Gregory J Galloway
email: galloway@math.miami.edu

José M M Senovilla
email: josemm.senovilla@ehu.es

Singularity theorems based on trapped submanifolds of arbitrary co-dimension

Keywords: numbers: 04.20.Cv, 04.20.Dw, 04.50.-h, 02.40.Ky

Standard singularity theorems are proven in Lorentzian manifolds of arbitrary dimension n if they contain closed trapped submanifolds of arbitrary codimension. By using the mean curvature vector to characterize trapped submanifolds, a unification of the several possibilities for the boundary conditions in the traditional theorems and their generalization to arbitrary co-dimension is achieved. The classical convergence conditions must be replaced by a condition on sectional curvatures, or tidal forces, which reduces to the former in the cases of co-dimension 1, 2 or n.

Introduction

The celebrated Hawking-Penrose singularity theorem [START_REF] Hawking | The singularities of gravitational collapse and cosmology[END_REF], see also [START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF][START_REF] Wald | General Relativity[END_REF], proves causal geodesic incompleteness of 4-dimensional spacetimes under strong-energy, causality and generic conditions if there is one of the following:

• a closed achronal set without edge,

• a closed trapped surface,

• a point with re-converging light cone.

It is interesting to note that the first case corresponds to a hypersurface (co-dimension 1 submanifold), the second case to a surface (co-dimension 2 submanifold) and the last case to a point (co-dimension 4, in 4-dimensional spacetimes). The co-dimension 3 case -a closed spacelike curve-, however, is missing in this list. One can wonder why.

In this communication we will show that the missing case can be included by using the geometrical characterization of trapped submanifolds provided by the mean curvature vector, and thereby one can further prove the singularity theorems in spacetimes of arbitrary dimension with closed trapped submanifolds of arbitrary codimension. This was conjectured in [START_REF] Senovilla | Novel results on trapped surfaces[END_REF], where the main ideas and the strategy to be used were presented.

The mean curvature vector and trapped submanifolds

To fix nomenclature and notation, let (V, g) be an n-dimensional causally orientable manifold with Lorentzian metric g µν of signature -, +, +, . . . , + (µ, ν, . . . = 1, . . . , n). Let ζ be a (for simplicity) connected (n-m)-dimensional submanifold with local intrinsic coordinates {λ A } (A, B, . . . = m + 1, . . . , n) imbedded in V by the smooth parametric equations x α = Φ α (λ A ) where {x α } are local coordinates for V. The tangent vectors e A of ζ are locally given by

e A ≡ e µ A ∂ ∂x µ ζ ≡ ∂Φ µ ∂λ A ∂ ∂x µ ζ
and the first fundamental form of ζ in V is:

γ AB ≡ g µν | ζ ∂Φ µ ∂λ A ∂Φ ν ∂λ B .
We assume that ζ is spacelike and therefore γ AB is positive definite. Any one-form n µ defined on ζ and orthogonal to the tangent vectors (n µ e µ A = 0) is called a normal one-form to ζ. At each point on ζ there are m linearly independent normal one-forms. If m > 1 all of these can be chosen to be null if desired.

The orthogonal splitting into directions tangential or normal to ζ leads to the standard formula [START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF]:

∇ e A e B = Γ C AB e C -K AB
where Γ C AB are the symbols of the Levi-Civita connection ∇ of γ and K AB is the shape tensor of ζ in (V, g). Observe that K AB = K BA and it is orthogonal to ζ. The contraction of K AB with any normal one-form

K AB ( n) ≡ n µ K µ AB = -n µ e ν A ∇ ν e µ B = e µ B e ν A ∇ ν n µ
is the second fundamental form with respect to n µ of ζ in (V, g).

The mean curvature vector of ζ in (V, g) [START_REF] Neill | Semi-Riemannian geometry[END_REF][START_REF] Kriele | Spacetime[END_REF] is the trace of the shape tensor

H ≡ γ AB K AB where γ AB is the contravariant metric on ζ: γ AC γ CB = δ A B .
By definition, H is orthogonal to ζ. Its contraction with any normal one-form

θ( n) ≡ n µ H µ = γ AB K AB ( n)
is called the expansion of ζ along n, and is the trace of the corresponding second fundamental form.

A spacelike submanifold ζ is said to be future trapped (f-trapped from now on) if H is timelike and future-pointing everywhere on ζ, and similarly for past trapped. This is equivalent to the condition that the expansions relative to all possible future-pointing normal one-forms are negative. Observe that the extreme case where n = m (ζ is a point) can be somehow considered to be included here if the expansion along every possible null geodesic emanating from ζ becomes negative -as required in the Hawking-Penrose singularity theorem. If H is future-pointing causal everywhere on ζ then ζ is said to be weakly future-trapped, while if it is null and future-pointing on ζ then ζ is called marginally future-trapped. In these cases the expansions are non-positive. The extreme case with H = 0 on ζ corresponds to a minimal submanifold. (Weakly, marginally) trapped submanifolds of arbitrary co-dimension m were considered in [START_REF] Mars | Trapped surfaces and symmetries[END_REF][START_REF] Senovilla | Novel results on trapped surfaces[END_REF][START_REF] Senovilla | On the existence of horizons in spacetimes with vanishing curvature invariants[END_REF] and they appear to have many common properties independent of m.

Existence of points focal to submanifolds of arbitrary dimension

In order to prove singularity theorems based on trapped submanifolds of arbitrary codimension the only intermediate result that needs to be generalized is the existence of focal points along normal geodesics. This will bring in the restrictions on the curvature that produce the necessary convergence of neighbouring geodesics. We will concentrate on future-trapped submanifolds, but the past case can be considered similarly.

Let n µ be any future-pointing normal to the spacelike submanifold ζ, normalized such that n µ u µ = -1 for a fixed timelike unit vector field u, and let γ denote a geodesic curve tangent to n µ at ζ. Let u be the affine parameter along γ, and set u = 0 at ζ. We denote by N µ the geodesic vector field tangent to γ and by E A the vector fields defined by parallelly propagating e A along γ. Observe that E A | u=0 = e A , N µ | u=0 = n µ , and that N µ E µ A = 0 for all u. By construction g µν E µ A E ν B is independent of u, so that g µν E µ A E ν B = g µν e µ A e ν B = γ AB and one can define along γ

P νσ ≡ γ AB E ν A E σ B
such that P νσ = P σν and P µ µ = n -m. At u = 0 this is the projector to ζ. Proposition 1. Let ζ be a spacelike submanifold of co-dimension m, and let n µ be a future-pointing normal to ζ. If θ( n) ≡ (m -n)c < 0 and the curvature tensor satisfies the inequality

R µνρσ N µ N ρ P νσ ≥ 0 ( 3 . 1 )
along γ, then there is a point focal to ζ along γ at or before γ| u=1/c , provided γ is defined up to that point.

Proof. As null geodesics are also considered here, the energy index form (or Hessian) [START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF] -rather than the length index form-is appropriate. The energy index form of a geodesic γ orthogonal to ζ is the symmetric bilinear form I(•, •) acting on vector fields that vanish at (say) u = b and are orthogonal to γ defined by

I( V , W ) ≡ b 0 [(N ρ ∇ ρ V µ )(N σ ∇ σ W µ ) -N ρ R ρ µντ V µ N ν W τ ] du + K AB ( n)v A w B (3.2) where v = V | u=0 , v A = v µ e µ
A is the part of v tangent to ζ, and analogously for W and w.

Let X A ≡ (1 -cu) E A , that is X A are vector fields orthogonal to γ such that X A | u=0 = e A and X A | u=1/c = 0. Then I( X A , X B ) = 1/c 0 c 2 γ AB -(1 -cu) 2 N ρ R ρ µντ E µ A N ν E τ B du + K AB ( n) and using γ AB K AB ( n) = θ( n) = (m -n)c one easily gets γ AB I( X A , X B ) = - 1/c 0 (1 -cu) 2 N ρ R ρ µντ N ν P µτ du . (3.3)
Condition (3.1) implies that this is non-positive. However, standard results [START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF] state that there is no focal point along γ for u ∈ (0, 1/c] if and only if the energy index form

I(•, •) is positive semi-definite, with I( V , V ) = 0 only if V is proportional to N on γ.
Given that γ AB is positive definite it follows from γ AB I( X A , X B ) ≤ 0 that the energy index form cannot have such a property, and therefore there exists a point focal to ζ along γ for u ∈ (0, 1/c].

Remarks:

(i) Notice that for spacelike hypersurfaces, when the co-dimension m = 1, there is a unique timelike orthogonal direction n µ . Then

P µν = g µν -(N ρ N ρ ) -1 N µ N ν and the condition (3.1) reduces to simply R µν N µ N ν ≥ 0, that is to say, the timelike convergence condition [2, 7 , 9, 12, 16, 20] along γ.
(ii) For co-dimension m = 2, there are two independent null normal directions at each point of ζ, say n µ and µ . By propagating µ along γ one can define the null vector field L µ on γ. Then,

P µν = g µν -(N ρ L ρ ) -1 (N µ L ν + N ν L µ )
and again the condition (3.1) reduces to simply R µν N µ N ν ≥ 0, that is to say, the null convergence condition [START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF][START_REF] Wald | General Relativity[END_REF] along γ.

(iii) For co-dimension m > 2, the interpretation of condition (3.1) can be given physically in terms of tidal forces, or geometrically in terms of sectional curvatures [START_REF] Eisenhart | Riemannian Geometry[END_REF][START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF]. A completely analogous condition was used in [START_REF] Galloway | Some results on the occurrence of compact minimal submanifolds[END_REF] for the proper Riemannian case in connection with minimal submanifolds (of arbitrary dimension). For a timelike unit normal n µ one has

R µνρσ n µ e ν n ρ e σ = k(n, e)(n ρ n ρ )(e ρ e ρ ) = -k(n, e)(e ρ e ρ )
where n µ e µ = 0, and k(n, e) is called the sectional curvature relative to the plane spanned by n and e. Therefore, by choosing n -m mutually orthogonal vectors e A tangent to ζ condition (3.1) states that the sum of the n -m sectional curvatures relative to a set of independent and mutually orthogonal timelike planes aligned with n µ is non-positive, and remains so along γ. In physical terms, this is a statement about the attractiveness of the gravitational field, on average. The tidal force -or geodesic deviation-in directions initially tangent to ζ is attractive on average, in such a way that the overall result is a tendency to converge.

For a null normal n µ one may consider analogously,

R µνρσ n µ e ν n ρ e σ = k(n, e)(e ρ e ρ )
where n µ e µ = 0, and k(n, e) is called the null sectional curvature relative to the plane spanned by n and e. Note, as per the standard definition in this case [START_REF] Beem | Global Lorentzian Geometry[END_REF], the minus sign is omitted. Thus, the interpretation above remains essentially the same (but with "non-positive" now replaced by "non-negative") by considering both the sectional curvatures and planes to be null.

The curvature condition (3.1) in Proposition 1 can in fact be weakened. It is sufficient that it hold on the average in a certain sense. Only a milder, integrated version, is needed, as we now briefly discuss. Thus, the condition that tidal forces should be attractive along γ can be substantially relaxed.

For the following, let the notation be as in Proposition 1.

Proposition 2. Let ζ be a spacelike submanifold of co-dimension m, and let n µ be a future-pointing normal to ζ. If, along γ (assumed to be future complete) the curvature tensor satisfies,

∞ 0 R µνρσ N µ N ρ P νσ du > θ( n) , (3.4)
then there is a point focal to ζ along γ.

Observe that there is no restriction on the sign of θ( n). We also note that, unlike Proposition 1, this proposition does not restrict the location of the focal point, but this will not be needed to prove singularity theorems under weaker curvature conditions (such as (4.5) below.)

Proof. Let f = f (u), u ≥ 0 be the solution to the initial value problem, (n -m)f + r(u)f = 0 , f (0) = 1, f (0) = θ( n) n -m ,
where r ≡ R µνρσ N µ N ρ P νσ . Given that (3.4) holds, it follows immediately from Lemma 3 in [START_REF] Galloway | Some results on the occurrence of compact minimal submanifolds[END_REF] that f has a zero on (0, ∞), i.e., there exists b > 0 such that

f (b) = 0. Now let X A ≡ f E A on [0, b]. Substituting into (3.2), gives, I( X A , X B ) = b 0 f 2 γ AB -f 2 N ρ R ρ µντ E µ A N ν E τ B du + K AB ( n) .
Contracting with γ AB and using

f 2 = (f f ) -f f , we obtain, γ AB I( X A , X B ) = - b 0 [(n -m)f + r(u)f ]f du + (n -m)f f | b 0 + θ( n) = 0 .
The remainder of the argument is just as in the proof of Proposition 1.

Main results: singularity theorems

We start by proving the generalization of the Penrose singularity theorem [START_REF] Penrose | Gravitational collapse and space-time singularities[END_REF], see also [START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF][START_REF] Wald | General Relativity[END_REF], which is the first of the "modern" theorems. Then, we will also [START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF][START_REF] Wald | General Relativity[END_REF] imply that these null geodesics enter I + (ζ) from the focal point on. Let K be the set of points reached by all these null geodesics up to the affine parameter 1/C inclusive, so that K is compact. Obviously E + (ζ) ⊂ K, and thus it is enough to prove that E + (ζ) is closed. Let {p i } be an infinite sequence of points p i ∈ E + (ζ) and let p be their accumulation point on the compact K. p cannot be in I + (ζ) as otherwise, I + (ζ) being open, there would be a neighbourhood of p within I + (ζ) containing some of the p i , which is impossible as

p i ∈ E + (ζ). But p certainly is in K ⊂ J + (ζ). Hence, p ∈ E + (ζ) showing that E + (ζ) is closed and thus compact.
The analogue of the Penrose singularity theorem can now be proven. Proof. The proof is standard [START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF][START_REF] Penrose | Gravitational collapse and space-time singularities[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF][START_REF] Wald | General Relativity[END_REF], so we just sketch it. If (V, g) were null geodesically complete E + (ζ) would be compact due to Proposition 3. But the spacetime is globally hyperbolic so that

(i) E + (ζ) = ∂J + (ζ)
is the boundary of the future set J + (ζ) and therefore a proper achronal boundary, which are known to be imbedded submanifolds (without boundary) [START_REF] Penrose | Techniques of Differential Topology in Relativity[END_REF][START_REF] Hawking | The singularities of gravitational collapse and cosmology[END_REF][START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Kriele | Spacetime[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF][START_REF] Wald | General Relativity[END_REF]; and

(ii) the manifold is the product V = R × Σ, see [START_REF] Bernal | Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions[END_REF] and references therein.

Then the canonical projection on Σ of the compact achronal E + (ζ) would have to have a boundary, ergo the contradiction.

Theorem 2. The conclusion of Propositon 3, and hence, of Theorem 1, remains valid if the curvature condition and the trapping condition assumed there are jointly replaced by Let us finally consider the more powerful Hawking-Penrose singularity theorem [START_REF] Hawking | The singularities of gravitational collapse and cosmology[END_REF]. As is known, this theorem is based on the Hawking-Penrose Lemma, proven in [START_REF] Hawking | The singularities of gravitational collapse and cosmology[END_REF], see also [START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Kriele | Spacetime[END_REF][START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF], stating that the following three statements cannot hold simultaneously in a given spacetime

a 0 R µνρσ N µ N ρ P νσ du > θ( n) , (4.5
• every endless causal geodesic has a pair of conjugate points,

• there are no closed timelike curves (chronology condition),

• there is an achronal set η such that E + (η) is compact. ‡ See [START_REF] Kupeli | Null cut loci of spacelike surfaces[END_REF] for relevant definitions and properties in the co-dimension two case; the higher co-dimension case works similarly.

This result holds independently of the dimension, and the achronal set η can also be anything as long as E + (η) is compact. Thus, one only has to ensure that the standard result leading to the compactness of E + (η) is achieved if there is a f-trapped submanifold of arbitrary dimension. This is proven now -for the definition of strong causality refer to any standard reference [START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Kriele | Spacetime[END_REF][START_REF] Neill | Semi-Riemannian geometry[END_REF][START_REF] Penrose | Techniques of Differential Topology in Relativity[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF][START_REF] Wald | General Relativity[END_REF]-. Proof. Assume that (V, g) is null geodesically complete, hence E + (ζ) is compact due to Proposition 3 so that η is also compact. The set η is also non-empty, as otherwise ζ ⊂ I + (ζ) against the assumption of strong causality [START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Hawking | The large scale structure of spacetime[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF]. One only needs to prove that actually E + (η) = E + (ζ), ergo compact. This can be done in standard fashion [START_REF] Beem | Global Lorentzian Geometry[END_REF][START_REF] Senovilla | Singularity Theorems and Their Consequences[END_REF] by covering the compact ζ with convex normal neighbourhoods such that the piece of ζ on each of them is achronal, extracting a finite sub-cover, and then showing that

I + (ζ) ⊂ I + (η), leading immediately to I + (ζ) = I + (η). Consider now q ∈ J + (ζ). From the previous equality, if q ∈ I + (ζ) then q ∈ I + (η) ⊂ J + (η). If q / ∈ I + (ζ) = I + (η), thene there is a point p ∈ ζ with q ∈ E + (p). It is obvious that p / ∈ I + (ζ) because q / ∈ I + (ζ), hence p ∈ ζ -I + (ζ) ergo p ∈ E + (ζ) ∩ ζ = η so that q ∈ J + (η), implying that J + (ζ) = J + (η). Finally, E + (η) = J + (η) -I + (η) = J + (ζ) -I + (ζ) = E + (ζ) as required.
From this result the analogue of the Hawking-Penrose theorem follows at once. Theorem 3. If the chronology, generic and strong energy conditions hold and there is a closed f-trapped submanifold ζ of arbitrary co-dimension such that condition (3.1) holds along every null geodesic emanating orhogonally from ζ then the spacetime is causal geodesically incomplete.

Remark: Of course, for co-dimension m = 1 there are no null geodesics orthogonal to ζ and there is no need to assume (3.1) nor anything concerning the mean curvature vector of ζ, as in Remark (i) to Proposition 3. For co-dimension m = 2 the condition (3.1) is actually included in the strong energy condition as explained in Remark (ii) to Proposition 1. The same happens for co-dimension m = n. These three cases cover the original Hawking-Penrose theorem.

In principle, any other singularity theorem assuming a closed trapped surface can be appropriately generalized to arbitrary co-dimension by means of Propositions 1-4.

Discussion with some applications

The main application of these theorems is, of course, to higher dimensional spacetimes, and therefore they can be used in the fashionable Kaluza-Klein/string/supergravity/Mtype theories. In dimension 11, say, there are now 10 different possibilities for the boundary condition in the theorems, in contrast with the classical three possibilities. All this should be explored and can have relevance in connection with the compactified extra-dimensions.

For example, the singularity theorems proven in this paper reinforce the arguments put forward by Penrose [START_REF] Penrose | On the instability of extra space dimensions[END_REF] about the classical instability of spatial extra-dimensions, as they might develop singularities within a tiny fraction of a second. In a 10-dimensional spacetime, the argument in [START_REF] Penrose | On the instability of extra space dimensions[END_REF] needs some ad-hoc splittings, and some restrictions on the Ricci tensor, that can be avoided by using the Theorems proven herein. It is enough that the compact extra-dimensional space, or any of its compact less-dimensional subsets, satisfy the trapping condition, and the restriction on Ricci curvatures can be replaced by the appropriate (averaged) condition on tidal forces. All in all, the basic argument of Penrose acquires a wider applicability and requires less restrictions by using the singularities proven to develop under the existence of compact submanifolds of any dimension.

Even in the traditional 4-dimensional Lorentzian manifolds of any classical gravitational theory, the new theorem may have some applications when considering the case of a closed trapped curve. Observe that these are just curves whose acceleration vector is timelike. An obvious relevant example, as explained in [START_REF] Senovilla | Novel results on trapped surfaces[END_REF], is the case of spacetimes with whole cylindrical symmetry [START_REF] Stephani | Exact Solutions to Einstein's Field Equations 2[END_REF], expressed in local coordinates by the line-element

ds 2 = -A 2 dt 2 + B 2 dρ 2 + F 2 dϕ 2 + E 2 dz 2 , (5.6)
where ∂ ϕ , ∂ z are spacelike commuting Killing vectors. The coordinate ϕ is closed with standard periodicity 2π. The cylinders given by constant values of t and ρ are geometrically preferred 2-surfaces, however, these cylinders are not compact in general, so that they have no direct implication in the development of geodesic incompleteness. Nevertheless, the spacelike curves defined by constant values of t, ρ and z are also geometrically distinguished and certainly closed. Their mean curvature vector is easily seen to be proportional to dF . Thus, the causal character of the gradient of only the function F , which is the norm of the circular Killing vector ∂ ϕ , characterizes the trapping of these closed circles. Thereby, many results on incompleteness of geodesics can be found. Moreover, there arises a new hypersurface, defined as the set of points where dF is null, which is a new type of horizon, being a boundary separating the trapped from the untrapped circles, and containing marginally trapped circles.

We now consider an application of Theorem 1 (or, more precisely, its time-dual) to the existence of initial singularities in asymptotically de Sitter spacetimes with compact Cauchy hypersurfaces. This, as we shall see, also involves trapped circles. In [START_REF] Andersson | dS/CFT and spacetime topology[END_REF][START_REF] Galloway | Cosmological spacetimes with Λ > 0[END_REF], results were obtained that establish a connection between Cauchy hypersurfaces with nontrivial topology and the occurrence of past singularities in such spacetimes. This behavior is well illustrated by dust filled FLRW solutions to the Einstein equations [START_REF] Stephani | Exact Solutions to Einstein's Field Equations 2[END_REF] with positive cosmological constant, Λ > 0; cf., the discussion [START_REF] Galloway | Cosmological spacetimes with Λ > 0[END_REF]. Such models with spherical spatial topology (de Sitter spacetime being the limiting case) need not have past singularities, whereas models which have, e.g., toroidal spatial topology, all have past singularities. We now present another general result along these lines.

We shall say that a group G is sufficiently large provided it has a nontrivial normal subgroup H such that |G/H| = ∞ (i.e., the cardinality of G/H is not finite.) The fundamental group π 1 (T k ) = Z k of the k-torus T k , k ≥ 2, is an example of a sufficiently large group. Theorem 4. Let (V, g) have dimension n ≥ 3, with all null sectional curvatures nonnegative. Suppose Σ is a compact Cauchy hypersurface for (V, g) which is expanding to the future in all directions, i.e., which has positive definite second fundamental form with respect to the future pointing normal. Then, if π 1 (Σ) is sufficiently large, (V, g) is past null geodesically incomplete.

Remark. We note that the curvature condition in Theorem 4 is satisfied strictly in the FLRW models mentioned above, as well as in sufficiently small perturbations of those models. Since the timelike convergence condition [7, Section 4.3] is not assumed in Theorem 4, and indeed does not in general hold in spacetimes with positive cosmological constant, many of the standard singularity theorems (such as Theorem 4 in [7, p. 272]) are not applicable to the setting considered in Theorem 4.

Proof. Let H be a nontrivial normal subgroup of π 1 (Σ) such that |π 1 (Σ)/H| = ∞. Since Σ is compact we can minimize arc length in the free homotopy class of a nontrival element of H to obtain a closed geodesic σ in Σ. Since Σ has negative definite second fundamental form with respect to the past pointing unit normal, one easily verifies that σ is a past-trapped co-dimension n -1 submanifold in (V, g). If (V, g) were past null geodesically complete then the time-dual of Proposition 3 and the curvature condition would imply that E -(σ) is compact. Since all the Cauchy hypersurfaces of (V, g) are compact [START_REF] Bernal | Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions[END_REF], this does not directly lead to a contradiction. But now we pass to a covering spacetime. By standard covering space theory there exists a covering manifold Σ of Σ, with covering map p : Σ → Σ, such that the induced map on fundamental groups p * : π 1 ( Σ) → H ⊂ π 1 (Σ) is an isomorphism. Since |π 1 (Σ)/H| = ∞, this is an infinite sheeted covering, and hence Σ is non-compact. We know by the global hyperbolicity of (V, g) that V is diffeomorphic to R × Σ [START_REF] Bernal | Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions[END_REF], and hence that the fundamental groups of Σ and V are isomorphic. From this it follows that there is a covering spacetime ( Ṽ, g), with covering map P : Ṽ → V, such that (i) P is a local isometry, (g is the pullback of g via P ), and (ii) Ṽ contains Σ as a Cauchy hypersurface, such that P | Σ = p. Now, since p * is an isomorphism, σ lifts to a closed geodesic σ in Σ, which, because P is a local isometry, is past trapped in ( Ṽ, g). Then, since the curvature condition lifts to ( Ṽ, g), the time-dual of Theorem 1 implies that there exists a past incomplete null geodesic in ( Ṽ, g). This projects, via P , to a past incomplete null geodesic in (V, g).

Of course, this theorem has a dual version to the future, if the compact Cauchy hypersurface is contracting.

Proposition 3 .

 3 prove the generalization of the more elaborated Hawking-Penrose theorem. Recall that for any set ζ, E + (ζ) ≡ J + (ζ)\I + (ζ), using the standard notation for the causal J + (ζ) and chronological I + (ζ) futures of ζ, see e.g. [2, 8, 7, 9, 14, 16]. Let ζ be a closed f-trapped submanifold of co-dimension m > 1, and assume that the curvature tensor satisfies the inequality R µνρσ N µ N ρ P νσ ≥ 0 for any future-pointing null normal one-form n µ . Then, either E + (ζ) is compact, or the spacetime is future null geodesically incomplete, or both. Remark: The case with m = 1 is not included here because it is trivial. If ζ is a spacelike hypersurface, then E + (ζ) ⊂ ζ -and actually E + (ζ) = ζ if ζ is achronal-, and the compactness of E + (ζ) follows readily without any further assumptions. Proof. Assume that (V, g) is future null geodesically complete. As ζ is f-trapped one has θ( n) = (m -n)c < 0 for any future-pointing null normal one-form n µ . Let (m -n)C be the maximum value of all possible θ( n) on the compact ζ. Due to Proposition 1 every null geodesic emanating orthogonally from ζ will have a focal point at or before the affine parameter reaches the value 1/C. Standard results

Theorem 1 .

 1 If (V, g) contains a non-compact Cauchy hypersurface Σ and a closed ftrapped submanifold ζ of arbitrary co-dimension, and if condition (3.1) holds along every future directed null geodesic emanating orthogonally from ζ, then (V, g) is future null geodesically incomplete.

)

  along each future inextendible null geodesic γ : [0, a) → V, a ∈ (0, ∞], emanating orthogonally from ζ with initial tangent n µ .Proof. We shall assume that ζ is acausal. (The case in which ζ is not acausal can be handled by considering a suitable finite cover of ζ by acausal subsets.) To prove Theorem 2, it is sufficient to show that E + (ζ) is compact under the assumption of future null geodesic completeness. Under this assumption (in which case a = ∞ in (4.5)), Proposition 2 implies that there is a focal point along each future directed null geodesic emanating orthogonally from ζ. Since cut points to ζ ‡ come at or before focal points, there is a null cut point along each of these null geodesics. The compactness of ζ and the fact that the affine distance to each null cut point is upper semi-continuous, as a function of points in ζ, imply that there exists C * > 0 such that each cut point occurs at or before the affine value C * . Let K be the set of points reached by all these null geodesics up to the affine parameter C * inclusive, so that K is compact. Since E + (ζ) ⊂ K, and, as argued in Proposition 3, E + (ζ) is closed, we have that E + (ζ) is compact.Thus, for example, even if ζ is only weakly or marginally f-trapped, or minimal, the conclusion of future null geodesic incompleteness in Theorem 1 still holds, provided the inequality (3.1) is strict at least at one point on each future directed null geodesic γ emanating orthogonally from ζ.

Proposition 4 .

 4 If (V, g) is strongly causal and there is a closed f-trapped submanifold ζ of arbitrary co-dimension m > 1 such that condition (3.1) holds along every null geodesic emanating orthogonally from ζ, then either E + (E + (ζ) ∩ ζ) is compact, or the spacetime is null geodesically incomplete, or both. Remark: Thus, η ≡ E + (ζ) ∩ ζ provides the needed set in the Hawking-Penrose Lemma above, the achronality of η being a direct consequence of the achronality of E + (ζ).

Acknowledgements

Comments from M Mars and M Sánchez are gratefully acknowledged. Supported by grants FIS2004-01626 (MICINN), GIU06/37 (UPV/EHU), and DMS-0708048 (NSF).


Andrés E Aceña
email: acena@aei.mpg.de

Minimal data at a given point of space for solutions to certain geometric systems

published or not. The documents may come

Minimal data at a given point of space for solutions to certain geometric systems

Introduction

Einstein's equations form the core of General Relativity. Solutions to this system of equations represent physically feasible spacetimes. Therefore, analyzing the properties of such solutions is one of the most important tasks in General Relativity. It is, at the same time, one of the utmost difficulty. One of the ways to be able to analyze properties of the solutions is to make simplifying assumptions, which restrict the possible solutions. Typical assumptions are those of staticity or stationarity, where the existence of a timelike Killing field is assumed, allowing for a dimensional reduction of the problem. The literature is full of examples of such procedure, the simplest being the static Einstein vacuum field equations. More complicated examples are given by the stationary Einstein-Maxwell fields in arbitrary dimensions [START_REF] Ida | Stationary Einstein-Maxwell fields in arbitrary dimensions[END_REF] (here stationary means that the dimensionally reduced problem is three dimensional) and static Einstein-Maxwell-dilaton fields [START_REF] Mars | On uniqueness of static Einstein-Maxwell-dilaton black holes[END_REF]. Also if dimensions higher than four are considered, as in Kaluza-Klein type theories, dimensional reduced versions of the theories are usually also analyzed. As said, this practice is due to the complicated structure of Einstein's equations and because the static or stationary solutions provide important information about some properties of the solutions.

Even though simpler, the arising systems of equations are still complicated and need detailed analysis. The main objective of the present work is to answer the question of existence and characterization of solutions to the arising systems of equations. That is, we want to show that given suitable data, this data is enough to characterize uniquely the solution and that there is a solution with such a data. As we do not want to enter in a case by case discussion, the first step is to look at the shared properties of those systems. In general they have the following features:

• The base manifold is three dimensional.

• The metric is Riemannian.

• On the base manifold there are some scalar fields.

• The metric satisfies an equation whose principal part is given by the Ricci tensor of the metric.

• The fields satisfy Laplace type equations.

There are two features that are central for our analysis. The first is that in a suitable gauge the systems are elliptic. This implies that the solutions are analytic in this gauge. Secondly, in this gauge the systems of equations are diagonal, i.e. for each system the principal symbol has only diagonal elements (the off-diagonal elements vanish) and all the diagonal elements are the same. This implies that the complex null cone (characteristic cone) is the same for all the unknowns in each system. This is very important for the geometrical analysis of the problem.

We consider now a more general system of equations that includes all the named features. The setting is the following.

Let N be a three dimensional manifold with negative definite metric h and n scalar fields on it, φ α , α = 1, . . . , n, that satisfy the following system of equations:

∆ h φ α = f α (φ γ , D c φ γ ), α= 1, . . . , n, (1)
R ab [h] = F α (φ γ )D a D b φ α + F αβ (φ γ )D a φ α D b φ β + f (φ γ , D c φ γ )h ab , (2)
where D is the covariant derivative associated with h, ∆ h is the Laplace operator with respect to h and R ab [h] is the Ricci tensor of h. F α (φ γ ), F αβ (φ γ ), α, β = 1, . . . , n, are analytic functions of the scalar fields and f (φ γ , D c φ γ ), f α (φ γ , D c φ γ ), α = 1, . . . , n, are analytic functions of the scalar fields and its first derivatives. In order not to burden the text, the span of greek indices is not always stated. We assume that the summation rule for repeated indices holds for both latin and greek indices, the latter being written always in the upper position, as no metric is associated with them. We assume F αβ = F βα , so we do not to have to symmetrize on the a, b indices. The functions f , f α , F α , F αβ are scalars if considered as functions on N through their dependence on the fields and their first derivatives. This is necessary for (1), [START_REF] Beig | Time-independent gravitational fields[END_REF] to be a geometrically meaningful system of equations. Later on, we will see that the functions f , f α , F α and F αβ are not independent but have to satisfy some relations. We refer to (1), [START_REF] Beig | Time-independent gravitational fields[END_REF] as the field equations.

The setting above includes all the named static or stationary cases of Einstein's equations. A particular class of systems that has received increasing attention over the years and that is also included corresponds to harmonic maps coupled to gravity (also called σ-models) whose base space is a three dimensional manifold or where the spacetime is stationary [START_REF] Schimming | Harmonic maps from spacetimes and their coupling to gravitation[END_REF][START_REF] Breitenlohner | 4-Dimensional black holes from Kaluza-Klein theories[END_REF]. Of course the present listing of included systems of equations is far from complete, but it clearly shows that many physically interesting systems of equations are included. Now we can state clearly the objective of this work: to give a complete characterization of the solutions to the field equations above in a neighbourhood of a given point.

So, let us take a point q in N . As said before, in a suitable gauge the system of equations is elliptic with analytic coefficients, therefore the solution, if it exists, is analytic [START_REF] Morrey | On the Analyticity of the Solutions of Analytic Non-Linear Elliptic Systems of Partial Differential Equations: Part I. Analyticity in the Interior[END_REF] (cf. [START_REF] Müller | On the analyticity of stationary vacuum solutions of Einstein's equation[END_REF], [START_REF] Beig | Time-independent gravitational fields[END_REF] for applications of the result in [START_REF] Morrey | On the Analyticity of the Solutions of Analytic Non-Linear Elliptic Systems of Partial Differential Equations: Part I. Analyticity in the Interior[END_REF] to the stationary vacuum Einstein equations). This suggest prescribing as data the coefficients of the Taylor expansion of the fields at q. This is not possible, as the Taylor expansion coefficients are not independent of each other. Instead we consider the following n sequences of symmetric trace-free tensors at q,

D α = {φ α (q), D a 1 φ α (q), C(D a 2 D a 1 φ α )(q), C(D a 3 D a 2 D a 1 φ α )(q), . . .}, (3)
where C means taking the symmetric trace-free part of the tensor to which it is applied. These sequences are called the null data for our system of equations. It turns out that the null data is indeed a minimal set of data for the field equations, that is, it determines a formal expansion of the solution and its components are independent of each other. Our purpose can then be stated as to derive necessary and sufficient conditions for the null data to determine apart from gauge conditions (unique) real analytic solutions of (1) and ( 2). This will show that the null data do indeed provide the searched characterization.

Prescribing a minimal set of data at a point as characterization for the solutions to a gauge-elliptic problem is certainly different from posing a standard boundary value problem. A particular advantage of our approach is that the data has a geometrical meaning and does not depend on the choice of an arbitrary hypersurface or coordinate system. This makes the characterization intrinsic to the geometry of the solution.

Another advantage arises if there is a geometrically distinguished point. Our approach then allows a complete control and analysis of the solution at the given point. The existence of a geometrically distinguished point happens in the case of asymptotically flat static or stationary spacetimes (cf. Friedrich [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] and Aceña [START_REF] Aceña | Convergent null data expansions at space-like infinity of stationary vacuum solutions[END_REF]), where there is a point in the manifold that represents infinity. Unfortunately those cases are not included in our present treatment because besides the metric there are further tensor fields among the unknowns, while we only consider scalar fields. An extension of the present result to include tensors in the unknowns is possible although not entirely straightforward, as one has possibly to take into account integrability conditions that may arise for the system of equations to be consistent. We do not want to get involved in such a discussion here.

It is convenient to express the tensors in D α in terms of an h-orthonormal frame c a , a = 1, 2, 3, centered at q. Denoting by D a the covariant derivative in the direction of c a ,

D α * = {φ α (q), D a 1 φ α (q), C(D a 2 D a 1 φ α )(q), C(D a 3 D a 2 D a 1 φ α )(q), . . .}. (4)
These tensors will be called the null data in the frame c a , and are defined uniquely up to rigid rotations in R 3 .

If the metric h and the potentials φ α exist, then they are real analytic near q and one has Cauchy estimates on the derivatives of the potentials. The Cauchy estimates imply that there exist positive constants M, r, such that the components of the null data satisfy

|C(D ap ...D a 1 φ α )(q)| ≤ Mp! r p , p≥ 0, a p , . . . , a 1 = 1, 2, 3. (5)
Our main result corresponds to the statement that these estimates are not only necessary but also sufficient to have an analytic solution, and is presented in the following theorem.

Theorem 1.1. Suppose

Dα = {ψ α a 1 , ψ α a 2 a 1 , ψ α a 3 a 2 a 1 , ...}, α= 1, . . . , n, (6)
are n infinite sequences of symmetric, trace free tensors given in an orthonormal frame at the origin of a 3-dimensional Euclidean space. If there exist positive constants M, r such that the components of these tensors satisfy the estimates

|ψ α ap...a 1 | ≤ Mp! r p , p≥ 0, a p , ..., a 1 = 1, 2, 3, α= 1, . . . , n,
then there exists an analytic solution h, φ α , α = 1, . . . , n, of the field equations near q, unique up to isometries, so that the null data implied by it in a suitable frame c a as described above satisfy

C(D ap ...D a 1 φ α )(q) = ψ α ap...a 1 , p ≥ 0, a p , ..., a 1 = 1, 2, 3, α = 1, . . . , n.
The sequences [START_REF] Mars | On uniqueness of static Einstein-Maxwell-dilaton black holes[END_REF], not necessarily satisfying any estimates, will be referred to as abstract null data. As the type of estimates imposed here on the abstract null data does not depend on the orthonormal frame in which they are given, and since these estimates are necessary as well as sufficient, then all possible solutions of (1), (2) are characterized by the null data.

A few words on the requirement on the manifold N to be three dimensional are in order here. The proof that we present of the theorem above relies heavily in the use of space spinors, which can be used only on a three dimensional manifold. This may seem as the main reason for such a requirement, in fact it is not. There is a one to one correspondence between space spinors analysis and the more usual tensor analysis on a three dimensional manifold. Therefore, all the proofs could be translated and expressed in terms of tensor analysis, and then one could try to extend them to higher dimensions. The main reason for the requirement is that in three dimensions the Riemann tensor is completely determined by the Ricci tensor. If one wants to extend the result to higher dimensions one is forced to include in the system of equations an equation for the Weyl tensor, which makes the system of equations more complicated and introduces also complications regarding the integrability conditions.

The use of null data in General Relativity was first presented by Penrose and Rindler [START_REF] Penrose | Spinors and space-time[END_REF], where it is used as data for fields coupled to gravity in spacetime (four dimensional Lorentzian manifold). It is important to note that although there are similarities in the use of the null data in the Lorenzian case and in the Riemannian case, treated in the present work, there is a profound difference. Namely, in the Lorenzian case one has to impose analyticity on the solution to be able to characterize it using the null data, while in the Riemannian case analyticity is a natural consequence of the equations.

In the context of three dimensional Riemannian spaces the null data was first introduced by Friedrich [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] as a way to characterize static asymptotically flat solutions to the vacuum Einstein field equations. They were also used by Aceña [START_REF] Aceña | Convergent null data expansions at space-like infinity of stationary vacuum solutions[END_REF] for the stationary asymptotically flat vacuum case. The techniques that we use to prove the result of the present work are similar to those introduced by Friedrich and used by Aceña. Therefore we will not present the procedure in full detail, but we will state the important steps and the features that are distinctive for the case that we are treating here.

The exact sets of equations argument

We have defined the null data D α (3) and we want to use them to characterize solutions to the field equations. Therefore an important first step is to show that the null data can actually be used to construct formal solutions to the field equations. For this we construct expansions of the fields in normal coordinates.

We assume from now on N to be small enough to coincide with a convex hnormal neighbourhood of q. Let c a , a = 1, 2, 3, be an h-orthonormal frame field on N which is parallelly transported along the h-geodesics through q and let x a denote normal coordinates centered at q so that c b a ≡ dx b , c a = δ b a at q. We refer to such a frame as a normal frame centered at q. Its dual frame will be denoted by

χ c = χ c b dx b .
In the following all tensor fields, except the frame field c a and the coframe field χ c , will be expressed in terms of this frame field, so that the metric is given by h ab ≡ h(c a , c b ) = -δ ab . With D a ≡ D ca denoting the covariant derivative in the c a direction, the connection coefficients with respect to c a are defined by D a c c = Γ a b c c b . An analytic tensor field T a 1 ...a k on N has in the normal coordinates x a a normal expansion at q, which can be written

T a 1 ...a k (x) = p≥0 1 p! x bp ...x b 1 D bp ...D b 1 T a 1 ...a k (q), (7)
where we assume from now on that the summation convention does not distinguish between bold face and other indices. Since h ab = -δ ab , it remains to be seen how to obtain normal expansions for the φ α 's using the field equations and the null data. That is, we need to see how to obtain

D ap ...D a 1 φ α (q), p≥ 0, a p , . . . , a 1 = 1, 2, 3.
The algebra necessary for doing this simplifies considerably in the space-spinor formalism. How to do the transition is explained in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF]. Here we recall a few important properties.

• A space spinor field T A 1 B 1 ...ApBp = T (A 1 B 1 )...(ApBp) arises from a real tensor field T a 1 ...ap if and only if

T A 1 B 1 ...ApBp = (-1) p τ A 1 A 1 ..τ Bp B p TA 1 B 1 ...A p B p , (8)
where

τ AA = 0 A 0 A + 1 A 1
A . AB is the constant -spinor, which satisfies AB = -BA , 01 = 1 and it is used to move indices according to the rules

ι B = ι A AB , ι A = AB ι B .
• Any spinor field T A...H admits a decomposition into products of totally symmetric spinor fields and -spinors which can be written schematically in the form

T A...H = T (A...H) + s × symmetrized contractions of T. (9)
• The operation of taking the symmetric trace-free part of a tensor translates into taking the totally symmetric part of the corresponding spinor. So the null data translates into n sequences of totally symmetric spinors.

• We also have a complex frame field c AB (related to c a ), such that h(c AB , c CD ) = h ABCD ≡ -A(C D)B , and its dual 1-form field χ AB (related to χ a ). The covariant derivative of a spinor field ι C in the direction of c AB is given by

D AB ι C = c AB (ι C ) + Γ AB C D ι D
, where Γ ABCD = Γ (AB)(CD) are the spinor connection coefficients.

• The commutator of derivatives are given in terms of the curvature spinor,

(D CD D EF -D EF D CD )ι A = r A BCDEF ι B , (10)
where

r ABCDEF = 1 2 s ABCE - 1 6 rh ABCE DF + s ABDF - 1 6 rh ABDF CE , (11)
being r the Ricci scalar of h and s ABCD = s (ABCD) the trace free part of the Ricci tensor of h. In tensor notation the decomposition of the Ricci tensor reads 1), (2) take in the space-spinor formalism the form

R ab [h] = s ab [h] + 1 3 r[h]h ab . Equations (
D P A D BP φ α = - 1 2 AB f α , (12)
r ABCDEF = 1 2 S ABCE - 1 6 Rh ABCE DF + S ABDF - 1 6 Rh ABDF CE , (13)
where

S ABCD = F α D AB D CD φ α + F αβ D AB φ α D CD φ β + f h ABCD , (14)
R = f , ( 15
) being f = - 1 3 (F αβ D a φ α D a φ β + F α f α ), f = F αβ D a φ α D a φ β + 3f + F α f α .
Using these equations and the theory of 'exact sets of fields' is possible to prove the following result.

Lemma 2.1. Let there be n given sequences

Dα = {ψ α , ψ α A 1 B 1 , ψ α A 2 B 2 A 1 B 1 , ψ α A 3 B 3 A 2 B 2 A 1 B 1 , .
..} of totally symmetric spinors satisfying the reality condition [START_REF] Müller | On the analyticity of stationary vacuum solutions of Einstein's equation[END_REF]. Assume that there exists a solution h, φ α , to the field equations so that the spinors given by Dα coincide with the null data D α * given by (4) in terms of an h-orthonormal normal frame centered at q, i.e.

ψ α ApBp...A 1 B 1 = D (ApBp ...D A 1 B 1 ) φ α (q), p≥ 0,
Then the coefficients of the normal expansions (7) of the fields φ α , i.e.

D ApBp ...D A 1 B 1 φ α (q), p≥ 0,
are uniquely determined by the data Dα and satisfy the reality condition.

Proof. The proof is by induction. It holds φ α (q) = ψ α , D AB φ α (q) = ψ α AB . To discuss the induction step we assume that the expansion coefficients of φ α up to order p are known and start with D A p+1 B p+1 ...D A 1 B 1 φ α (q) and its decomposition in the form [START_REF] Penrose | Spinors and space-time[END_REF]. By assumption, the totally symmetric part of it is given by ψ α A p+1 B p+1 ...A 1 B 1 . The other terms in the decomposition contain contractions. Let us consider a general contraction, say A i contracted with A j . We can commute the operators D A i B i and D A j B j with other covariant derivatives, generating by ( 10) and (13) only terms of lower order, until we have 12) then shows how to express the resulting term by quantities of lower order that are already known.

D A p+1 B p+1 ...D A i+1 B i+1 D A i-1 B i-1 ...D A j+1 B j+1 D A j-1 B j-1 ...D A 1 B 1 D P B i D P B j φ(q). Equation (
That the expansion coefficients satisfy the reality condition is a consequence of the formalism and the fact that they are satisfied by the data.

In order to show the convergence of the formal series determined in the previous lemma we need to impose estimates on the free coefficients given by Dα . For the necessary part we have the following result.

Lemma 2.2.

A necessary condition for the formal series determined in Lemma 2.1 to be absolutely convergent near the origin is that the data given by Dα satisfy estimates of the type

|ψ α ApBp...A 1 B 1 | ≤ p!M r p , p≥ 0, ( 16
)
with some positive constants M, r.

The proof of the last lemma is very similar to the respective lemma in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] and is not repeated here.

A maybe more clarifying way of stating the last two lemmas is the following.

• If we have two solutions of the field equations and the null data (4) of one of the solutions is related to the null data of the other solution by a rigid rotation in R 3 , then the two solutions are related by an isomorphism and are therefore geometrically equivalent.

• If we have a solution of the field equations, then its null data (4) satisfy estimates of the form (5).

The characteristic initial value problem

After showing that the null data determine the solution, one would have to show that the estimates (16), imply Cauchy estimates for the expansion coefficients

|D ApBp ...D A 1 B 1 φ α (q)| ≤ p!M r p , p≥ 0.
This would ensure the convergence of the normal expansion in a neighbourhood of q and the existence of the solution. But deriving estimates on the expansion coefficients from estimates on the null data using the procedure described in the proof of Lemma 2.1 has not been possible. Instead, one can use the intrinsic geometric nature of the problem and the data to formulate the problem as a boundary value problem to which Cauchy-Kowalevskaya type arguments apply. The formalism necessary for this has been developed in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] and used in [START_REF] Aceña | Convergent null data expansions at space-like infinity of stationary vacuum solutions[END_REF]. Here we present the important facts for the present work, following the notation in [START_REF] Aceña | Convergent null data expansions at space-like infinity of stationary vacuum solutions[END_REF]. The reader is referred to [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF][START_REF] Aceña | Convergent null data expansions at space-like infinity of stationary vacuum solutions[END_REF] for details. The fields h, φ α can be extended near q by analyticity into the complex domain and considered as holomorphic fields on a complex analytic manifold N c . Under the analytic extension, and choosing N c to be a sufficiently small neighbourhood of q, the main differential geometric concepts and formulae remain valid. The extended coordinates and the extended frame, again denoted by x a and c AB satisfy the same defining equations and the extended fields, denoted again by h, φ α satisfy the field equations as before.

The null cone at q is defined as the set

N q = {p ∈ N c |Γ(p) = 0}
where Γ = δ ab x a x b . It is the cone swept out by the complex null geodesics through q. For the type of systems we consider one very specific and important feature of this cone is that it is the characteristic cone not only for the Ricci operator in a harmonic gauge but also for the Laplace operators that act on the scalar fields. The coincidence of the characteristic cones for all the unknowns is due to the special form of the principal symbol of the system, it is diagonal and all the diagonal elements are the same. If this were not the case then the construction used in the present work would not be possible. It turns out for our problem that knowing the null data is equivalent to knowing the restriction of the holomorphic functions φ α to the null cone. The identification is made explicit later on.

Our problem can thus be formulated as the boundary value problem for the field equations with data given on the null cone by the functions φ α | Nq . The difficulty with this formulation is that N q is not a smooth hypersurface but an analytic set with a vertex at the point q, therefore an adapted coordinate and frame field is needed. The construction is done rigorously in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] in terms of the principal bundle of spin frames over N c , SL(N c ). There a three-dimensional submanifold N of SL(N c ) is constructed in such a way that together with the projection map π : SL(N c ) → N c it induces coordinates on N c . Here we describe how the construction is seen on N c .

We start by taking the null vector c 11 at q and constructing the null geodesic γ with affine parameter w whose tangent vector at q is c 11 . That is, γ(w) is the affinely parametrized null geodesic that has γ(0) = q and γ (0) = c 11 .

We define now a family of frames e AB at q in terms of the frame c AB and a parameter v ∈ C in the following way

e 00 = c 00 + 2vc 01 + v 2 c 11 , e 01 (v) = c 01 + vc 11 , e 11 (v) = c 11 .
These frames are orthonormal in the sense that h(e AB , e CD ) = h ABCD . As v varies e 00 covers all null directions at q except c 11 .

We parallelly propagate the frames e AB (v) over γ(w). So now at each point of γ(w) we have a family of frames e AB (v, w).

Consider a fixed value for v = v 0 and w = w 0 . At γ(w 0 ) we construct the null geodesic that goes through this point and has tangent vector e 00 (v 0 , w 0 ) at γ(w 0 ). We call u the affine parameter on this null geodesic that vanishes at γ(w 0 ) and we assign to a point on this geodesic the corresponding value of u and the values v = v 0 and w = w 0 . We also parallelly propagate the frame e AB (v 0 , w 0 ) over the null geodesic. Doing this for all possible values of v and w covers all of N c and defines the frame field e AB there.

The functions z 1 = u, z 2 = v, z 3 = w define holomorphic coordinates on N . We denote again π the restriction of the projection to N . The map π induces a biholomorphic diffeomorphism of N ≡ N \U 0 , where U 0 ≡ {u = 0}, onto π( N ), but the gauge is singular in the sense that the set U 0 , which is a two-surface on N , projects into the curve γ on N c . We also need to distinguish the set I ≡ {u = 0, w = 0}, where π(I) = q. Finally, the set W 0 ≡ {w = 0} projects onto N q \γ and will therefore define the initial data set for our problem.

It is important to recall that although the coordinates z a are well adapted to the geometry of the problem we are dealing with, and are holomorphic on N , they are not a good coordinate system in all of N c . The set {u = 0} corresponds to the curve γ instead of being a hypersurface on N c , and thus there is no way of assigning a v coordinate to points on γ. For this the null curve γ will be referred to as the the singular generator of N q in the gauge determined by the frame c AB at q. The singularity of the gauge is reflected in the following properties, deduced in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF], and which are important for the present work.

   =    1 O(u 2 ) O(u 2 ) 0 1 2u + O(u) O(u) 0 0 1    as u → 0.
We shall write e a AB = e * a AB + êa AB , with singular part

e * a AB = δ a 1 A 0 B 0 + δ a 2 1 u (A 0 B) 1 + δ a 3 A 1 B 1 ,
and holomorphic functions êa AB on N which satisfy êa

AB = O(u) as u → 0. ( 17
)
The connection coefficients Γ ABCD = Γ (AB)(CD) satisfy

Γ 00AB = 0 on N, Γ 11AB = 0 on U 0 ,
and also

Γ ABCD = Γ * ABCD + ΓABCD , with singular part Γ * ABCD = - 1 u (A 0 B) 1 C 0 D 0 ,
and holomorphic functions ΓABCD on N which satisfy

ΓABCD = O(u) as u → 0. ( 18
)
We need a couple more of definitions and properties to be able to deal with spinor functions on N .

In general, a holomorphic spinor field ψ on N c is represented on SL(N c ) by a holomorphic spinor-valued function ψ A 1 ...A j , given by the components of ψ in corresponding spin frame. We shall use the notation ψ k = ψ (A 1 ...A j ) k , k = 0, .., j, where (......) k denotes the operation 'symmetrize and set k indices equal to 1 the rest equal to 0'. These functions completely specify ψ if ψ is symmetric. They are then referred to as the essential components of ψ.

As the induced map π of N into N c is singular on U 0 , not every holomorphic function of the z a can arise as a pull-back to N of a holomorphic function on N c . The former must have a special type of expansion in terms of the z a which reflects the particular relation between the 'angular' coordinate v and the 'radial' coordinate u. The following definition and lemma, taken from [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF], are needed for manipulating spinor functions on N . Definition 3.1. A holomorphic function g on N is said to be of v-finite expansion type k g , with k g an integer, if it has in terms of the coordinates u, v, and w a Taylor expansion at the origin of the form

g = ∞ p=0 ∞ m=0 2m+kg n=0 g m,n,p u m v n w p
where it is assumed that g m,n,p = 0 if 2m + k g < 0.

Lemma 3.1. Let φ A 1 ...A j be a holomorphic, symmetric, spinor-valued function on SL(N c ). Then the restrictions of its essential components

φ k = φ (A 1 ...A j ) k , 0 ≤ k ≤ j, to N satisfy ∂ v φ k = (j -k)φ k+1 , k = 0, ..., j, on U 0 ,
(where we set φ j+1 = 0) and φ k is of expansion type j -k.

As stated at the beginning of this section, prescribing the null data is equivalent to knowing φ α | Nq . Following [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] it is possible to see how this fit into our particular gauge. Consider the normal frame c AB on N c near q and denote the null data of h in this frame by

D α * = {D (ApBp ...D A 1 B 1 ) φ α (q), p = 1, 2, 3, ...}, then on W 0 φ α (u, v) = ∞ m=0 2m n=0 ψ α m,n u m v n , (19)
where

ψ α m,n = 1 m! 2m n D (AmBm ...D A 1 B 1 )n φ α (q), 0 ≤ n ≤ 2m.
This shows how to determine φ α (u, v) from the null data D α * and vice versa.

The field equations on N

Now, having the coordinates and frame field, we can use the frame calculus in its standard form. Given the fields φ α and using the frame e AB and the connection coefficients Γ ABCD on N , we set

r ABCDEF ≡ e CD (Γ EF AB ) -e EF (Γ CDAB ) + Γ EF K C Γ DKAB + Γ EF K D Γ CKAB -Γ CD K E Γ KF AB -Γ CD K F Γ EKAB + Γ EF K B Γ CDAK -Γ CD K B Γ EF AK -t CD GH EF Γ GHAB
R ABCDEF ≡ r ABCDEF - 1 2 S ABCE - 1 6 Rh ABCE DF + S ABDF - 1 6 Rh ABDF CE , σ α AB ≡ D AB φ α -φ α AB , Σ α AB ≡ D P A φ BP + 1 2 AB f α .
It is important to note that R and S ABCD , defined in (15), ( 14), represent functions of the fields that need not bear relation with the geometric Ricci scalar and trace-free part of the Ricci tensor. The equality among the quantities defined in terms of the fields φ α and its derivatives and the geometric quantities comes when the field equations are solved.

The tensor fields on the left hand side have been introduced as labels for the equations and for discussing in an ordered manner their interdependencies. In terms of these tensor fields, the field equations read

t AB EF CD e a EF = 0, R ABCDEF = 0, σ α AB = 0, Σ α AB = 0.
The first equation is Cartan's first structural equation with the requirement that the metric connection be torsion free. The second equation is equation (13). The third equation define the symmetric spinors φ α AB . The last equation is the field equation (12) in therms of φ α AB . We want to see how to calculate in our particular gauge a formal expansion of the fields using the initial data in the form φ α (u, v). As the system of field equations is overdetermined we have to choose a subsystem of it. In the rest of this section we choose a particular subsystem, writing the chosen equations in our gauge, and at the end we see how a formal expansion is determined by these equations and the initial data.

The σ α 00 = 0 equations

The first set of equations that needs particular attention are the equations σ α 00 = 0. In our gauge they read

∂ u φ α = φ α
00 . These equations are used in the following to calculate φ α 00 each time we know φ α as a function of u. In particular, as φ α will be prescribed on W 0 as part of the initial data, this equation allows us to calculate φ α 00 there immediately.

The '∂ u -equations'

We now present what we will refer to as the '∂ u -equations'. These equations are chosen because they have the following features. They are a system of PDE's for the set of functions êa A1 , ΓA1CD and φ α A1 , which comprise all the unknowns with the exceptions of the free data φ α and the derived functions φ α 00 . They are all interior equations on the hypersurfaces {w = w 0 } in the sense that only derivatives in the directions of u and v are involved, in particular, if we consider the hypersurface W 0 , they are all inner equations in N q . The possibility of choosing such a subsystem of inner equations in N q is due to the coincidence of the characteristic cones for all the unknowns. Also they split into a hierarchy that will be presented in the following section. The '∂ u -equations' are: Equations t AB EF 00 e a EF = 0 :

∂ u ê1 01 + 1 u ê1 01 = -2 Γ0101 + 2 Γ0100 ê1 01 , ∂ u ê2 01 + 1 u ê2 01 = 1 u Γ0100 + 2 Γ0100 ê2 01 , ∂ u ê1 11 = -2 Γ1101 + 2 Γ1100 ê1 01 , ∂ u ê2 11 = 1 u Γ1100 + 2 Γ1100 ê2 01 .
Equations R AB00EF = 0 :

∂ u Γ0100 + 2 u Γ0100 -2 Γ2 0100 = 1 2 (F α ∂ u φ α 00 + F αβ φ α 00 φ β 00 ), ∂ u Γ0101 + 1 u Γ0101 -2 Γ0100 Γ0101 = 1 2 (F α ∂ u φ α 01 + F αβ φ α 00 φ β 01 ), ∂ u Γ0111 + 1 u Γ0111 -2 Γ0100 Γ0111 = 1 2 (F α ∂ u φ α 11 + F αβ φ α 00 φ β 11 + f ) - 1 12 f, ∂ u Γ1100 + 1 u Γ1100 -2 Γ0100 Γ1100 = F α ∂ u φ α 01 + F αβ φ α 00 φ β 01 , ∂ u Γ1101 -2 Γ1100 Γ0101 = F α ∂ u φ α 11 + F αβ φ α 00 φ β 11 + f + 1 12 f, ∂ u Γ1111 -2 Γ1100 Γ0111 = F α 1 2u ∂ v φ α 11 + ê1 01 ∂ u φ α 11 + ê2 01 ∂ v φ α 11 -2 Γ0111 φ α 01 + 2 Γ0101 φ α 11 + F αβ φ α 01 φ β 11 . Equations Σ α A0 = 0 : ∂ u φ α 01 = 1 2u (∂ v φ α 00 -2φ α 01 ) + ê1 01 ∂ u φ α 00 + ê2 01 ∂ v φ α 00 -2 Γ0101 φ α 00 + 2 Γ0100 φ α 01 , ∂ u φ α 11 = 1 2u (∂ v φ α 01 -φ α 11 ) + ê1 01 ∂ u φ α 01 + ê2 01 ∂ v φ α 01 -Γ0111 φ α 00 + Γ0100 φ α 11 + 1 2 f α .

The ∂ u -equations hierarchy

The system of ∂ u -equations splits into a hierarchy of subsystems as follows H1) R 000001 = 0, H2) t 01 EF 00 e 2 EF = 0, H3) t 01 EF 00 e 1 EF = 0, R 010001 = 0, Σ α 00 = 0, H4) R 000011 = 0, H5) t 11 EF 00 e 2 EF = 0, H6) R 110001 = 0, Σ α 10 = 0, H7) R 010011 = 0, H8) R 110011 = 0, H9) t 11 EF 00 e 1 EF = 0. The defining property of the hierarchy is the following. If φ α are prescribed on {w = w 0 } and using σ α 00 = 0, then H1 reduces to an ODE. Once we have its solution, H2 reduces to an ODE. Given its solution, H3 reduces to a system of ODE's, with coefficients that are calculated by operations interior to {w = w 0 } from the previously known or calculated functions. This procedure continues till the end of the hierarchy. So, given φ α on {w = w 0 } and the appropriate initial data on U 0 ∩ {w = w 0 }, all the unknowns can be determined on {w = w 0 } by solving a sequence of ODE's in the independent variable u.

The '∂ w -equations'

The initial data, φ α are prescribed on W 0 , and to determine their evolution off W 0 we need the equations σ α 11 = 0, referred to as the ∂ w -equations, which read

∂ w φ α = φ α 11 -ê1 11 ∂ u φ α -ê2 11 ∂ v φ α .

Calculating the formal expansion

Let us call X any of the unknowns that we are solving for, i.e. êa AB , ΓABCD , φ α , φ α AB . We want to see that it is possible to obtain a formal expansion of X in a neighbourhood of q using the σ α 00 -equations, the ∂ u -equations and the ∂ w -equations, given φ α on W 0 as our datum. We give here an inductive argument showing that with our setting ∂ k w X| W 0 can be determined for all k.

We need the following conditions, obtained from the gauge requirements (17), (18)

∂ k w êa A1 | I = 0, a= 1, 2, A= 0, 1, k≥ 0, ∂ k w ΓA1CD | I = 0, A,C,D= 0, 1, k≥ 0,
and from the σ α 00 = 0 equations and the spinorial behaviour as discussed in Lemma 3.1,

∂ k w φ A1 | I = 1 2 ∂ u ∂ 1+A v ∂ k w φ| I , A= 0, 1, k≥ 0,
which are our initial conditions for the ∂ k w -derivatives of the ∂ u -equations. Using the initial conditions for k = 0 and following what has been said in Subsection 4.3 we successively integrate the subsystems H1 to H9 to determine all of X on W 0 .

As inductive hypothesis we assume as known

∂ p w X| W 0 , 0 ≤ p ≤ k -1, k ≥ 1. Applying formally ∂ k-1 w
to the ∂ w -equations, and restricting them to W 0 , we find ∂ k w φ α | W 0 in terms of known functions. We apply formally ∂ k w to the ∂ u -equations. This is a system of PDE's where the unknowns are ∂ k w X. Keeping the discussed hierarchy and considering the functions that we already know on W 0 , it again becomes a sequence of ODE's, which together with the initial conditions on I can be integrated on W 0 . Thus we know ∂ k w X| W 0 and the induction step is completed.

The procedure just stated shows that we know ∂ k w X| W 0 for all k. Expanding these functions around q = {u = 0, v = 0, w = 0} gives

∂ m u ∂ n v ∂ p w X| q ∀ m, n, p,
and the procedure gives a unique sequence of expansion coefficients for all the functions in X.

Lemma 4.1. The procedure described above determines at the point O = (u = 0, v = 0, w = 0) from the data φ α given on W 0 according to (19) a unique sequence of expansion coefficients

∂ m u ∂ n v ∂ p w X(O), m,n,p= 0, 1, 2, ...
where X stands for any of the functions êa AB , ΓABCD , φ α , φ α AB . If the corresponding Taylor series are absolutely convergent in some neighbourhood Q of O, they define a solution to the equation σ α 00 = 0, to the ∂ u -equations and to the ∂ w -equations on Q.

By Lemma 3.1 we know that all spinor-valued functions should have a specific vfinite expansion type. The following lemma, whose proof is quite similar to the proof in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF], is important for handling the estimates in the following section.

k ê1 AB = -A -B, k ê2 AB = 3 -A -B, AB = 01, 11, k Γ01AB = 2 -A -B, k Γ11AB = 1 -A -B, A, B = 0, 1, k α φ = 0, k φ α AB = 2 -A -B, A, B = 0, 1.

Convergence of the formal expansion

In the previous section we have seen how to calculate a formal expansion for êa AB , ΓABCD , φ α , φ α AB given φ α | W 0 , or, what is the same, given the null data. From Lemma 2.2 we know which are the necessary conditions on the null data in order to have analytic solutions of the conformal field equations. In this section we show that those conditions are also sufficient for the formal expansion determined in the previous section to be absolutely convergent.

We consider the abstract null data as given by n sequences

Dα = {ψ α A 1 B 1 , ψ α A 2 B 2 A 1 B 1 , ψ α A 3 B 3 A 2 B 2 A 1 B 1 , .
..} of totally symmetric spinors satisfying the reality condition (8) and we construct φ α | W 0 by setting in the expansions ( 19)

D (AmBm ...D A 1 B 1 ) φ α (q) = ψ α AmBm...A 1 B 1 , m≥ 0.
Observing Lemma 2.2 one finds as a necessary condition for the functions φ α on W 0 to determine an analytic solution to the conformal static vacuum field equations that its non-vanishing Taylor coefficients at the point O satisfy estimates of the form

|∂ m u ∂ n v φ α (O)| ≤ 2m n m!n! M r m , m≥ 0, 0 ≤ n ≤ 2m. ( 20
)
This conditions are also sufficient for φ α (u, v) to be holomorphic functions on W 0 . So the null data give rise to n analytic functions φ α on W 0 . From σ α 00 = 0 we have φ α 00 = ∂ u φ α , so having φ α | W 0 we have φ α 00 | W 0 , which is also an analytic function on W 0 .

Following Lemma 6.1 in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF], we can derive from (20) slightly different type of estimates for φ α (u, v) which are more convenient in our case. Lemma 5.1. Let e be the Euler number. For given ρ φ α in R, such that 0 < ρ φ α < e 2 , there exist positive constants c φ α , r φ α so that (20) imply estimates of the form

|∂ m u ∂ n v φ α (O)| ≤ c φ α r m φ α m!ρ n φ α n! (m + 1) 2 (n + 1) 2 , m≥ 0, 0 ≤ n ≤ 2m. ( 21
)
We present our estimates.

φ α (u, v) = ∞ m=0 2m n=0 ψ α m,n u m v n
so that its Taylor coefficients at the point O satisfy estimates of the type (21) with some positive constants c φ α , r φ α , and ρ φ α < 1 8 . Then there exist positive constants r, ρ, c êa AB , c ΓABCD , c φ α AB so that the expansion coefficients determined from φ α in Lemma 4.1 satisfy for m, n, p = 0, 1, 2, ...

|∂ m u ∂ n v ∂ p w X(O)| ≤ c X r m+p+q X (m + p)!ρ n n! (m + 1) 2 (n + 1) 2 (p + 1) 2 (22)
where X stands for any of the functions êa AB , ΓABCD , φ α , φ α AB and q êa AB = q ΓABCD = -1, q φ α = q φ α AB = 0. Remark. Taking into account the v-finite expansion types of the functions X obtained in Lemma 4.2, we can replace the right hand sides in the estimates above by zero if n is large enough relative to m. This will not be pointed out at each step and for convenience the estimates will be written as above.

We take the following four lemmas from [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF]. The first states the necessary part of the estimates, and the other three are needed in order to manipulate the estimates in the proof of Lemma 5.2. Lemma 5.3. If g is a holomorphic function near O, then there exist positive constants c, r 0 , ρ 0 such that 2 , m, n, p = 0, 1, 2, ... for any r ≥ r 0 , ρ ≥ ρ 0 . If in addition g(0, v, 0) = 0, the constants can be chosen such that

|∂ m u ∂ n v ∂ p w g(O)| ≤ c r m+p (m + p)!ρ n n! (m + 1) 2 (n + 1) 2 (p + 1)
|∂ m u ∂ n v ∂ p w g(O)| ≤ c r m+p-1 (m + p)!ρ n n! (m + 1) 2 (n + 1) 2 (p + 1) 2 , m, n, p = 0, 1, 2, ... for any r ≥ r 0 , ρ ≥ ρ 0 .
Lemma 5.4. For any non-negative integer n there is a positive constant C, C > 1, independent of n so that

n k=0 1 (k + 1) 2 (n -k + 1) 2 ≤ C 1 (n + 1) 2 .
In the following C will always denote the constant above. Lemma 5.5. For any integers m, n, k, j, with 0 ≤ k ≤ m, and 0 ≤ j ≤ n resp.

0 ≤ j ≤ n -1 holds m k n j ≤ m + n k + j resp. m k n -1 j ≤ m + n k + j .
Lemma 5.6. Let m, n, p be non-negative integers and g i , i = 1, ..., N, be smooth complex valued functions of u, v, w on some neighbourhood U of O whose derivatives satisfy on U (resp. at a given point q ∈ U ) estimates of the form

|∂ j u ∂ k v ∂ l w g i (O)| ≤ c i r j+l+q i (j + l)!ρ k k! (j + 1) 2 (k + 1) 2 (l + 1) 2 for 0 ≤ j ≤ m, 0 ≤ k ≤ n, 0 ≤ l ≤ p,
with some positive constants c i , r, ρ and some fixed integers q i (independent of j, k, l). Then one has on U (resp. at q) the estimates

|∂ m u ∂ n v ∂ p w (g 1 • ... • g N )(O)| ≤ C 3(N -1) c 1 • ... • c N r m+p+q 1 +...+q N (m + p)!ρ n n! (m + 1) 2 (n + 1) 2 (p + 1) 2 . ( 23
)
Remark. This lemma remains true if m, n, p are replaced in (23

) by integers m , n , p with 0 ≤ m ≤ m, 0 ≤ n ≤ n, 0 ≤ p ≤ p.
The factor C 3(N -1) in ( 23) can be replaced by C (3-s)(N -1) if s of the integers m, n, p vanish.

From now on we consider that a function in a modulus sign is evaluated at the origin O.

Proof of Lemma 5.2. The proof is by induction, following the inductive procedure which led to Lemma 4.1. A general outline is as follows. We start leaving the choice of the constants r, ρ, c f open. We use the induction hypothesis and the equations that lead to Lemma 4.1 to derive estimates for the derivatives of the next order. These estimates are of the form

|∂ m u ∂ n v ∂ p w X| ≤ c X r m+p+q X (m + p)!ρ n n! (m + 1) 2 (n + 1) 2 (p + 1) 2 A X (24)
with certain constants A X which depend on m, n, p and the constants c X , r and ρ. Sometimes superscripts will indicate to which order of differentiability particular constants A X refer. In the way we will have to make assumptions on r to proceed with the induction step. We shall collect these conditions and the constants A X , or estimates for them, and at the end it will be shown that the constants c X , r and ρ can be adjusted so that all conditions are satisfied and A X ≤ 1. This will complete the induction proof.

In order not to write long formulae that do not add to the understanding of the procedure, we state here some properties that are used to simplify the estimates:

• During the procedure we need estimates on the derivatives of the functions f , f α , F α and F αβ , which are analytic functions of the fields φ α and φ α AB . The functions are also scalars if we consider them as functions on the manifold through their dependence on the fields. This and the Cauchy estimates for derivatives of analytic functions allow us to get the following result. Let us denote by g any of the functions f , f α , F α and F αβ . If, for r ≥ r 0 , 0

≤ j ≤ m, 0 ≤ k ≤ n, 0 ≤ l ≤ p, |∂ j u ∂ k v ∂ l w φ α | ≤ c φ α r j+l (j + l)!ρ k k! (j + 1) 2 (k + 1) 2 (l + 1) 2 , |∂ j u ∂ k v ∂ l w φ α AB | ≤ c φ α AB r j+l (j + l)!ρ k k! (j + 1) 2 (k + 1) 2 (l + 1) 2 then there exists positive constants c g , R 0 ≥ r 0 such that for R ≥ R 0 |∂ m u ∂ n v ∂ p w g| ≤ c g R m+p (m + p)!ρ n n! (m + 1) 2 (n + 1) 2 (p + 1) 2 . ( 25
)
Using this for calculating the estimates leads to terms with the factor R r . We can then fix R by making R = R 0 and add to the requirements for r that r ≥ R 0 . We have then R r ≤ 1.

(26)

• After calculating the estimates and using (25) and (26) we find that all the A's satisfy inequalities of the form

A ≤ α + α r ,
where α, α are constants that do not depend on r. If α = 0 then we have to show that we can make α ≤ 1. If the α's are not zero we can take a constant a, 0 < a < 1, and require that α ≤ a and then choose r large enough such that α r ≤ 1 -a. In the estimates that follows, we shall not write the explicit expressions for the α's, as they do not play any role if we are able to make r big enough at the end of the procedure.

As the φ α 's are analytic functions of u and v on W 0 , also the φ α 00 's are. Then there exist positive constants 0 < ρ φ α , ρ φ α 00 ≤ 1 8 and c φ α , c φ α 00 , r φ α , r φ α 00 such that

|∂ m u ∂ n v φ α | ≤ c φ α r m φ α m!ρ n φ α n! (m + 1) 2 (n + 1) 2 , |∂ m u ∂ n v φ α 00 | ≤ c φ α 00 r m φ α 00 m!ρ n φ α 00 n! (m + 1) 2 (n + 1) 2 .
As the inequalities do not change if the constants are changed for bigger constants, we choose (but leaving the precise value open)

r ≥ max{r φ α , r φ α 00 }, ρ≥ max{ρ φ α , ρ φ α 00 }, then |∂ m u ∂ n v ∂ 0 w φ α | ≤ c φ α r m m!ρ n n! (m + 1) 2 (n + 1) 2 , |∂ m u ∂ n v ∂ 0 w φ α 00 | ≤ c φ α 00 r m m!ρ n n! (m + 1) 2 (n + 1) 2 .
Using how the frame fields and the coordinates where constructed,

êa AB | U 0 = 0 ⇒ |∂ 0 u ∂ n v ∂ p w êa AB | = 0, ΓABCD | U 0 = 0 ⇒ |∂ 0 u ∂ n v ∂ p w ΓABCD | = 0.
From the required spinorial behaviour we have

φ 01 = 1 2 ∂ v φ 00 , φ 11 = ∂ v φ 01 , on U 0 , then |∂ 0 u ∂ n v ∂ 0 w φ 01 | = 1 2 |∂ n+1 v φ 00 | ≤ 1 2 c φ α 00 ρ n+1 (n + 1)! (n + 2) 2 = c φ α 01 ρ n n! (n + 1) 2 A m=0,p=0 φ α 01 , where A m=0,p=0 φ α 01 = 1 2 c φ α 00 c φ α 01 ρ (n + 1) 3 (n + 2) 2 ≤ 1 2 c φ α 00 c φ α 01 ρ,
as k φ α 01 = 1 and then n = 0, 1. Now that we have this inequality, and in the same way, we get

A m=0,p=0 φ α 11 ≤ 1 4 c φ α 01 c φ α 11 ρ.

Now we have estimates for

∂ 0 u ∂ n v ∂ 0 w X, with n ∈ N 0 .
We assume as inductive hypothesis that we have estimates for

∂ l u ∂ n v ∂ 0 w X, with 0 ≤ l ≤ m -1, n ∈ N 0 .
We already have those estimates for φ α and φ α 00 . We use the ∂ u -equations to get estimates for ∂ m u ∂ n v ∂ 0 w X for the rest of the unknowns. The estimates obtained in this step are less restrictive than the estimates for general p, so we skip the enumeration of them and go on to the next step.

We assume that we have estimates for ∂ m u ∂ n v ∂ l w X, with 0 ≤ l ≤ p -1, m ∈ N 0 , n ∈ N 0 , and use the system of equations and the properties stated on the lemmas of this section to get estimates for ∂ m u ∂ n v ∂ p w X. Using the ∂ w -equations we get

A p≥1 φ α ≤ 4C 3 c ê1 11 + αp≥1 φ α r .
Using the equation D 11 φ α 00 = D 00 φ α 11 , which follows from σ α 00 = 0 and σ α 11 = 0,

A p≥1 φ α 00 ≤ 4 c φ α 00 c φ α 11 + C 3 c ê1 11 c φ α 00 + αp≥1 φ α 00 r .
For m = 0 and p ≥ 1 it is not possible to use the ∂ u -equations to get estimates, so it is necessary to use the spinorial behaviour to get, as before r . We take a constant a, 0 < a < 1, whose precise value will be fixed later. We see that if we can make r arbitrarily large, then all the A's are less or equal to 1 if the following inequalities are satisfied: We define

A m=0 φ α 01 ≤ 1
c Γ0100 2c ê2 01 ≤ a, c Γ1100 c ê2 11 ≤ a, C 3 c F α c φ α 00 2c Γ0100 ≤ a, C 3 c F α c φ α 01 2c Γ0101 ≤ a, C 3 c F α c φ α 11 2c Γ0111 ≤ a, C 3 c F α c φ α 01 c Γ1100 ≤ a, C 3 c F α c φ α 11 c Γ1101 ≤ a, C 3 (ρ+C 3 c ê1 01 )c F α c φ α 11 c Γ1111 ≤ a,
ρ ≡ max{ρ φ α , ρ φ α 00 } ≤ 1 8 and a ≡ (32ρ 2 ) 1 3 < 1.
Considering the last two inequalities, we define

c ê1 01 ≡ ρ C 3 , c φ α 01 ≡ 2ρc φ α 00 a , c φ α 11 ≡ 2ρc φ α 01 a .
With the previous definitions and c ê1 11 ≡ 4ρ 2 C 3 a 2 the third to last inequality is satisfied. The rest of the inequalities are satisfied by defining

c Γ0100 ≡ C 3 c F α c φ α 00 2a , c Γ0101 ≡ C 3 c F α c φ α 01 2a , c Γ0111 ≡ C 3 c F α c φ α 11 2a , c Γ1100 ≡ C 3 c F α c φ α 01 a , c Γ1101 ≡ C 3 c F α c φ α 11 a , c Γ1111 ≡ C 3 (ρ+C 3 c ê1 01 )c F α c φ α 11 a , c ê2 01 ≡ c Γ0100 2a , c ê2 11 ≡ c Γ1100
a . Now we can take r big enough so that all the A's are less or equal than 1.

The following lemma states the convergence result. The proof follows as the one given in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF].

Lemma 5.7. The estimates (22) for the derivatives of the functions X and the expansion types given in Lemma 4.2 imply that the associated Taylor series are absolutely convergent in the domain |v| < 1 αρ , |u| + |w| < α 2 r , for any real number α, 0 < α ≤ 1. It follows that the formal expansions determined in Lemma 4.1 define indeed a (unique) holomorphic solution to the ∂ u -equations, the ∂ w -equations and the σ α 00 = 0 equations, which induces the data φ α on W 0 .

The complete set of equations on N

We have seen in Section 4 how to calculate a formal expansion for our fields using a subset of the field equations. In the previous section we have shown that these formal expansions are convergent in a neighbourhood of q. In this section we shall show that these fields satisfy the complete system of field equations. First, we prove that the field equations are satisfied in the limit as u → 0. Second, we derive a subsidiary system of equations, for which the first result provides the initial conditions, and which allows us to prove that the complete system is satisfied.

Lemma 6.1. The functions êa AB , ΓABCD , φ α , φ α AB , whose expansion coefficients are determined by Lemma 4.1, with expansions that converge on an open neighbourhood of the point O, neighbourhood that we assume to coincide with N , satisfy the complete set of field equations on the set U 0 in the sense that the fields t AB CD EF , R ABCDEF , σ α AB , Σ α AB calculated from these functions on N \U 0 have vanishing limit as u → 0.

Proof. Taking into account that the equations used to calculate the expansion coefficients are already satisfied, it is left to show that t 01

In the equations R AB00EF = 0, which where used to calculate the unknowns as part of the ∂ u -equations, the quantities S 0000 , S 0001 , S 0011 , S 0111 and R are equal to the corresponding components of the trace-free part of the Ricci spinor and the Ricci scalar. Then the equations D EF S 00EF -1 6 D 00 R = 0 and D EF S 01EF -1 6 D 01 R = 0 are automatically satisfied, as they only include S 0000 , S 0001 , S 0011 , S 0111 and R, and if these quantities are replaced for their expression in terms of the field frame coefficients and the connection coefficients, then the two equations are the components of the contracted Bianchi identity. The equation that is not automatically satisfied is D EF S 11EF -1 6 D 11 R = 0, as it includes S 1111 , which has not been used as part of the procedure to calculate the unknowns. So we need to include

D EF S 11EF - 1 6 D 11 R = 0 (29)
as requirement for the complete system of field equations to be satisfied. So we have that if and only if (29) is satisfied then

t AB EF CD = 0, R ABCDEF = 0.
We assume from now on (29) to be satisfied and analyze the implications in the following section.

From the definition of σ α AB ,

D AB σ α CD -D CD σ α AB = -t AB EF CD D EF φ α + BC Σ α AD + AD Σ α CB , then ∂ u + 1 u σ α 01 = 2 Γ0100 σ α 01
, which together with the initial condition for u = 0 gives σ α 01 = 0, and together with the quantities that are already known to be zero from the ∂ u -equations

σ α AB = 0. The last equation imply that Σ α AB = -Σ α BA + t AG EF G
B D EF φ α , using the ∂ u -equations and what has already been deduced, Σ α AB = 0. So the full system of field equations is satisfied.

Conditions on

f , f α , F α , F αβ
We need to consider now the requirement (29). If we write it in full using the field equations, it takes the form

J αAB D 11 D AB φ α + K α D 11 φ α = 0, (30)
where J αAB is the spinor version of

J αa = - 1 2 ∂f ∂(D a φ α ) + 1 2 F β ∂f β ∂(D a φ α ) + ∂F α ∂φ β + F α F β D a φ β (31)
and

K α = - 1 2 ∂f ∂φ α + f F α + 1 2 F β ∂f β ∂φ α - 1 2 f β ∂F β ∂φ α + f β F βα (32) + ∂F αβ ∂φ γ - 1 2
∂F βγ ∂φ α + F αγ F β D a φ β D a φ γ . As D AB D CD φ α = D (AB D CD) φ α + 1 3 f α h ABCD ,
we can write (30) in the following form,

J α 00 D (11 D 11) φ α -2J α 01 D (01 D 11) φ α + J α 11 1 3 f α + D (00 D 11) φ α + K α D 11 φ α = 0.
If we now evaluate this expression at the origin all the quantities involved depend only on the free initial data. That is, J α AB , K α and f α are functions of φ α | 0 and D AB φ α | 0 . As D (AB D CD) φ α | 0 are also part of the free initial data, in order for the equality to be satisfied for all initial data we need that

J α AB = 0. (33)
So from (30) we are left with

K α D 11 φ α = 0.
Again evaluating this expression at the origin, as K α is a scalar function, and considering that the equality should be satisfied for all orthonormal frames at the origin, we also need that

K α = 0. (34)
Conditions (33) and (34) imply that the contracted Bianchi identity needs to be satisfied by both sides of (2) from the beginning. Let us consider (33). F α does not depend on D a φ α , then the condition using the expression (31) can be written as

∂ ∂(D a φ α ) (f -F β f β ) = 2 ∂F α ∂φ β + F α F β D a φ β .
This equation can only be integrated if

∂F α ∂φ β = ∂F β ∂φ α , (35)
and then

f -F α f α = ∂F α ∂φ β + F α F β D a φ α D a φ β + G(φ γ ),
where G(φ γ ) is a free function of φ γ . We can consider this equation as an expression for f in terms of the other functions, having

f = F α f α + ∂F α ∂φ β + F α F β D a φ α D a φ β + G(φ γ ).
If we put this into (34) using the expression (32) we get

1 2 ∂G ∂φ α -GF α + f β ∂F β ∂φ α -F α F β -F αβ + 1 2 ∂ 2 F γ ∂φ α ∂φ β -F α ∂F β ∂φ γ + F β ∂F γ ∂φ α - ∂F αβ ∂φ γ + 1 2 ∂F βγ ∂φ α -F α F β F γ -F αβ F γ D a φ β D a φ γ = 0.
This can be written in the following form

G αβ f β = G α + G αβγ D a φ β D a φ γ , (36)
where

G α = - 1 2 ∂G ∂φ α + F α G, G αβ = ∂F β ∂φ α -F α F β -F αβ , G αβγ = - 1 2 ∂ 2 F γ ∂φ α ∂φ β + F α ∂F β ∂φ γ -F β ∂F γ ∂φ α + ∂F αβ ∂φ γ - 1 2
∂F βγ ∂φ α + F α F β F γ + F αβ F γ . If det(G αβ ) = 0 then f ρ = (G ρα ) -1 (G α + G αβγ D a φ β D a φ γ )
Equation ( 35) means that the form corresponding to F α is closed, and therefore there exists a function F (φ γ ) such that F α = ∂F ∂φ α . So we are left with the possibility of choosing 1 2 (n 2 + n + 4) free functions of the fields φ γ , namely G, F , F αβ . Once we have these functions, we calculate F α = ∂F ∂φ α , and if det(G αβ ) = 0 we calculate f α . If det(G αβ ) = 0 then some of the f α 's are free functions, and we have conditions on the r.h.s. of (36).

Analyticity at q

The last needed step is to show that the holomorphic solution of Lemma 5.7 can be extended as to cover a full neighbourhood of the point q. This is not obvious from our construction as our gauge is singular. The proof that we can indeed get a holomorphic solution in a hole neighbourhood of q is in all similar to the respective proofs in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF][START_REF] Aceña | Convergent null data expansions at space-like infinity of stationary vacuum solutions[END_REF], therefore here we only explain the steps of the proof. The first step is to show that the obtained solution can be expressed in terms of the normal coordinates x a and the frame field c AB based on the frame c AB at q. This can be done by showing that for small |x c | the coordinate transformation x a → z a (x c ), where defined, is nondegenerate. This means that all the tensor fields entering the field equations can be expressed in terms of the normal coordinates x a and the normal frame field c AB . The coordinates x a cover a domain U in C 3 on which the frame vector fields c AB = c a AB ∂ x a exist, are linearly independent and holomorphic. Also in U the other tensor fields expressed in terms of the x a and c AB are holomorphic. However, by the singularity of our gauge, U does not contain the hypersurface x 1 + ix 2 = 0 but the boundary of U becomes tangent to this hypersurface at x a = 0.

The second step is to notice that the construction of the submanifold N was done based on the frame c AB at q. Starting with a different frame cAB at q all the previous constructions and derivations can be repeated if the estimates for the null data in the c AB -gauge can be translated to the same type of estimates for the null data in the cABgauge. Indeed, the estimates do translate into each other, and hence all the statements made about the solution in the c AB -gauge apply to the solution in the cAB -gauge, in particular statements about domains of convergence. Now, the solution in the cAB -gauge can be expressed in terms of normal coordinates xa based on the frame cAB at q, which cover a domain Ũ in C 3 . Again Ũ does not contain the hypersurface x1 + ix 2 = 0 but the boundary of Ũ becomes tangent to this hypersurface at xa = 0.

By the uniqueness statements made so far, the solution in the cAB -gauge and in normal coordinates xa is related to the solution in the c AB -gauge and in normal coordinates x a , on the domain U ∩ Ũ , by the rotation that takes cAB to c AB , corresponding to the rotation of normal coordinates. We can extend this as a coordinate and frame transformation to the solution obtained in the cAB -gauge to express all fields in terms of x a and c AB . Then the solution obtained in the c AB -gauge and the solution in the cAB -gauge are genuine holomorphic extensions of each other, as one covers the singular generator of the other one away from the origin in a regular way.

Therefore, the set U can be extended in such a way as to contain a punctured neighbourhood of the origin in which the solution is holomorphic in the normal coordinates x a and the normal frame c AB . Then the solution is in fact holomorphic on a full neighbourhood of the origin x a = 0, which represents the point q, as holomorphic functions in more than one dimension cannot have isolated singularities.

By Lemma 2.1 we have from null data satisfying the reality conditions a formal expansion of the solution with expansion coefficients satisfying the reality conditions. By the various uniqueness statements obtained in the lemmas, this expansion must coincide with the expansion in normal coordinates of the solution obtained above. This implies the existence of a 3-dimensional real slice on which the tensor fields satisfy the reality conditions. It is obtained by requiring the coordinates x a to assume values in R 3 . This completes the proof of Theorem 1.1.

Conclusions

We have seen how to determine a formal expansion of the solution to certain types of elliptic systems of equations at a given point using a minimal set of freely specifiable data, the null data. We have also obtained necessary and sufficient conditions on the null data for the formal expansion to be absolutely convergent, thus showing that the null data characterize all possible solutions in a neighbourhood of the given point.

The system of equations is general enough as to include as particular cases stationary Einstein-Maxwell fields, static Einstein-Maxwell-dilaton fields and harmonic maps coupled to gravity whose base space is three dimensional or where the spacetime is stationary.

One interesting outcome of the analysis are the conditions that the functions of the fields that enter the field equations need to satisfy for the existence of solutions. These conditions can be read as that the contracted Bianchi identity needs to be fulfilled by the system of equations. This seems to indicate that only geometrically well behaved systems of equations allow for general data and solutions. It could be interesting to analyze if this forces the system of equations to be the Euler-Lagrange equations for some Lagrangian. Along the same line, it would also be interesting to analyze whether the considered system of equations can generally arise from a dimensional reduction. This would be the case for example if one starts with a four dimensional stationary spacetime or in Kaluza-Klein type theories.

  Regarding the frame field, as seen on N , if one writes e AB = e a AB ∂ z a , then on N ,

  and define there the quantities t AB EF CD , R ABCDEF , σ α AB , Σ α AB by t AB EF CD e a EF ≡ 2Γ AB E (C e a D)E -2Γ CD E (A e a B)E -e a CD,b e b AB + e a AB,b e b CD ,

Lemma 4 . 2 .

 42 If the data φ α are given on W 0 as in (19) the formal expansions of the fields obtained in Lemma 4.1 correspond to ones of functions of v-finite expansion types given by

Lemma 5 . 2 .

 52 Assume φ α = φ α (u, v) are holomorphic functions defined on some open neighbourhood U of O = {u = 0, v = 0, w = 0} in W 0 = {w = 0} which have expansions of the form

C 3 (

 3 ρ+C 3 c ê1 01 )c F α c φ α

4(c φ α 11 +C 3 c ê1 11 c φ α 00 )

 1100

Acknowledgements

I would like to thank Helmut Friedrich for helpful discussions.

σ α 01 = 0 and Σ α AB = 0 in the limit u → 0. From the definition (see [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] for details) and using the way in which the coordinates and the frame field were constructed lim u→0 t 01 AB 11 = 0.

Using the definitions from R ABCDEF , r ABCDEF , t AB CD EF and the way in which the coordinates and the frame field were constructed, we have near u = 0

Taking the limit u → 0 and using on U 0 the ∂ u -equations, the ∂ w -equation and the spinorial behaviour of the quantities involved, we get

This finishes the proof showing that the complete system of field equations are satisfied in the limit as u → 0.

Lemma 6.2. The functions êa AB , ΓABCD , φ α , φ α AB , corresponding to the expansions determined in Lemma 4.1, satisfy the complete set of field equations on the set N .

Proof. We have seen that the field equations are satisfied in the limit u → 0, we proceed to deduce a system of equations that those quantities satisfy.

Following the proof of Lemma 5.5 in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] we find that

Also following the proof of Lemma 5.5 in [START_REF] Friedrich | Static vacuum solutions from convergent null data expansions at space-like infinity[END_REF] we get in our case that

Let us consider the coupled system of PDE's ( 27) and ( 28