Global dynamics of asymptotically locally AdS spacetimes with negative mass

The Einstein vacuum equations in five dimensions with negative cosmological constant are studied in biaxial Bianchi IX symmetry. We show that if initial data of Eguchi-Hanson type, modelled after the four-dimensional Riemannian Eguchi-Hanson space, have negative mass, the future maximal development does not contain horizons, i.e. the complement of the causal past of null infinity is empty. In particular, perturbations of Eguchi-Hanson-AdS spacetimes within the biaxial Bianchi IX symmetry class cannot form horizons, suggesting that such spacetimes are potential candidates for a naked singularity to form. The proof relies on an extension principle proven for this system and a priori estimates following from the monotonicity of the Hawking mass.

The system (1.1) is of hyperbolic nature, and studying the dynamic evolution of initial data is very difficult in general, leading us to take recourse to settings with high degrees of symmetry. In particular, it is desirable to reduce the dimension of the dynamical problem to the simplest case of 1 + 1 dimensions. This approach has a longer history for Λ = 0. There, for = 4, the only symmetry group achieving the reduction to a 1 + 1-dimensional problem whilst consistent with the spacetime being asymptotically flat is spherical symmetry. However, the well-known Birkhoff theorem prevents any dynamical consideration since such a four-dimensional spacetime is necessarily static, embedding locally into a subset of a member of the Schwarzschild family.
To study spherically symmetric gravitational dynamics in four dimensions, one can follow the approach of the seminal work by Christodoulou and couple gravity to matter. In a sequence of papers -see his own survey article [Chr99] for references -he initiated the rigorous analysis of spherically symmetric gravitational collapse for Λ = 0 by studying the Einstein-scalar field system. The model of a real massless scalar field was chosen because, on the one hand, this matter model does not develop singularities in the absence of gravity and, on the other hand, its wave-like character resembles the character of general gravitational perturbations of Minkowski space. Christodoulou's work led to a complete understanding of weak and strong cosmic censorship for this model. His approach has later been extended to other matter models; see [Kom13] for a systematic overview and references.
Christodoulou's approach was adapted to the context of Λ < 0 by Holzegel and Smulevici in [HS11a] and by Holzegel and Warnick in [HW13], who show well-posedness of the Einstein-Klein-Gordon system with the scalar field satisfying various reflecting boundary conditions at infinity. The work [HS11b] shows stability of Schwarzschild-AdS in this symmetry class for Dirichlet boundary conditions. A recent breakthrough has been achieved by Moschidis in [Mos17a] and [Mos17b]; in his work, he shows instability of exact anti-de Sitter space as a solution to the Einstein-null dust system in spherical symmetry with an inner mirror.
Another possibility of evading the restrictions of Birkhoff's Theorem is to study (1.1) in higher dimensions. Working in five dimensions and imposing biaxial Bianchi IX symmetry, a symmetry corresponding to a subgroup of (4), still reduces the system to 1+1 dimensions and introduces a dynamical variable , not dissimilar to the scalar field in the coupled system. This model was introduced by Bizón, Chmaj and Schmidt. In [BCS05], they initiated the study of gravitational collapse for Λ = 0 in this symmetry class by numerical computations; investigations along those lines were continued in [DH06b] and [BCS06].
The study of this system in the realm of negative Λ has been initiated by . In [DH06a] -now mostly cited for the conjecture of the instability of exact AdS space -, our Corollary 1.11 has been put forward without rigorous proof. Back then, the problem of proving local well-posedness for the system in biaxial Bianchi IX symmetry was not solved, thus no extension principle sufficiently strong was available. The present paper can be seen as a completion of [DH06a], building on the insight into problems in asymptotically locally AdS spacetimes obtained over the past decade; for an overview of this work, see [HW13], [EK14] and references therein.
In five dimensions and for Λ < 0, (1.1) has many static solutions which are asymptotically locally AdS. A spacetime is asymptotically locally AdS if the asymptotics of the metric towards conformal infinity ℐ is modelled after AdS space, but ℐ need not be R × 3 topologically. Prominent examples of such static solutions are exact AdS 5 space with spherical conformal infinity 1 and the AdS soliton of [HM98b] with toric ℐ. Eguchi-Hanson-AdS spacetimes form another such family with ℐ ∼ = R × ( 3 /Z ) for ≥ 3. 2

Spaces of Eguchi-Hanson type
We introduce four-dimensional Riemannian manifolds modifying Eguchi-Hanson space to the asymptotically locally AdS context. These will serve as initial data for the five-dimensional Einstein vacuum equations via the local well-posedness Theorem 1.7. Our data also exhibit an (2) × (1) symmetry, thus giving rise to spacetimes with biaxial Bianchi IX symmetry. Then Eguchi-Hanson-AdS spacetimes form particular examples of the spacetimes thus obtained.
Definition 1.1. We say that an initial data set ( , , ) to (1.2) exhibiting ( for fixed > 0 and ≥ 3 satisfying where ( 1 , 2 , 3 ) is a basis of left-invariant one-forms on (2) (see below) and , , : ( , ∞) → R are smooth functions such that the following conditions hold: (i) Around the centre = , the functions satisfy the regularity conditions 1 Numerical studies within the biaxial Bianchi IX symmetry class for perturbations of AdS5 were carried out recently in [BR17]. 2 The space 3 /Z is defined in the usual way as the lense space ( , 1).
(iv) For an > , we have We require that where is the mean curvature of the symmetry orbits, is finite; we call the mass of ( , , ) at infinity.
In terms of the left-invariant one-forms, the Minkowski metric on R 5 is given by The Euler angles ( , , ) parametrise the 3-sphere away from the poles. By restricting to have period 4 / , we obtain coordinates on 3 /Z . 2. By identity (A.1), the notion of mass at infinity is consistent with the renormalised Hawking mass introduced in Definition 1.5.
Prima facie it seems as if had a singularity at = . However, one should compare this situation to that of spherical symmetry in spherical coordinates. This intuition is made more precise in the following to leading order. For fixed ( , ), the restriction on the range of (see Remark 1.2) guarantees that the metric can be extended smoothly to = . By adding an 2 at = , we obtain a manifold without boundary that has local topology R 2 × 2 .
Remark 1.4. We immediately see that at infinity, ( , ) is asymptotically locally AdS.
Given Eguchi-Hanson-type initial data, the Einstein vacuum equations (1.2) are well-posed in the biaxial Bianchi IX symmetry class -see [Dol17] for a proof. We merely state the well-posedness theorem here. Here ℎ is a Lorentzian metric on , and and are smooth real-valued functions on . The value ( ) is the area radius of the squashed sphere through ∈ , i. e.
where 3 is the sphere at . In this symmetry class, we introduce the renormalised Hawking mass (henceforth referred to as Hawking mass) Definition 1.6. A spacetime (ℳ, ) exhibiting biaxial Bianchi IX symmetry is asymptotically locally AdS with radius ℓ and conformal infinity ℐ if it is conformally equivalent to a manifold (M,˜) with boundary ℐ :=M such that (i) Conformal infinity ℐ has topology R × ( 3 /Γ).
(iii) The rescaled metric −2 is a smooth metric on a neighbourhood of ℐ inM.
(iv) For small˜> 0, there exist coordinates ( ,˜) on such that, locally, in a neighbourhood of ℐ. (v) The quantity satisfies a Dirichlet boundary condition, i. e. = 0 on ℐ.
Theorem 1.7. Let ( , , ) be an asymptotically locally AdS initial data set with mass at infinity such that ( , , ) is of Eguchi-Hanson type. Then there is a > 0 and a manifold ℳ := (− , ) × equipped with a metric exhibiting biaxial Bianchi IX symmetry such that (ℳ, ) is asymptoticall locally AdS, solves (1.2) and {0} × has induced metric and second fundamental form . Moreover, (ℳ, ) is the unique asymptotically locally AdS solution to (1.2) with initial data ( , , ).
1. The local well-posedness theorem for an initial data set ( , , ) yields the existence of a unique maximal development in the sense of [HS11a]. 2. The proof of the local well-posedness theorem in [Dol17] proceeds along the lines of [HW13]. Well-posedness of the Einstein vacuum equations for Λ < 0 in four dimensions without symmetry assumptions was shown by Friedrich in [Fri95], and a recent generalisation to higher dimensions by Enciso and Kamran is also available; see [EK14]. In particular, Theorem 1.7 follows from their work. However the theorem as stated is too general for an extension principle (Section 3.2); so to exploit the monotonicity of the Hawking mass (see Section 3.3), a local well-posedness result in norms propagated by the mass (as in Section 2) is required.
The explicit examples behind this well-posedness theorem are Eguchi-Hanson-AdS spacetimes, constructed in [CM06]. They form a family of solutions (ℳ EH, , EH, ) to (1.1) in five dimensions. For fixed Λ = −6/ℓ 2 < 0, they form a one-parameter family of static spacetimes exhibiting biaxial Bianchi IX symmetry. If we define and choose coordinates such that the metric takes the form in ( , , , , ) variables with ∈ ( , ∞). In the limit ℓ → ∞, the metric EH, restricted to hypersurfaces of constant yields the Riemannian Eguchi-Hanson metric, which was first presented in [EH78]. We immediately note: is negative. At the centre = , the Hawking mass is ill-defined, tending to −∞.

The main result
The main novel result of this paper consists in showing that for initial data of Eguchi-Hanson type with negative mass, a Penrose diagram such as Figure 2 cannot arise.
Theorem 1.10. Let ( , , ) be of Eguchi-Hanson type with negative mass < 0 at infinity. Then there is no future horizon in the maximal development, i. e. the causal past of null infinity is empty.
Corollary 1.11. Small perturbations of Eguchi-Hanson-AdS spacetimes do not contain future horizons.
It is important to stress that the absence of a horizon is a stronger statement than the absence of trapping -shown in Proposition 3.2 -for a horizon concerns the causal past of null infinity and hence the global geometry of the spacetime, whereas trapping is a local phenomenon.
Combining Theorem 1.10 with the arguments in Section 3.1 leaves us with the following dichotomy: either the future development of Eguchi-Hanson-type data with negative mass contains a first singularity in Γ∖Γ, where Γ is the centre -see Figure 3 -, or no first singularities form at all.
In virtue of the properties and conjectures described in the next section, our result, restricted to perturbations of Eguchi-Hanson-AdS spacetimes, can corroborate the conjecture put forward in in [DH06a]: uation of a horizon. The endpoint of ℐ, denoted by + , is either in ℐ or its completion; the latter case corresponds to future complete ℐ, the former one to an incomplete ℐ. We will show the absence of a horizon, that is the impossibility of either of those two cases.

The significance of Eguchi-Hanson-AdS spacetimes
The main motivation that sparked recent interest in asymptotically locally AdS solutions to the Einstein vacuum equations within the physics community is a putative connection between spacetimes of this form and conformal field theories defined on their respective boundaries: the AdS-CFT correspondence. It is of interest to understand what the positivity of gravitational energy means in the conformal field theory and thus 'ground states', lowest energy configurations classically allowed, deserve consideration -see [GSW02] for more details and references on the issue of gravitational energy in this context.
A ground state depends heavily on the topology at infinity. If the spacetime is asymptotically AdS, this ground state is exact anti-de Sitter space with vanishing mass -see [BGH84]. For asymptotically locally AdS spacetimes with toroidal topology at infinity, the works [HM98a], [HM98b] and [GSW02] lend support to the conjecture that the so-called AdS soliton is the ground state in a suitable class of spacetimes.
The article [CM06] was motivated by searching for a spacetime that asymptotically approaches AdS 5 /Γ, where Γ is any freely acting discrete group of isometries, but has energy less than that of AdS 5 /Γ. This led to the Eguchi-Hanson-AdS solution in five dimensions. These spacetimes have also been conjectured in [CM06] to have minimal mass among asymptotically locally AdS spacetimes with topology AdS 5 /Z at infinity: In a neighbourhood of Eguchi-Hanson-AdS solutions, this was indeed shown to be true: Theorem 1.14 ( [DH06a], see also [CM06]). Given any > 0, assume initial data ( , , ) of Eguchi-Hanson type with = ( , ∞)×( 3 /Z ) which are a sufficiently small, but non-zero perturbation of the data induced by the Eguchi-Hanson-AdS spacetime with parameter , then the mass at infinity satisfies Thus, fixing , the Eguchi-Hanson-AdS spacetime satisfying (1.4) can be seen as the ground state in the biaxial Bianchi IX symmetry class. There is a folklore statement that such ground states would be stable under gravitational perturbations. However, in contrast, the present work, paired with the above conjectures, heuristically hints at an instability: Perturbing an Eguchi-Hanson-AdS spacetime slightly increases its mass at infinity, whilst remaining negative; therefore, by Corollary 1.11, the future maximal development cannot contain a black hole, but by Conjecture 1.15, there is no static end state for the perturbation, which intimates that a first singularity forms, emanating from the centre. Therefore, such perturbations are potential candidates for examples of the formation of naked singularities.
It is interesting to note that a dual situation is found for perturbations of AdS 3 , as investigated in [BJ13]. There, small perturbations of three-dimensional AdS space were studied numerically as solutions to the Einstein-scalar field system. The parallel to our case is that in three dimensions, there exists a mass threshold below which no black holes can form. In contrast, while the numerical computations of [BJ13] suggest turbulence which cannot be terminated by a black hole formation, they provide evidence that small perturbations remain globally regular in time since the turbulence is too weak.
Finally, studying five-dimensional static spacetimes for various values of Λ or, more precisely, classifying their four-dimensional Riemannian counterparts is still an active field of research in geometry. It is known that there are exactly four complete non-singular four-dimensional Ricci flat Riemannian spaces: Euclidean space, Eguchi-Hanson space, self-dual Taub-NUT space and Taub-Bolt space. See [Gib05] for further details. Moreover, Eguchi-Hanson space has been used in geometric gluing constructions; see [Biq13] and [BK17]. For more results in this realm, both classical and recent, see [BGPP78], [LeB88], [EH79], [BK17] and references therein.

Outline of the paper
From the local well-posedness theorem (Theorem 1.7), we obtain the existence of a maximal development of Eguchi-Hanson-type data, with satisfying a Dirichlet boundary condition at infinity. The global geometry of spacetimes arising from such data is described in Section 3.1. We also prove in that section that the spacetime is either globally regular without a horizon, or forms a horizon, or evolves into a first singularity at the centre. We proceed to show that no horizons can form in the dynamical evolution.
Proving the absence of horizons will take the structure of an argument by contradiction (Section 3.3). Suppose that the Penrose diagram of the spacetime looks like  Figure 4: We can achieve that the initial data slice touches null infinity and does not reach Γ by moving from a slice such as 1 to 2.
soft arguments relying on the well-posedness result, we show that one can always find a null hypersurface such as 2 that does not intersect the initial hypersurface, but reaches from the horizon to null infinity. In Section 3.2, an extension principle is shown for triangular regions around null infinity -such as the one enclosed by ℋ + , 2 and ℐ -, which permits to extend the solutions to the future to a strictly larger triangle, provided uniform bounds hold in the triangular region. The proof of the extension principle uses the local existence result proved in [Dol17] and reviewed in Section 2. We then proceed (Section 3.3) to establish that those quantities can indeed be bounded only in terms of their values on 2 , where they hold by compactness of 2 and the local well-posedness result. Thus we can extend the solution along ℐ beyond ℋ + , which is a contradiction.

Local well-posedness
Before beginning with the proof of the main result, we review the local well-posedness result around null infinity of [Dol17] since the extension principle of Section 3.2 relies on it. The exposition of this section parallels that of [HW13]. This will allow the reader familiar with that argument to gain quick access to the problem at hand. Proving local well-posedness in the context of negative cosmological constant around infinity has been achieved for the four-dimensional Einstein-Klein-Gordon system in [HS11a] and [HW13]. Several differences arise in the present context, which are outlined in [Dol17]. However, we can follow the general strategy of [HW13]. We first define the triangle We treat these as defining auxiliary variables (2.1) The general discussion of Appendix A.2 show that the correct notion of a solution to the Einstein vacuum equations in the triangular region is encapsulated in the following Definition 2.1. A weak solution to the Einstein vacuum equations in Δ , 0 is a triple such that˜, , ∈ 0 loc and the equations =Ω 2˜3 are satisfied in the interior of Δ , 0 in a weak sense.
From this, we obtain a complete initial data set Remark 2.4. Imposing higher regularity on the initial data, we obtain a classical solution to (1.2) in Δ , 0 ; see [Dol17] for a precise statement.
We conclude this section with a remark about the Hawking mass. The mass is a dynamical variable and does not have to be conserved at infinity a priori. However, a geometric version of conservation holds:

The global problem
This section is devoted to studying the global dynamics arising from Eguchi-Hanson-type initial data. The existence of a maximal development is guaranteed by Theorem 1.7 and Remark 1.8. In Section 3.1, we specify our choice of coordinates on the orbits of the (2) × (1) action and derive some geometric properties; here, we follow the exposition of [Daf04a] and [Daf04b] mutatis mutandis. Proving that the existence of a horizon would be contradictory is the content of Sections 3.2 and 3.3.

Global biaxial Bianchi IX symmetry
Let ( , , ) be of Eguchi-Hanson type with negative mass at infinity. Then there is a unique maximal forward development (ℳ + , ) by Theorem 1.7 and Remark 1.8 which is asymptotically locally AdS. There is a projection map : ℳ + → + onto a twodimnsional manifold with boundary + such that every ∈ + represents an orbit under the (2) × (1) symmetry. The manifold + can be embedded smoothly into (R 2 , Mink ) and its boundary consists of a one-dimensional curve Σ (initial hypersurface) and a onedimensional curve Γ (central worldline, where = 0). Choosing standard null coordinates ( , ) on R 1+1 , + shall be endowed with a metric .
We choose such that the curves of constant are outgoing and such that as well as are increasing to the future along Γ. A coordinate chart ( ′ , ′ ) preserves these assumptions if and only if With respect to ℎ, Σ is spacelike and Γ timelike. Conformal infinity ℐ ⊆ + ∖ + is defined as follows: Set For each ∈ , there is a unique * ( ) such that ( , * ( )) ∈ + ∖ + .
Note here that the closure is always taken with respect to the topology of R 2 . Now define null infinity as ℐ := ⋃︁ ∈ ( , * ( )).
Since the spacetime is asymptotically locally AdS, null infinity ℐ is timelike. We have i. e. + is in the future domain of dependence of Σ and ℐ. By a simple change of coordinates satisfying (3.1), we achieve that = on ℐ. Note that in general, we cannot achieve that both ℐ and Γ are straightened out in this way. We know that extends continuously to ℐ and vanishes there. Moreover, we know that the Hawking mass extends continuously to ℐ and equals a constant value < 0.
Lemma 3.1. The following hold: > 0 at one point in Σ and Proof. By the monotonicity of the mass and < 0, we immediately obtain that < 0 on Σ. The radius is unbounded on Σ by the definition of Eguchi-Hanson-type data. For the Hawking mass to be finite at infinity, as → ∞. Moreover, by the above choice of ( , ) → −1 as → ∞. The conformal factor Ω 2 grows as 2 since the spacetime is asymptotically locally AdS. Therefore, we deduce that /Ω 2 is positive and finite as → ∞.
Proposition 3.2. The above manifold ℳ + does not have any trapped or marginally trapped surfaces, i. e.
Proof. From we conclude that < 0 wherever is finite. Since there is a point on Σ where > 0, the conclusion follows. ( 1 , 1 ) This fact already allows us to prove a weak geometric statement about the potential singularities that can arise in the time evolution.
Definition 3.4. Let ∈ + . The indecomposable past subset − ( ) ∩ + is said to be eventually compactly generated if there exists a compact subset ⊆ + such that Here we denote by − ( ) the causal past of a subset .
Definition 3.5. A point ∈ + ∖ + is a first singularity if − ( ) ∩ + is eventually compactly generated and if any eventually compactly generated indecomposable past proper subset of − ( ) ∩ + is of the form − ( ) for a ∈ + .
Proof. Suppose / ∈ Γ. Since the compact set of Definition 3.5 has to be wholly contained in + , we know that / ∈ ℐ. In particular, is the future endpoint of a rectangle, whose remainder is completely contained in the interior of + ; see Figure 6. By Proposition 3.2, < 0 and > 0 in this rectangle. Therefore, we can apply the standard extension principle away from infinity and the central worldline -in a manner as e. g. in [Daf04a] to conclude that ∈ + , a contradiction.
Proof. The horizon is given by Let be the future endpoint of the horizon. Since there is an open set such that ∩ + ̸ = ∅ If / ∈ Γ ∪ ℐ, then is a first singularity and we have a contradiction to Lemma 3.6. Therefore ∈ ℐ.

An extension principle
We formulate and prove an extension principle tailored to extending a solution beyond a supposed horizon. and that there is a > 0 such that Then there is a > 0 such that the solution (˜, , ) can be extended to the strictly larger triangle Δ + , 0 .
Proof. Extending beyond the domain of existence means using the local well-posedness result to extend the solution further into the future. We will first need to make sure that on each constant -slice, the function˜satisfies the correct boundary conditions. Reformulating equation (2.2), we obtaiñ Therefore˜= 0 on ℐ aux . Using a coordinate change, we want to fix the value of˜. By the assumptions and since˜= −˜on ℐ aux , |˜|, |˜| ≥ > 0 and where ′ = and ′ = on ℐ, defines a regular change of coordinates that preserves the biaxial Bianchi IX symmetry. Moreover, in ( ′ , ′ ) coordinates, Hence we can assume without loss of generality that (3.3) already holds in the original ( , ) coordinates.
To increase the domain of existence, we also need initial -slices of increased length. This can be achieved by an application of the standard local existence theorem away from infinity in double null coordinates whose proof proceeds by the same methods as for Λ = 0, which is standard by now. Prescribing data on the slice ( 0 , 0 + + ′ ] of constant = 0 (for a ′ > 0), we infer that for every > 0, there is a ′ > 0 such that the solution can be extended to the set For each constant -ray in Δ , we have a initial data set, whose functions have norms uniformly bounded by 2 . Note that the condition 1 − > 2 /ℓ 2 holds everywhere because < 0. Therefore, there is a * independent of such that each slice of constant in Δ yields a solution in a triangular domain of size * . Now we choose = * /2 and see that the solution (˜, , ) extends to a strictly larger triangle Δ + , 0 , where = .
The proof above yields a another version of the extension principle that we formulate separately for the sake of clarity.
Corollary 3.9. Suppose the assumptions of Theorem 3.8 hold. Moreover, let us assume that the classical solution on Δ has an extension to the extended initial data slice˜= ( 0 , 0 + + ]. Then there is a > 0 such that the solution (˜, , ) can be extended to Δ + , 0 such that it agree on˜∪ Δ + 0 , 0 with the given values. Furthermore, the extension is unique for sufficiently small > 0

A priori estimates
In this section, we first establish what was described through Figure 4 in Section 1.5 as the soft argument. This is the content of Lemma 3.10. The remainder of the section contains the argument by contradiction, using the extension principle in form of Corollary 3.9.
Proof. If Δ 1 1 touches Γ, then by moving the initial slice of constant to the future -as depicted in Figure 4 -, we achieve that ≥ 0 > 0 since > 0 globally by Proposition 3.2. By assumption, the bound on /Ω 2 holds on Σ. Set The set ℛ is closed and touches ℋ + and Σ. The continuous function /Ω 2 is positive in + by Proposition 3.2. Therefore the bound on /Ω 2 holds in ℛ. We will show that (3.4) holds in the causal future of such that the constants 1 and 2 depend on the values of /Ω 2 on . We rewrite the constraint equation (A.3) as (3.5) Given ( , ) ∈ + ( ), there is a ( ′ , ′ ) ∈ such that ( , ) and ( ′ , ′ ) are connected by a null curve. We integrate (3.5) along a ray of constant to find For the first inequality, we have used 1 − > 1 and ≥ 0 . For the second inequality, we have used (2.4) and have dropped a non-negative term. Therefore, we obtain .
Remark 3.11. A bound of the form (3.6) can always be achieved, independently of the exact value of . Now assume for the sake of contradiction that + possesses a horizon ℋ + . According to Lemma 3.10, we find a Δ := Δ 0 0 such that ( 0 + , 0 ) ∈ ℋ + and such that the conclusions of the lemma hold. In particular, the constants and bounds will be fixed henceforth. Again, let ℐ aux := Δ , 0 ∖{Δ , 0 ∪ {( 0 + , 0 + )}}. Let := 0 0 . We always have that We will show that all the assumptions of the extension principle hold. Let us first turn to estimating the norms of . The mass achieves its minimum at ( 1 , 0 ) and its maximum on ℐ aux . Therefore, upon integration over constant , we obtain Note that we have 1 − 2 3 ≥ min{ 2 /2, 1}.
We need to estimate the coefficients in the integral. From and see that˜is uniformly bounded above and below by a constant depending only on data on . Therefore . Therefore there is a constant depending only on values of˜,˜and on = 0 such that It follows that˜− 2 | | ≤ pointwise (3.10) uniformly. Together with (3.9), this yields ∫︁ 1 3 (︀ 2 + ( ) 2 )︀ d < ′ . (3.11) In a similar way, one also obtains ∫︁ 0 3 (︀ 2 + ( ) 2 )︀ d < ′ for ∈ [ 0 , 1 ) from integrating and then deriving a bound on /Ω 2 as (3.6). Here one uses that Using the wave equation for in the form Since Ω 2 / 2 is bounded by virtue of the bounds established above, the third term is easily seen to be bounded. The second term is estimates as where depends on values on and on 1 , 2 ,, ′ . Since is a classical solution up to and including the horizon, there is an > 0 such that = ( 0 ( ) + 1 ( ) + ( )) (3.13) for smoothly differentiable functions 0 and 1 of = ( + )/2. From above, the asymptotics of , , and Ω 2 are known as we approach the boundary. Inserting (3.13) into (3.12), we obtain = 4. Therefore, is bounded on and we have established the desired pointwise bound on in Δ. An application of the extension principle (Theorem 3.8) yields Theorem 1.10 if it also holds true that This has been established already in (3.7), thus finishing the proof of Theorem 1.10. from (2.1). This yields The second constraint equation is obtained analogously. To obtain the equation for (log Ω) , we multiply (A.2) by Ω 2 and differentiate with respect to : Applying (A.6), the desired equation follows.