics are known in the physics literature as Plebański-Demiański metrics (PD metrics).PD metrics are algebraically special of Petrov type D meaning (in the Riemannian setting) that at every point the selfdual and anti-selfdual parts of the Weyl curvature have repeated eigenvalues.This also equivalent to the ambiKähler condition of [ACG16]: the metric is conformally Kähler or Hermitian in both orientations.This curvature condition forces toric symmetry by [Gol94].
The metrics of the PD family have a remarkably compact form (2) and depend solely on two related quartic polynomials P and Q of one variable.Still, despite their simplicity and their discovery in the early 70's, these explicit metrics, once extended to the Riemannian setting contain in some limits most known examples of Einstein metrics (S 4 , S 2 × S 2 , Fubini-Study, Page's metric, Taub-NUT, Taub-Bolt, Eguchi-Hanson, Schwarzschild, Kerr and their AdS counterparts...) that were often discovered much later with complicated ansatz, see [LP81] where smooth Ricci-flat and compact Einstein PD metrics are classified.Extensions of these families more generally solve the Einstein-Maxwell equations and include known metrics such as Lebrun's scalar-flat metrics [LeB88].
This family also contains families developing orbifold singularities in the so-called AdS-Taub-Bolt family.It moreover contains continuous families of metrics exhibiting global collapsing bubbling out (Ricci-flat) Taub-NUT or Schwarzschild metrics in the so-called AdS-Taub-NUT (or Pedersen's) metrics or AdS-Schwarzschild families.We will focus on cusp formation here.

Degeneration in the family of AdS C-metrics
It is now classical in the physics literature that a limit "without rotation or twisting" of the PD metrics leads to the well-known AdS C-metrics whose Ricci-flat versions were found by Levi-Civita [LC18] and Weyl [Wey17] in the 1910's(!).In this family, we first find a 2-dimensional moduli space of smooth Poincaré-Einstein metrics on R 4 containing the hyperbolic 4-metric and whose limiting behaviors include metrics forming one or two cusps.A significant asymptotic quantity of Poincaré-Einstein metrics is the renormalized volume defined in [Gra00].Despite the drastic degenerations presented in this article, the renormalized volume stays bounded.

Theorem 0.3 (Section 2).
There exists a smooth family of smooth Poincaré-Einstein metrics on R 4 parametrized by an open region Ω in R 2 .Approaching some points at the boundary ∂Ω, the metrics converge smoothly to the hyperbolic space or degenerate forming one or two codimension 2 cusps.These cusps have asymptotic behaviors : for a, b > 0 in the bulk of the manifold, and dr 2 + ae −r dθ 2 1 + bdθ 2 2 at conformal infinity with r ∈ [0, +∞).These examples have uniformly bounded renormalized volume.

An important question left open is the following one.
Question 0.4.Does there exist a continuous family of Poincaré-Einstein metrics forming cusps separating the manifold into a complete finite volume piece and another complete Poincaré-Einstein metric?Remark 0.5.Unfortunately, this is impossible in our family of metrics and there is little hope to find such a family of metrics explicitly given in coordinates.Indeed, in our case, one limit of such a degeneration has to be an Einstein metric with negative Ricci curvature and with at least one Killing vector field with finite length, which is impossible by Bochner's formula; see [Yor84] for instance.

Degeneration in the Carter-Plebański family of metrics
The limits "without acceleration" of the PD metrics consitute the Carter-Plebański family of metrics.In Section 3, we exhibit a subfamily of smooth Poincaré-Einstein metrics with topology CP 2 \D 4 forming cusp in some limits, and discuss how other topologies may be reached.

The families of metrics considered 1.Plebański-Demiański family of metrics
A "Euclideanized" Plebański-Demiański (PD) metric has the following form where Q(y) and P (x) are polynomials of degree 4 which can be chosen depending on the value of a ∈ R, physically understood as a rotation parameter, so that g P D is an Einstein metric with Ric g P D = −3g P D following the (Riemannian version of the) computations in [PD76].Up to rescaling, we can assume a ∈ {0, 1}.
Let us first consider the larger family with a = 1 from which the other ones can be obtained from various limiting procedures.The Einstein condition (1) with Λ = −3 is equivalent to P and Q having the form for b, c, d, e ∈ R where we note the identity Q(y) = P (y) + y 4 − 1.The local metric (2) is then Einstein and Riemannian on ranges depending on roots of P and Q.When "closing-up" at roots of P and Q it may have codimension 2 cone-edge singularities (which we will avoid) or cusp ends which are discussed in Appendix A. These metrics are moreover Poincaré-Einstein since they are conformal to a metric with boundary: the boundary is given by {x = y} and the conformal factor is The conformal infinity of these metrics is the conformal class of the metric induced on {x = y} by (x − y) 2 g P D .We will see in different instances, especially in Section 4 that these conformal infinities may degenerate.The possible degenerations of the conformal infinity are collected in the Appendix B.
Without loss of generality, we can write . The pointwise norm of the Riemannian tensor of g P D is given by The volume element in these coordinates is −1+x 2 y 2 (x−y) 4 dxdydϕdψ and one checks that W g P D L 2 (g P D ) is finite for the domains we consider, hence, by [And01], the renormalized volume is controlled for our examples.

Non rotating limit: AdS C-metrics
For this section, we will follow [CLT15a] and will adopt their notation.Our study and goals are purely geometric and differ from theirs.The AdS C-metrics are obtained from the general Plebański-Demiański family (2) by taking the non-rotating limit a → 0. These metrics are the Riemannian analogues of the metrics considered in [CLT15b], and have the form where we will assume that Q and P are parametrized by two variables µ, ν as and This ensures Einstein condition (1) is satisfied.The pointwise norm of the Riemannian tensor of g C is given by and more precisely, one has Ric(g C ) = −3g C and the eigenvalues of both the selfdual and anti-selfdual parts of W g C are equal to µ 4 (y − x) 3 (2, −1, −1) which as expected go to zero as x → y.Moreover, when µ = 0, the metric is locally hyperbolic.A direct computation ensures again that W g C 2 L 2 (g C ) is bounded.In particular, from [And01], these examples have bounded renormalized volume.

Non accelerating limit: Carter-Plebański metrics
The Carter-Plebański family of metrics is a special limit of the Plebański-Demiański family of metrics (2) after a change of coordinates.To do this, start from (2) in the coordinates of [LP81] and perform a rescaling by b > 0 (acceleration parameter) of coordinates as in [GP06, Section 2.2], which yields the following metric: for polynomials P b and Q b depending on b > 0 chosen to satisfy (1) with Λ = −3.Taking the "no acceleration limit" b → 0 as in [GP06, Section 5], we obtain from (6) the metric where the limiting polynomials P and Q are of the form: following the notations of [MP13] for some real numbers E, M , N and α.We will consider intervals where P(p) 0 and Q(q) 0. This time, the range in p will be compact of the form [p − , p + ] for p ± roots of P and the range in q will be of the form [q + , +∞) for q + root of Q.

Degenerations of AdS C-metrics
In this section, we study a specific 2-dimensional family of AdS C-metrics on R 4 forming one or two cusps in different limits.The cusps forming here effectively separate the manifold into two or three Poincaré-Einstein metrics with cusps ends in their bulk and their conformal infinities.We prove Theorem 0.3.
As in Section 1.2 we consider the metric (4) where Q(y) = y 1 + ν + (µ + ν)y + µy 2 and P (x) = (1 + x) 1 + νx + µx 2 .The roots of P and Q respectively are as follows , and In order to approach metrics with cusp ends in this family by smooth metrics, we consider the case when x ± , y ± are complex conjugate roots which we will let approach a real double root -leading to a cusp degeneration by Section A.4.In the (µ, ν) plane, this condition means that (µ, ν) lies in the region bounded by the curves We then consider −1 < x < y < 0 where the conformal infinity is at {x = y}, see Figure 1b.For the metric to be smooth, we require that 1−ν+µ 2 ϕ and 1+ν 2 ψ be 2π-periodic, see Proposition A.3.We further impose that µ > max(ν/2, −ν).This corresponds to forcing the real part of x ± and y ± to be in (−1, 0), this way the double root degeneration (when the imaginary part of the roots tends to zero) happens where the metric is defined and is geometrically meaningful.We end up with the region D4 in [CLT15a] shaded in Figure 1a  Remark 2.1.In the limit (µ, ν) → 0 from our region shaded in Figure 1a, our metrics converge smoothly to the hyperbolic 4-space.Indeed, the metric is already locally hyperbolic by our curvature computations and the change of variables x = − sin 2 ((u − π)/2) , at the conformal infinity {x = y}, the restriction of the metric (x − y) 2 g CLT with µ = ν = 0 takes the form Thus we recover the metric of the round 3-sphere in Hopf's coordinates since ϕ and ψ are 4π-periodic.This in particular ensures that the topology we consider is R 4 .
From (10) we see that, for (µ, ν) in the shaded region in Figure 1a, if one of P, Q has a double root, then (µ, ν) lies on at least one of the boundary curves ν = 2 √ µ or ν = µ − 2 √ µ respectively in blue and red in Figure 1, see the first two columns of Figure 2 for the associated polynomials and geometric representation.The intersection of these curves, (µ, ν) = (16, 8), is the unique case when P and Q have double roots at x = −1 4 and y = −3 4 ) as described in Figures 2c and 2f respectively leading to two cusps dividing the manifold in three regions, while the point (µ, ν) = (0, 0) corresponds to hyperbolic 4-space from Remark 2.1.The possible double roots of P and Q respectively lie in the intervals (−1, −1 4 ] and [ −3 4 , 0).

Degenerations in Carter-Plebański family of metrics
In this section, we indicate how to find families of metrics forming cusps with different topologies.We take the simplest example here on CP 2 \D 4 with conformal infinity S 3 .We follow [MP13, Sections 2.1, 2.2 and 2.3] for our regularity conditions: we impose τ and σ to be as [MP13, Sections 2.13 and 2.17].This also requires N = M , which is equivalent to the metric being self-dual, and forcing P = Q.
We will moreover parametrize our polynomial by the roots and looking for a metric with a cusp, we will consider a polynomial with a double root: for p 3 , p 4 , p 0 ∈ R (following notations of [MP13]): We will then consider the range (p, q) ∈ [p 3 , p 4 ] × [p 4 , +∞], where the associated metric is indeed Riemannian.
Remark 3.1.Recall that from Remark 0.5 we cannot have a double root in Q on (p 4 , +∞).All we will find instead is a double root of P on (p 3 , p 4 ) corresponding to a cusp in the manifold extending to infinity.
We need our double root p 0 , to lie in (p 3 , p 4 ) so that it is reflected in our metric.Since the sum of the roots is 0 (the cubic coefficient of the polynomial is zero), p 0 = − p3+p4 2 and so p 0 ∈ (p 3 , p 4 ) imposes p 3 < 0 < p 4 , and We can find this polynomial (11) as a limit of polynomials with two complex conjugate roots: for 0  where we get the double root mentioned above when → 0 and we also let Q = P to satisfy the above regularity condition of [MP13].Since the roots of the polynomials are the same, the intervals in which these are defined stay the same.Geometrically, in the limit → 0, the metrics (7) associated to Q = P develop a cusp along {p = −(p 3 + p 4 )/2} separating the manifold in two parts by an argument similar to Section A.3.
The topology of the manifold is that of CP 2 minus a ball and the conformal infinity is S 3 .The "bolt" of the metric is reached at [p 3 , p 4 ] × {q = 1} which is a codimension 2 submanifold (a 2-sphere) because of the degeneration of the metric there, see [MP13] or the discussion in Section A.

Remark 3.2. It is likely possible to obtain infinitely many different topologies from the Carter-Plebański family
of metrics by having a larger and larger "self-intersection" for the 2-sphere while obtaining a conformal infinity S 3 /Z k for Z k a cyclic subgroup of SU (2) acting freely on S 3 .See [CT10, Section 5.1] for a discussion of the regularity conditions and possible topologies.In the larger Plebański-Demiański family of metrics, we believe that there is also a large class of additional possible topologies, with two "bolts" (and a "NUT").The conformal infinity, could this time be an arbitrary lens space.See [CT10, Section 5.2] for a discussion of the regularity conditions and possible topologies.

Degenerations in the Plebański-Demiański family of metrics
We will now turn to the general PD family of metrics.The above degenerations of Sections 2 and 3 can be found in the full family of Plebański-Demiański, but we focus on exhibiting new behaviors of complete metrics whose conformal infinities develop unexpected types of singularities.We prove Theorem 0.6.
Remark 4.1.This is true on a larger range of values of α 1 which we do not attempt to describe.For instance, the graphs below show examples where the sign conditions for 2 to be Riemannian are satisfied for α 1 > 0, but the example α 1 = 0.01, α 2 = α 3 = 0 and α 4 = 7 does not yield the right sign.
Lastly, we assume that ϕ and ψ satisfy the periodicity conditions imposed in Proposition A.3 to ensure that we find smooth metrics.The metrics obtained in this way can be approached by perturbing the parameters α 2 and α 3 in various ways around (0, 0).This gives the following different types of degenerations degenerations, which we describe below.
Degeneration 1: from a smooth metric to a naked singularity.By taking α 3 > 0 and keeping α 2 = 0, the double root of P ∞ at 1 is replaced with two complex roots, see Figure 5a.Taking the limit α 3 → 0 yields the above naked singularity.Similarly, by taking α 2 < 0 and α 3 = 0, the double root of P ∞ is moved past the conformal infinity y = x, see Figure 5b.Taking the limit α 2 → 0 yields the above naked singularity.
Both of these situations yield a smooth metric at conformal infinity by Proposition B.1.Indeed, P ∞ does not have any root close to the root of Q ∞ .Degeneration 2: from a conical singularity to a naked singularity.By taking α 3 < 0 and α 2 = 0, see Figure 6a, the double root of P ∞ at 1 is split in two real roots x − < 1 < x + .This changes the topology and creates a codimension 2 cone-edge singularity along {x = x − } by Lemma A.1, extending to the conformal infinity {x = y}.As α 3 → 0, the angle tends to zero and a naked singularity appears while the singularities in the bulk are "sent to infinity".By setting α 3 < 0 and α 2 = − √ −α 3 as in Figure 6b the double root of P ∞ at 1 is split into a single root at 1 and a root larger than 1.This gives a conical singularity in the metric at the conformal infinity only this time.As α 3 → 0, the angle tends to zero and a naked singularity appears in the limit.Degeneration 3: from cusp to naked singularity.By taking α 2 > 0 and α 3 = 0, the double root 1 − α 2 of P ∞ is moved to the left of the root in Q ∞ .This creates a cusp in the bulk metric as well as in its infinity by Sections A and B in the appendix.
As in Section 1.2, this cusp at {x = 1 − α 2 } separates the manifold in two regions infinitely far apart, and the conformal infinity in two finite volume manifolds with cusp ends.
When α 2 → 0, the volume of {1 − α 2 < x = y < 1} tends to zero and the region disappears, the metric on {α 1 < x = y < 1 − α 2 } has infinite diameter for α 2 > 0 but finite diameter in the limit α 2 → 0 (these remarks do not depend on the representative of the conformal class).This is a manifestation of cusp degenerations in the bulk manifold comparable to those of Section 1.2.Indeed, in the family of metrics obtained from (14), there is a 4-dimensional family of smooth Poincaré-Einstein metrics with a (3-dimensional) boundary constituted of metrics with one cusp separating the manifold in two set, and a 2-dimensional family with two cusps separating the manifold in three.

Remark 4.2.
There are important differences with Section 1.2.The cusps from (2) for a = 1 are "twisted" (see (16)) and do not look like mere products of surfaces in the limit.Moreover, as described above, as α 2 → 0 the cusps "escapes" to infinity creating the above unexpected naked singularity at infinity.This was impossible in the family (4) because the double root in P could not approach 0 and the double root in Q could not approach −1.

Two cusps at conformal infinity only
This time, we exhibit a metric with codimension 2 cusps ends at the conformal infinity only -in particular, the conformal infinity is not compact.Unlike the example of Section 2 these cusps do not cut the manifold in different pieces.Consider which are limit of the polynomials in (14) for α 1 = −1, α 2 = 0, α 3 = 0 and α 4 = 1.These polynomials have the desired signs on the region −1 x 1, −1 y 1 making the metric (2) with a = 1 Riemannian.Its infinity has two cusp ends at the points (−1, −1) and (1, 1) thanks to (21).This is a limiting case for all the previous degenerations as well as a limit of naked singularities at either 1 or −1.

B Possible behaviors of the conformal boundary metrics
As for the bulk metric, the conformal infinity {x = y} has different possible asymptotic behaviors close to roots of P or Q.We consider (2), whose conformal metric at infinity is: dx 2 − Q(x) 1 − a 2 x 4 (dψ − ax 2 dϕ) 2 + P (x) 1 − a 2 x 4 (dϕ − ax 2 dψ) 2 . (18) We will moreover assume that the regularity conditions for the bulk of Proposition (A.3) are satisfied whenever applicable.Simpler arguments than Sections A.1 and A.3 imply the following result.
Proposition B.1.Under the assumptions of Proposition A.3, the conformal metric is smooth.Moreover, if P (or Q) has a double root at x 0 , then the conformal boundary metric of (18) has a codimension 2 separating cusp as described in (16).
We will now focus on the case a = 1 of (2) and we will see that allowing the roots to be at ±1 leads to different degenerate behavior for the conformal infinity alone.
B.1 At ±1 a simple root of both P and Q: conical singularity Let us assume that 1 is a simple root of both P and Q for the metric (18).The case of −1 is treated similarly.As x → 1, we obtain that the metric (18) is asymptotic to where θ 1 (x) → dϕ − dψ, θ 2 (x) → dϕ + dψ as x → 1, and C 1 = 4

Figure 1 :
Figure 1: Parameter ranges considered.The dashed {x = y} is the conformal infinity.

Figure 2 :
Figure 2: Different configurations of double roots.

)
Example of double root in P with N = M .Intervals where P and Q are defined are highlighted.Double root in P (in blue) extending infinity q → +∞.Range in (p, q) shaded.

Figure 3 :
Figure 3: Polynomial and range of coordinates.
Figure 4: Example polynomials and region for metric with naked singularity at infinity.

Figure 5 :
Figure 5: Smooth Metric to Naked Singularity

Figure 7 :
Figure 7: Cusp to Naked Singularity Figure 8: Two cusps at conformal infinity only.