Charles Frances

Karin Melnick

Karin Melnick The

Lorentzian Lichnerowicz

THE LORENTZIAN LICHNEROWICZ CONJECTURE FOR REAL-ANALYTIC, THREE-DIMENSIONAL MANIFOLDS

published or not. The documents may come

Introduction

The Lichnerowicz Conjecture in conformal Riemannian geometry was proved simultaneously by J. Ferrand and M. Obata. Recall that the conformal transformations of a semi-Riemannian manifold (M, g) form the group

Conf(M, [g]) = {f ∈ Diff(M ) : f * g = e 2λ g, λ ∈ C ∞ (M )}
and that this is a Lie group provided dim M ≥ 3. A subgroup H ≤ Conf(M, [g]) is called essential if it does not act isometrically with respect any g = e 2λ g in the conformal class [g] of g. The identity component of a Lie group H is denoted by H 0 . Theorem 1.1 (Ferrand '71 [26]/ Obata '71 [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF]). Let (M, g) be a compact, Riemannian manifold with dimension n ≥ 2. If Conf 0 (M, [g]) is essential, then (M, g) is conformally diffeomorphic to the round sphere S n .

The second author was partially supported by a Joan and Joseph Birman Fellowship for Women Scientists and NSF Award DMS-2109347.

The first attempt to characterize the round sphere by this property of its conformal group seems to have been by A. Lichnerowicz [START_REF] Lichnerowicz | Sur les transformations conformes d'une variété riemannieinne compacte[END_REF] around 1964. Ferrand actually proved a stronger result for all n ≥ 2, that the above conclusion holds whenever Conf(M, [g]) is essential. (In dimension 2, the theorem is a straightforward consequence of the uniformization theorem for Riemann surfaces.) She later proved a statement for noncompact M in [START_REF]The action of conformal transformations on a Riemannian manifold[END_REF]. The reader will find in [START_REF] Ferrand | Histoire de la réductibilité du groupe conforme des variétés riemanniennes[END_REF] a nice account by her of the subject.

Obata's proof is based on techniques from differential geometry and transformation groups, while Ferrand's is based on quasiconformal analysis. Two more, totally different proofs, also covering the noncompact case, were given in 1995 by Schoen [START_REF] Schoen | On the conformal and CR automorphism groups[END_REF], based on geometric PDEs, in particular, scalar curvature theory, and in 2007 by the first author [START_REF]Sur le groupe d'automorphismes des géométries paraboliques de rang 1[END_REF], using Cartan connections and dynamical techniques.

The question whether there is a higher-signature analogue of theorem 1.1 has been around for about thirty years (see [START_REF] Gromov | Lectures on transformation groups: geometry and dynamics, Surveys in Differential Geometry[END_REF]Sec 6.7]). Note that essentiality of a conformal action on a compact Riemannian manifold is equivalent to noncompactness of the group. In higher-signature, there is a wide, largely uncharted array of compact pseudo-Riemannian manifolds with noncompact isometry group. Accordingly, there is not such a simple characterization of which conformal groups can act essentially.

Even compact Lorentzian manifolds with essential conformal group occur in a wide variety of global geometries. The first author found infinitely-many topological types of compact manifolds, for each n ≥ 3, supporting infinitelymany nonequivalent Lorentzian conformal structures admitting an essential conformal flow, in [START_REF] Frances | Sur les variétés lorentziennes dont le groupe conforme est essentiel[END_REF]. Locally, however, they are all conformally equivalent to Minkowski space-that is, all known essential Lorentzian examples are conformally flat. The conjecture is: Lorentzian Lichnerowicz Conjecture (LLC). Let (M n , g) be a compact Lorentzian manifold with n ≥ 3. If Conf(M, [g]) is essential, then (M, g) is conformally flat.

For pseudo-Riemannian metrics of type (p, q) with p, q ≥ 2, there are rather simple, polynomial deformations g of the flat, (p, q)-Minksowski metric such that a compact quotient of (R n , g)\{0}, diffeomorphic to S 1 × S n-1 for n = p + q, is not conformally flat and admits an essential flow [START_REF]About pseudo-Riemannian Lichnerowicz conjecture[END_REF]. Thus it seems there is no version of the Lichnerowicz Conjecture true in signature higher than Lorentzian.

In this article, we prove the Lichnerowicz Conjecture for 3-dimensional, realanalytic Lorentzian manifolds: Theorem 1.2. Let (M, g) be a 3-dimensional, compact, real analytic, Lorentzian manifold. If Conf 0 (M, [g]) is essential, then (M, g) is conformally flat.

1.1. Previous work on the conjecture. It is nearly understood which connected Lie groups can act conformally and essentially on compact Lorentzian manifolds. The expectation is that any such group admits a local monomorphism into O (2, n).

Let (M, g) be a compact, pseudo-Riemannian manifold of type (p, q), p+q ≥ 3, and assume p ≤ q. Let H ≤ Conf(M, [g]) be a connected subgroup.

For H semisimple, Zimmer proved that rk R H ≤ p + 1 [START_REF] Zimmer | Split rank and semisimple automorphism groups of G-structures[END_REF]; moreover, if this rank is attained, then H necessarily acts essentially. Bader-Nevo proved that if H is simple and attains the maximal R-rank, then it is locally isomorphic to O(p+1, k+1), for p ≤ k ≤ q [START_REF] Bader | Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds[END_REF]. Under the same assumptions, the first author and Zeghib subsequently proved that M is conformally flat, and in fact conformally equivalent to a certain compact, conformally homogeneous model space, up to covering spaces [START_REF] Frances | Some remarks on conformal pseudo-Riemannian actions of simple Lie groups[END_REF]. For g Lorentzian, Pecastaing has shown that if H is noncompact, simple, and essential, then (M, g) is conformally flat [START_REF]Lorentzian manifolds with a conformal action of SL(2,R)[END_REF].

For H nilpotent, the authors proved in [START_REF] Frances | Nilpotent groups of conformal flows on compact pseudo-Riemannian manifolds[END_REF] that the nilpotence degree of H is at most 2p + 1, and that, when this maximal degree is attained, (M, g) is conformally flat and again equivalent to the homogeneous model, up to covering spaces. Moreover, if H has the maximal nilpotence degree, it necessarily acts essentially.

A recent result of the second author and Pecastaing [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF], supporting the LLC, does not assume any structure on the group, as above, but rather topological properties of the space. The theorem states that the conformal group of a compact, simply connected, analytic Lorentzian manifold is compact. The proof shows that noncompactness of H implies conformal flatness. By D'Ambra's Theorem [START_REF] Ambra | Isometry groups of Lorentz manifolds[END_REF], H noncompact is equivalent to H essential for such spaces. Conformal flatness leads to a contradiction of the simple connectedness assumption. The proof reduces to the case that the group is abelian.

1.2. Compact three-dimensional Lorentzian manifolds. One of our motivations for theorem 1.2 was the thorough understanding of isometries of compact, 3-dimensional Lorentzian manifolds. In [START_REF] Zeghib | Killing fields in compact Lorentz 3-manifolds[END_REF], Zeghib classified all such spaces admitting an unbounded isometric flow. The first author recently improved this classification to all such spaces admitting any noncompact isometry group-including in particular the case where the isometry group is infinite and discrete [START_REF]Lorentz dynamics on closed 3-manifolds[END_REF].

There are moreover many useful classifications of homogeneous models for 3-dimensional Lorentzian manifolds (eg, [START_REF] Calvaruso | On the Ricci operator of locally homogeneous Lorentzian 3-manifolds[END_REF][START_REF] Coley | Lorentzian spacetimes with constant curvature invariants in three dimensions[END_REF][START_REF] Sekigawa | On some 3-dimensional curvature homogeneous spaces[END_REF][START_REF] Rahmani | Lorentz geometry of the Heisenberg group[END_REF]). In [START_REF] Dumitrescu | Géométries lorentziennes de dimension 3: classification et complétude[END_REF], Dumitrescu and Zeghib classified all metrically homogeneous Lorentzian spaces X such that there is a compact, 3-dimensional Lorentzian manifold locally modeled on X, and they proved that these are all complete. 1.3. Overview of proof. The proof rests on the approach we have developed to conformal Lorentzian transformations in our previous papers, which in turn is based on techniques involving the Cartan connection associated to a conformal structure and on Gromov's results on automorphisms of rigid geometric structures. We moreover draw on some of the work specific to 3-dimensional Lorentzian manifolds referenced above as well as the recent advance in [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF], from which we draw two major parts of our proof.

In section 2, we use Zeghib's classification of unbounded 3-dimensional Lorentzian flows [START_REF] Zeghib | Killing fields in compact Lorentz 3-manifolds[END_REF] to show that (M, g) has an essential conformal vector field. Denoting such a vector field by X, we ultimately focus on the Lie algebra z X of local conformal vector fields commuting with X.

In section 4 we gather local results yielding conformal flatness, based on our previous work and Gromov's theory, applied in this 3-dimensional, analytic context. These are used throughout the paper, and they immediately imply that the dimension of z X is at most 4.

The remainder of the paper comprises four more or less distinct proofs, for each of the cases, dim z X equals 4, 3, 2, or 1. The case dim z X = 4 corresponds to (M, [g]) being locally conformally homogenous, which quickly leads to the conclusion that it is conformally flat, or X is inessential, a contradiction.

When dim z X = 3, it can be R 3 , heis(3), or aff(R) ⊕ R. In the case of heis(3), we explicitly find a coordinate chart exhibiting g as conformally flat. For aff(R) ⊕ R, we find a complete (G, X)-structure on a closed, invariant surface, and use this to show that the flow along X on this surface gives rise to conformal flatness.

When dim z X = 2, it is isomorphic to R 2 . We show in section 7.1 that z X globalizes and integrates to a cylinder action on M . Then the situation strongly resembles that of [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF]Sec 6]; in the remainder of section 7, we follow the outline of that proof to reach the desired conclusion.

Finally, when dim z X = 1, we use fixed points of the flow along X, guaranteed by Gromov's theory, to alternately reach a contradiction or conclude conformal flatness. The proof in this case follows section 5 of [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF].

Existence of an essential vector field

The objective of this section is to prove that, under the hypothesis that Conf 0 (M, [g]) is essential, there exists an essential conformal vector field, namely, a vector field generating an essential conformal flow. We denote by X conf (M ) the space of all conformal vector fields on M . Our proof will be specific to 3 dimensions, but will not require analyticity. Proposition 2.1. Let (M, g) be a compact, smooth, 3-dimensional Lorentzian manifold. If Conf 0 (M, [g]) is essential, then it contains an essential 1parameter subgroup. In fact, every 1-parameter subgroup which is not relatively compact is essential.

Proof: If Conf 0 (M, [g]) is essential, then it is necessarily noncompact. Let {ϕ t X } < Conf 0 (M, [g]
) be an unbounded 1-parameter group. Suppose, for a contradiction, that {ϕ t X } is inessential, that is, contained in Isom 0 (M, g ) for some g ∈ [g]. In this case, Zeghib's classification of noncompact Lorentzisometric flows on compact 3-dimensional manifolds [START_REF] Zeghib | Killing fields in compact Lorentz 3-manifolds[END_REF]Thm. 2] gives two possibilities for (M, g ):

(1) (M, g ) is flat and complete-that is, a compact quotient of Minkowski space.

(2) M ∼ = G/Γ ρ for G a finite cover of PSL 2 (R), and Γ ρ the image of a uniform lattice Γ < G under a homomorphism Id Γ × ρ into G × G; the image of ρ is in a 1-parameter hyperbolic or unipotent subgroup

{h t } < G. The metric g lifts to a G × {h t }-invariant metric on G.
It is a general fact that the conformal group of a flat, complete, Lorentzian manifold (M, g ) is inessential. Indeed, any f ∈ Conf(M ), can be lifted to a conformal transformation of Minkowski space R 1,2 , namely, an element of Sim(R 1,2 ) ∼ = (R * × SO(1, 2)) R 3 . It follows that f is a homothetic transformation, one for which the conformal distortion is a constant λ. But a homothety on a compact manifold is necessarily an isometry (consider the formula M dvol g = M dvol f * g ).

To simplify the argument for case (2), we initially assume that G = SL 2 (R).

Denote by Λ the kernel of ρ : Γ → {h t }, and by Λ Z the Zariski closure in G. Note that Λ is not solvable, because Γ, which is commensurable to a surface group, is not solvable. It follows that Λ Z = G. Denote by M the cover of M diffeomorphic to G. Because G acts isometrically on the left on M , it acts linearly on the finite-dimensional vector space V of global conformal Killing fields of M . This representation is given by an algebraic homomorphism α : SL 2 (R) → GL(V). Denote by X conf ( M ) the subspace of V comprising lifts of vector fields in X conf (M ). The restriction of α(Λ) to this subspace is trivial, hence the same holds for Λ Z = G.

When G is a quotient or a connected finite cover of SL 2 (R), the previous arguments are easily adapted: we lift or project Λ to SL 2 (R), as appropriate.

The representation of G on X conf ( M ) lifts to, or factors through, a representation α of SL 2 (R), for which the subgroup corresponding to Λ is trivial on X conf ( M ). The same holds for the Zariski closure, hence, this subspace is a trivial summand of α. We conclude that G centralizes X conf ( M ). Now G commutes with all lifts of elements h ∈ Conf 0 (M ). Let h be such a lift. Choose a lift x0 of x 0 to M , and let h *

g x0 = λg h.x 0 . Given x ∈ M , let f ∈ G with f.x 0 = x. As f commutes with h, h * g x = h * f * g x0 = f * h * g x0 = f * λg h.x 0 = λg f h.x 0 = λg h.x
Thus h is a homothety, and so is h. Because M is compact, h must be an isometry. We conclude that Conf 0 (M ) is inessential. ♦ Remark 2.2. We expect proposition 2.1 to hold in any dimension, but this fact would obviously require a more general proof.

By proposition 2.1, under the assumptions of theorem 1.2, there is an essential conformal vector field on M . We fix such a vector field and call it X.

Because we assume (M, [g]) to be real-analytic, so will X be real-analytic.

In what follows, we will work with the geometric structure defined by the pair ([g], X).

Local and infinitesimal symmetries of ([g], X)

The conformal structure [g] on M determines a rigid geometric structure of algebraic type, in the sense of Gromov (see [START_REF] Gromov | Rigid transformations groups, Géométrie Différentielle[END_REF]). It is also fruitful to consider the canonically associated Cartan geometry, modeled on the 3-dimensional Lorentzian Einstein space Ein 1,2 . The latter space can be obtained as

Ein 1,2 = (S 1 × S 2 / ι , [-dθ 2 ⊕ g S 2 ])
where ι is the antipodal map on both factors. It is a conformally homogeneous space PO(2, 3)/P , for P the stabilizer of a null line in R 2,3 . Denote G = PO(2, 3) with corresponding Lie algebra g. The Cartan geometry comprises (see [START_REF] Sharpe | Differential geometry : Cartan's generalization of Klein's Erlangen program[END_REF] Ch V, [START_REF] Čap | Parabolic geometries I[END_REF] Sec 1.6):

• a principal P -bundle π : M → M ; and

• a Cartan connection ω ∈ Ω 1 ( M , g) satisfying, for all x ∈ M , (1) ω x : T x M → g is a linear isomorphism (2) ω x.g • R g * = Ad g -1 • ω x ∀g ∈ P (3) ω d dt 0 (x.e tY ) ≡ Y ∀ Y ∈ p The pair ([g], X
), with X as in section 2, is also a rigid geometric structure of algebraic type. It is not quite a Cartan geometry, but rather an enhanced Cartan geometry, a notion which was studied in [START_REF] Pecastaing | On two theorems about local automorphisms of geometric structures[END_REF]Sec 4.4.1]. It is proved there that the properties of the local orbit structure which we will use are the same for enhanced Cartan geometries as for usual Cartan geometries.

3.1.

Local transformations and vector fields. The conformal group of M lifts to a group of automorphisms of the Cartan bundle M preserving ω.

Because ω gives a parallelization of M , the action of Conf(M ) on M is free and proper (see, eg, [START_REF] Kobayashi | Transformation groups in differential geometry[END_REF]Thm. 3.2]).

We will denote by Conf loc (M ) the pseudogroup of local conformal transformations of M . As for global conformal transformations, any element of Conf loc (M ) defined on some open subset U ⊂ M lifts to an embedding of π -1 (U ) into M commuting with the principal P -action and preserving ω. We will work below with the sub-pseudogroup Z loc X ⊂ Conf loc (M ) centralizing X, where defined.

Let Is loc (x) ⊂ Conf loc (M ) be the stabilizer of a point x ∈ M ; it is a group. Any choice of x ∈ π -1 (x) gives a monomorphism ι x : Is loc (x) → P , the isotropy monomorphism with respect to x, defined implicitly by

h.x = x.ι x(h)
We will denote the image Îx . A different choice x = x.p gives Îx = p -1 Îx p. Denote Is loc X (x) the subgroup of Is loc (x) centralizing X. For x ∈ π -1 (x), denote ( ÎX ) x the image of Is loc (x) under ι x.

A theorem of Amores [START_REF] Amores | Vector fields of a finite type G-structure[END_REF] says that on real-analytic manifolds, germs of local conformal vector fields can be uniquely extended along paths. It follows that the algebra of germs of local conformal vector fields defined around a point x ∈ M is independent of x, up to isomorphism. Moreover, local conformal vector fields on the universal cover of M extend to global ones; note that these may not necessarily be complete.

We will work below with the local conformal vector fields commuting with X. Amores' theorem also implies that these form a well-defined subspace of the local conformal vector fields on M , which we will denote z X .

3.2. Gromov's Frobenius theorem and isotropy. Under the assumption that M is compact and C ω , Gromov's Frobenius theorem [START_REF] Gromov | Rigid transformations groups, Géométrie Différentielle[END_REF] ensures that, at each point x, a finite number of infinitesimal conditions are sufficient for the production of local conformal transformations at x. In the setting of analytic Cartan geometries, the second author showed that the jets of the curvature κ (see section 4.2 for the definition) provide this sufficient condition [START_REF] Melnick | A Frobenius theorem for Cartan geometries, with applications[END_REF]. The jet of order i can be captured by a P -equivariant map D (i) κ : M → U (i) , where U (i) is a finite-dimensional vector space derived from the curvature module, on which P acts linearly. For the enhanced Cartan structure ([g], X), there are a corresponding curvature κ X and corresponding P -equivariant maps D

(i) κ X : M → U (i)
X [START_REF] Pecastaing | On two theorems about local automorphisms of geometric structures[END_REF]Sec 4.4.1]. In this setting, the Frobenius theorem says: Theorem 3.1. [30, Prop 3.8], [24, 1.6.C and 1.7.A] Let (M, g) be a compact, real-analytic, pseudo-Riemannian manifold. Let X be an analytic vector field on M . There exists an integer i 0 ∈ N such that two points x and y are related by an element of Conf loc (M ) if and only if D (i 0 ) κ(x) = D (i 0 ) κ(ŷ) for some x ∈ π -1 (x) and ŷ ∈ π -1 (y). The same holds for Z loc X with κ X in place of κ.

From theorem 3.1, the group Îx coincides with the stabilizer of D (i 0 ) κ(x) for the representation of P on U (i 0 ) ; similarly, ( ÎX ) x is the stabilizer of D (i 0 ) κ X (x). This leads to the following result on the structure of isotropy groups (which in fact holds more generally, for real-analytic rigid geometric structures of algebraic type). Theorem 3.2 (Gromov [START_REF] Gromov | Rigid transformations groups, Géométrie Différentielle[END_REF] 3.4.A ). Let (M, g) be a real-analytic pseudo-Riemannian manifold, and let X be a real-analytic vector field. With respect to any x ∈ π -1 (x), the isotropy images Îx and ( ÎX ) x are real-algebraic subgroups of P ; in particular, Is loc (x) and Is loc X (x) have finitely-many components.

3.3.

Local orbit stratification. In this section, we will focus on the structure of the Conf loc (M )-orbits in M . By the Conf loc (M )-orbit of a point x ∈ M , we mean all points that can be reached from x by applying a finite sequence of local conformal maps; it will be denoted O(x). The Z loc X -orbit of the point x is defined analogously, and denoted O X (x).

Here are consequences of Gromov's stratification theorem which will be used below (see also [START_REF] Melnick | A Frobenius theorem for Cartan geometries, with applications[END_REF]Thm 4.1], [START_REF] Pecastaing | On two theorems about local automorphisms of geometric structures[END_REF]Thm 4.19]). This theorem stems from the Frobenius theorem 3.1 and properties of orbits for algebraic actions. Theorem 3.3 (Gromov [24] 3.1.A, 3.2). Let (M, g) be a compact realanalytic pseudo-Riemannian manifold.

(1) For all x ∈ M , the orbit O(x) is a semi-analytic subset of M . It is locally closed and has finitely-many components. The same holds for O X (x). (2) For all x ∈ M , the closure O(x) is locally connected and contains a closed Conf loc (M )-orbit. The analogous properties hold for O X (x).

Recall that a subset S of a topological space is locally closed if S is open in the closure S. A set is semianalytic if it is locally cut out by finitely many analytic equalities and inequalities; see [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF] for properties of these sets. The closure of a semianalytic set is again semianalytic. Local connectedness of semianalytic sets can be found in [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF]Cor 2.7].

In a slight abuse of language, we will call the z X -orbit of a point x the set of points reachable from x by flowing along finitely many local vector fields in z X . A consequence of the proof of theorem 3.3 is that the z X -orbit of x is the connected component of O X (x) containing x. We will implicitly make this identification several times below.

3.4. Recurrence produces isotropy. In [START_REF]Variations on Gromov's open-dense orbit theorem[END_REF] and [START_REF]Lorentz dynamics on closed 3-manifolds[END_REF], the first author combined the Frobenius theorem 3.

Proof:

Let x ∈ M be recurrent for {ϕ t X }, and consider x ∈ π -1 (x). There are t k → ∞ and {p k } ⊂ P such that ϕ t k X .x.p -1 k → x. Because Conf(M ) acts properly on M , the sequence {p k } necessarily tends to infinity. Let D i 0 κ X : M → U (i 0 ) be the P -equivariant map given by theorem 3.1. This map is also Conf(M )-invariant, so that p k .D (i 0 ) κ X (x) → D (i 0 ) κ X (x). Now P acts algebraically on U (i 0 ) with locally closed orbits, which implies p k .D (i 0 ) κ X (x) = k .D (i 0 ) κ X (x) for some k tending to the identity in P . This implies existence of a noncompact stabilizer of D (i 0 ) κ X (x), coinciding with ( ÎX ) x , again by theorem 3.1. Because this stabilizer is moreover algebraic in P , it has noncompact identity component. ♦

Conformal curvature and vanishing conditions

In this section, we gather several sufficient conditions for conformal flatness, which will be applied throughout our proof. Although some definitions and results later in this section will be valid in higher dimensions, we assume for now that (M, g) is a 3-dimensional, smooth Lorentzian manifold.

4.1.

Cotton-York tensor. Recall that in dimension 3, the Weyl curvature vanishes, and the obstruction to conformal flatness is the Cotton-York tensor C ∈ Γ(∧ 2 T * M ⊗ T * M ) (see [START_REF] Eisenhart | Riemannian geometry[END_REF]II.28]):

C x (u, v, w) = (∇ w P )(u, v) -(∇ v P )(u, w)
for u, v, w ∈ T x M , where P is the Schouten tensor

P x (u, v) = Ric x (u, v) - 1 4 Sc(x)g x (u, v)
and Ric and Sc denote the Ricci and scalar curvatures of g, respectively. The Cotton-York tensor is conformally invariant, meaning it is independent of a choice of metric in the conformal class [g]. For f ∈ Conf(M ), in particular,

C f (x) (f * x u, f * x v, f * x w) = C x (u, v, w) ∀x ∈ M, u, v, w ∈ T x M
This tensor moreover satisfies the Bianchi identity, meaning it is in the kernel of the map to ∧ 3 T * M , and is totally trace-free.

The Cotton-York tensor is a section of the vector bundle associated to the following module U. Choose

I =   1 1 1  
and write SO(1, 2) for SO(I). Write E 1 , E 2 , E 3 for the standard basis of R 3 , and take ξ i = E t i I as basis for R 3 * , i = 1, 2, 3. Denote

U = (∧ 2 R 3 * ⊗ R 3 * ) C the 5-dimensional SO(1,
2)-module of trace-free tensors satisfying the Bianchi symmetry. It is the sum of five 1-dimensional weight spaces, for weights w = -2, -1, 0, 1, 2. The positive weight spaces are The Cartan curvature

U +2 = R(ξ 1 ∧ ξ 2 ⊗ ξ 1 ) U +1 = R(ξ 1 ∧ ξ 3 ⊗ ξ 1 -ξ 1 ∧ ξ 2 ⊗ ξ 2 ) Denote U + = U +1 + U +2 . Denote G 0 = R * × SO(1, 2) = CO(1, 2). Letting d ∈ R * act
Ω(X, Y ) = dω(X, Y ) + [ω(X), ω(Y )]
is a semi-basic 2-form on M . It is the obstruction to (M, M , ω) being locally isomorphic to the model Cartan geometry, which in our case is (Ein 1,2 , G, ω G ), where ω G is the Maurer-Cartan form of G. The values of Ω lie in the nilpotent radical p + of p, which is identified as a G 0 -representation with R 1,2 * . Via the Cartan connection ω, the curvature Ω can be identified with a Pequivariant function κ : M → ∧ 2 (g/p) * ⊗ p + where the P -representation on the target is built from the adjoint representation of G restricted to P . It factors through

P/P + ∼ = G 0 . As G 0 -modules, ∧ 2 (g/p) * ⊗ p + ∼ = ∧ 2 R 1,2 * ⊗ R 1,2 *
The Cartan connection identifies T * M with M × P p + . In this way, κ corresponds to the Cotton tensor, read in the conformal frames given by M /P + (see [START_REF] Čap | Parabolic geometries I[END_REF]Cor 1.6.8]). In particular, κ factors through a G 0 -equivariant map to U.

Holonomy sequences.

Because M is compact, while local conformal transformations act freely and properly on M , we can associate to an unbounded sequence in Conf loc (M ) an unbounded sequence in the principal group P .

Definition 4.1. Let x ∈ M and {f k } ⊂ Conf loc (M ), with f k defined on a neighborhood of x = π(x) for all k. A holonomy sequence for {f k } at x with respect to xk → x, is {p k } ⊂ P such that f k .x k .p -1 k → ŷ for some ŷ ∈ M
A pointwise holonomy sequence is {p k } as above for which xk ∈ π -1 (x) ∀k.

Note that given a holonomy sequence as above, f k .x k → y, where x k = π(x k ), y = π(ŷ). Given any sequence {f k } ⊂ Conf loc (M ) and any x k → x ∈ M , we may assume, after passing to a subsequence, that f k .x k converges in M , since M is compact. Then there are holonomy sequences, including pointwise holonomy sequences, for {f k } at x.

A conformal transformations is, in dimension at least 3, determined by its 2-jet at a point (see, eg, [25, Sec IV.6]). A holonomy sequence captures the 2-jets of {f k } along the sequence {x k } and thus turns out to be a useful tool to understand the local behavior of {f k } around {x k }.

Since the 2-jet of a map can be read in different 2-frames, holonomy sequences are far from unique. Some of the choices involved in their construction are accounted for by vertical equivalence: two holonomy sequences [START_REF] Bader | Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds[END_REF][START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF] as the stabilizer of a pair of nonorthogonal isotropic lines in R 2,3 , the torus A equals the R-split Cartan subgroup of G. Up to vertical equivalence, a holonomy sequence in P can be written

{p k }, {q k } are vertically equivalent if q k = a k p k b k for {a k }, {b k } ⊂ P bounded. 4.3.1. Taxonomy of holonomy sequences. The reductive group G 0 ∼ = CO(1, 2) has a KAK decomposition, in which the R-split Cartan subgroup A is two- dimensional. Under the embedding of G 0 in G = PO
p k = d k τ k , with d k ∈ A and τ k ∈ P + .
Denote by a the subalgebra corresponding to A. The standard choice of simple roots spanning a * comprises a long root, γ, and a short root, β. The latter vanishes on the R-factor in the decomposition g 0 = R ⊕ o(1, 2), and it can be thought of as the generator of the root space of o(1, 2). We take α = γ -β; it corresponds to the negative log conformal dilation in the standard representation on R 1,2 . Explicitly, for the quadratic form 2x

0 x 4 +2x 1 x 3 +x 2 2 on R 2,3 , a =                  a b 0 -b -a       , a, b ∈ R           
with α and β dual to the a and b parameters, respectively.

The symmetry (α, β) → (α, -β) can be realized by conjugation in P . Denote A the semigroup comprising all d ∈ A with β(ln d) ≤ 0. Up to vertical equivalence, a holonomy sequence in P may be assumed to be in A P + . Definition 4.2. Let {d k τ k } be an unbounded sequence in A P + , and let

D k = ln d k .
The sequence is said to be

• of bounded distortion if α(D k ) is bounded while β(D k ) → -∞ • contracting if (α + β)(D k ) → ∞. • balanced if α(D k ) + β(D k ) is bounded, but each term is unbounded. • mixed if β(D k ) → -∞ and α(D k ) → ∞, while (α + β)(D k ) → -∞. It is called linear if τ k ≡ 1. 4.3.2.
Stability and propagation of holonomy. The following definition is inspired by [START_REF]Isometry groups and geodesic foliations of Lorentz manifolds I: Foundations of Lorentz dynamics[END_REF] (see also [

} in V is V AS (p k ) = {v = lim v k ∈ V : p k .v k is bounded}
The following proposition is a version for sequences of [5, Prop 2.9] (see [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF]Prop 3.13] for the one-line proof):

Proposition 4.4. Given a P -module V, represent a continuous, {f k }-invariant section of the associated bundle M × P V by a continuous, P -equivariant, {f k }-invariant map σ : M → V. Given any holonomy sequence {p k } for {f k } with respect to xk → x, the value σ(x) ∈ V AS (p k ).
An unbounded sequence {d k τ k } in A P + is called stable if it is linear and contracting or balanced. In general, holonomy sequences for a given {f k } ⊂ Conf loc (M ) can be of different types at different nearby points; however, stable sequences enjoy the property of local propagation of holonomy.

The exponential map of the Cartan connection is the vehicle for propagation of holonomy. Any X ∈ g defines a vector field X on M by ω( X) ≡ X.

Denote the flow along X by {ϕ

t X }. The exponential map at x ∈ M is exp x(X ) = ϕ 1 X (x) ∈ M
for X in a sufficiently small neighborhood of 0 in g. A holonomy sequence for {f k } at x turns out to be also valid along certain exponential curves from x (see [START_REF]Local dynamics of conformal vector fields[END_REF]Prop 6.3] or [31, Prop 3.9]):

Proposition 4.5. Let {p k } be a holonomy sequence for f k at x, with respect to xk ∈ π -1 (x k ). Suppose given Y k → Y ∈ g\p for which Ad p k (Y k ) con- verges. Then, provided Y is in the domain of exp x, {p k } is also a holonomy sequence for {f k } at x = π • exp(x, Y ) with respect to x k = exp(x k , Y k ).
One derives easily from proposition 4.5 the following corollary. Assuming (M, g) is not conformally flat, we may thus assume that the set of points admitting a balanced or a contracting holonomy sequence is nowhere dense.

Proposition 4.8. Assume that (M, g) is real-analytic. If there exists an unbounded sequence {f k } ⊂ Conf loc (M ), all defined on a neighborhood U of x ∈ M , admitting a stable holonomy sequence at x, then (M, g) is conformally flat.

Proof: A stable, unbounded holonomy sequence {p k = d k τ k } is balanced or contracting. As remarked above, proposition 4.5 implies that all points in a neighborhood V ⊆ U of x admit the same stable holonomy sequence. By proposition 4.7, the Cotton-York tensor vanishes on V (see also [START_REF]Causal conformal vector fields, and singularities of twistor spinors[END_REF]Prop 5]). By the analyticity assumption, (M, g) is conformally flat everywhere. ♦ 4.5. Big isotropy implies conformal flatness. Next we recall a key linearization theorem for conformal vector fields:

Theorem 4.9 (Frances-Melnick [START_REF]Formes normales pour les champs conformes pseudo-riemanniens[END_REF] Thm 1.2). Let (M, g) be a real-analytic Lorentzian manifold of dimension at least 3. Let x ∈ M and let X ∈ X conf (M ) vanish at x, with local flow {ϕ t X } < Is loc (x). If for some x ∈ π -1 (x), the image ι x({ϕ t X }) < G 0 , then {ϕ t X } is linearizable in a neighborhood of x. Otherwise, (M, g) is conformally flat.

Combining the theorem above with the previous propositions yields two useful corollaries.

Corollary 4.10. Assume that (M, g) is real-analytic. If for some x ∈ M , the isotropy algebra Is loc (x) contains a 2-dimensional abelian subalgebra, then (M, g) is conformally flat.

Proof: Consider the linear part of the isotropy: given x ∈ π -1 (x), compose the isotropy homomorphism (ι x) * e : Is loc (x) → p with the projection to g 0 , to obtain λ x : Is loc (x) → g 0 . Because (M, g) is real analytic, if λ x is not injective, then (M, g) is conformally flat by theorem 4.9. Assuming λ x is injective, it has 2-dimensional, abelian image in g 0 ∼ = R ⊕ o(1, 2), necessarily containing the center. There is thus a linear, contracting 1parameter subgroup of Is loc (x). Proposition 4.8 implies in this case that (M, g) is conformally flat. ♦ Now let X ∈ X conf (M ) and x ∈ M . Let Z loc X ⊂ Conf loc (M ), z X , and Is loc X (x) be as in section 3.2. The Lie algebra of Is loc X (x) is denoted Is loc X (x).

Corollary 4.11. Assume (M, g) is real-analytic. If the dimension of Is loc X (x) is at least 2 for some x ∈ M , then (M, g) is conformally flat.

Proof: If X(x) = 0, namely X ∈ Is loc X (x), then Is loc X (x) contains a twodimensional abelian subalgebra, and we conclude by Corollary 4.10.

Otherewise, Is loc X (x) annihilates X(x) = 0. Again by theorem 4.9, we can assume that the image (ι x) * e (Is loc (x)) is in g 0 for some x ∈ π -1 (x). A 0-eigenvector in R 1,2 of a 2-dimensional subalgebra of g 0 is necessarily lightlike, and the annihilator of a lightlike vector has dimension exactly 2. It includes a diagonal subgroup {d t = e tD } with α(D) = -β(D); any unbounded sequence {d t k } has balanced, linear holonomy with respect to x. Proposition 4.8 thus applies and ensures that (M, g) is conformally flat. ♦ 4.6. A new curvature vanishing result. We will need the following strengthening of [START_REF]Formes normales pour les champs conformes pseudo-riemanniens[END_REF]Thm 1.4]. The proof is somewhat shorter for 3dimensional manifolds, so we restrict to that case here. Theorem 4.12. Let (M, g) be a real-analytic, compact, 3-dimensional Lorentzian manifold. Let {f k } ⊂ Conf(M, [g]) be an unbounded sequence. If {f k } admits a holonomy sequence at x ∈ M contained in P + , then (M, g) is conformally flat.

Notice that the hypothesis that {f k } admits a holonomy sequence at x ∈ M contained in P + is equivalent to both sequences of differentials {D x f k } and {(D x f k ) -1 } being bounded. The proof of the theorem is somewhat technical, and is deferred to section 9, at the end of the paper.

Case z X is 4-dimensional

We begin the proof of the main theorem 1.2. Recall that (M, g) is 3dimensional, compact, real-analytic, and Lorentzian. The group Conf 0 (M, [g]) is assumed to be essential. By proposition 2.1, it admits an essential conformal vector field X ∈ X conf (M ). Denote by z X the algebra of local conformal vector fields on M commuting with X (see section 3.2). By corollary 4.11, the dimension of z X is at most four. Of course, X ∈ z X , so it has dimension at least one. We will prove theorem 1.2 by analyzing each possible value of this dimension.

Suppose z X has dimension four. Corollary 4.11 implies that all z X -orbits have dimension 3, hence there is only one such orbit. In particular, (M, g) is locally conformally homogeneous and X is nowhere vanishing. Given x ∈ M , the isotropy Is loc X (x) fixes the nonzero vector X(x). By theorem 4.9, (M, g) is conformally flat, or there exists x ∈ π -1 (x) with ( ÎX ) x < G 0 . This subgroup is 1-dimensional, and two cases may occur.

If ( ÎX ) x is not unimodular, then it is conjugate in G 0 to a diagonal subgroup {diag(1, λ, λ 2 ) | λ ∈ R * }.
Any unbounded sequence of this group is balanced and linear. Then proposition 4.8 ensures that (M, g) is conformally flat.

Next suppose ( ÎX ) x is unimodular. Then there is λ

x ∈ [g] x on T x M which is preserved by D x f for every f ∈ Is loc X (x). Given y ∈ M , choose f ∈ Z loc X
sending x to y, and define λ y = f * λ x . This does not depend on the choice of f , because for another choice, say h ∈ Z loc X , the difference

h -1 • f ∈ Is loc X (x), which preserves λ x . The result is a metric λ ∈ [g] which is Z loc X - invariant. Note that λ is analytic; indeed, given y ∈ M , there are Y, Z, T ∈ z X defined on a neighborhood of y, with values at y spanning T y M . The map (u, v, w) → ϕ u Y • ϕ v Z • ϕ w T (y)
provides a local analytic chart around y, in which λ is analytic. We conclude that {ϕ t X } is inessential, a contradiction.

Case z X is 3-dimensional

The center of z X is nontrivial because it contains X. Thus if z X is 3dimensional, it could be isomorphic to R 3 , heis(3), or aff(R) ⊕ R. For the sake of efficiency, we will assume for the rest of this section that (M, g) is not conformally flat, in order to arrive at a contradiction with the fact that {ϕ t X } is essential. The results collected thus far lead to: Proposition 6.1.

(1) The flow {ϕ t X } has no singularities. (2) All z X -orbits have dimension at least two.

(3) There is a closed z X -orbit Σ, which is a torus or a Klein bottle, on which X is lightlike.

Proof: If there were a singularity x, the differential {D x ϕ t X } at x would fix two linearly independent vectors, the values at x of two elements of z X linearly independent modulo X. From the fact that the differential preserves g x up to scale, simple linear algebra leads to the conclusion that D x ϕ t X = Id TxM for all t, a contradiction with theorem 4.9.

Point (2) follows from corollary 4.11.

If g(X, X) were nonvanishing, then {ϕ t X } woud preserve g/g(X, X) and be inessential. Thus the zero set Λ of g(X, X) is nonempty and closed. By theorem 3.3 (2), there exists x ∈ Λ such that the z X -orbit Σ of x is closed. If Σ is 3-dimensional then it equals M , and the identity component of Is loc X (y) is trivial at each y ∈ M . But since M is compact, the flow {ϕ t X } has recurrent points, which leads to a contradiction with proposition 3.4. Thus Σ must be a closed surface. It is a torus or a Klein bottle because X is nonvanishing. By construction, X is lightlike on Σ. Point (3) is proved. ♦ Arguments follow for each possible isomorphism type of z X .

6.1. Case z X is isomorphic to R 3 . Let Σ be a 2-dimensional orbit, as guaranteed by proposition 6.1 (3), and let x ∈ Σ. Because z X is abelian, the isotropy at x fixes two linearly independent vectors, spanning T x Σ. As in the proof of proposition 6.1 (1), we have a contradiction with theorem 4.9.

6.2. Case z X is isomorphic to heis(3). In this case, X generates the center of heis(3).

First suppose there exists an open z X -orbit Ω. On a sufficiently small open subset U ⊂ Ω, there is g 0 ∈ [g]| U such that (U, g 0 ) is isometric to an open subset of Heis(3) endowed with a left-invariant Lorentzian metric. Leftinvariant Lorentzian metrics on Heis(3) were classified in [START_REF] Rahmani | Lorentz geometry of the Heisenberg group[END_REF]; there are three isometry types, according to the sign of X, X . If X, X = 0, the metric is flat. By the analyticity assumption, (M, g) is conformally flat, contradicting our current hypothesis. If X, X = 0, the isometry group of the metric on Heis(3) is 4-dimensional and centralizes the center of Heis(3). Then z X has dimension at least 4, contradicting our current assumption that it is 3.

We conclude that the z X -orbits are all 2-dimensional. The following proposition, when combined with our analyticity assumption, concludes this case. Proposition 6.2. Let (M, g) be a smooth, 3-dimensional, Lorentzian manifold. Suppose there is a nonempty open subset Ω ⊆ M with a local conformal action of heis [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF], such that all pseudo-orbits are 2-dimensional. Then (M, g) is conformally flat.

Proof:

Let Y and Z be the further generators of heis Hence we can write

Z = (y + a(z)) ∂ ∂x + b(z) ∂ ∂y .
Observe that replacing Z by Z -a(0)X -b(0)Y will not affect the bracket relations between Z, Y and X, so we may assume that a(0) = b(0) = 0.

Given a point p = (p 1 , p 2 , p 3 ) in the domain of such a coordinate chart, the vector field

U = Z -(p 2 + a(p 3 ))X -b(p 3 )Y is nonzero and vanishes at p. At p, U, ∂ ∂x = 0, U, ∂ ∂y = - ∂ ∂x , U, ∂ ∂z = -a (p 3 ) ∂ ∂x -b (p 3 ) ∂ ∂y .
Since U belongs to heis(3), hence is conformal for g, the matrix

A =   0 -1 a (p 3 ) 0 0 b (p 3 ) 0 0 0   which is the matrix of ∇U (p) in the basis { ∂ ∂x , ∂ ∂y , ∂ ∂z }, must satisfy the identity g p (A•, •) + g p (•, A•) = αg p , α ∈ R
The matrix A generates a 1-parameter group {e tA } in R × O(1, 2), which is nontrivial because the rank of A is at least 1. If the rank of A were 1, then {e tA } would fix two linearly independent vectors. But no nontrivial flow in R × O(1, 2) has this property, so that we infer b (p 3 ) = 0.

As p was arbitrary, we may assume the derivative b does not vanish at any point of such a coordinate chart. Now let ψ be a smooth diffeomorphism on an interval around 0 such that ψ(0) = 0 and b(ψ(z)) = z. The transformation ϕ : (x, y, z) → (x, y -a(ψ(z)), ψ(z)).

then yields a local diffeomorphism fixing the origin. Applying (ϕ -1 ) * to the generators yields

Z = y ∂ ∂x + z ∂ ∂y , Y = ∂
B =   0 1 0 0 0 1 0 0 0   satisfies g p (B•, •) + g p (•, B•) = αg p , α ∈ R The matrix of g p in the basis ∂ ∂x , ∂ ∂y , ∂ ∂z is thus of the form   0 0 -β(p) 0 β(p) 0 -β(p) 0 γ(p)   , β(p) > 0
Replace g by 1 β g , which amounts to assuming β = 1. Now, X and Y are conformal Killing fields for g . But g (Y , Y ) = 1, and Y commutes with X and Y . It follows that X and Y are actually isometric Killing fields for g . In particular, the function γ only depends on the variable z, and the metric g is:

-2dxdz + dy 2 + γ(z)dz 2 . Now, if ζ(z)
is an antiderivative of γ(z)/2, then the change of coordinates (x, y, z) → (x + ζ(z), y, z) converts g to -2dxdz + dt 2 . We conclude that the conformal class [g], restricted to a nonempty open subset of Ω, contains a flat metric. ♦ 6.3. Case z X is isomorphic to aff(R) ⊕ R. Let Σ be as in proposition 6.1 (3) and x 0 ∈ Σ. Let Y and Z be further generators of z X on a neighborhood of x 0 such that [Y, Z] = Z and all other brackets are zero. The isotropy at x 0 can be of three types:

• isotropy generated by U ∈ z X transverse to Span(Z, X): The tangent vector X(x 0 ) is lightlike and fixed by the isotropy. Rescaling U if necessary gives

D x 0 ϕ t U =   1 0 0 0 e t 0 0 0 e 2t   .
By theorem 4.9 and proposition 4.8, (M, g) is conformally flat, a contradiction. • isotropy generated by Z: Then D x 0 ϕ t Z is trivial, which implies conformal flatness by theorem 4.9. We thus discard this case too.

• isotropy generated by Z +cX with c = 0: We handle this case below.

First we construct a model for the geometry on Σ. Consider R 2 with coordinates (x, y), and denote by H + the upper half-space defined by y > 0. On H + , let A = y ∂ ∂y , B = y ∂ ∂x , and C = ∂ ∂x . The only nontrivial bracket relation is [A, B] = B, so that the Lie algebra h generated by A, B, and C is isomorphic to aff(R) ⊕ R. These vector fields are complete and integrate to a genuine action of H Aff(R) × R on H + given by the affine transformations:

1 b 0 e a + c 0 , a, b, c ∈ R.
On H + , the isotropy for the local action of h is always generated by an element of Span(B, C) transverse to RC. Thus the local action of z X on Σ is locally modeled on that of h on H + .

Let G be the 4-dimensional Lie group given by the following transformations of H + :

(x, y) → (x + βy + γ ln(y) + τ, e α y) α, β, γ, τ ∈ R

In coordinates (α, β, γ, τ ), the subgroup comprising elements of the form (0, β, γ, τ ) is normal and abelian. The action of (α, 0, 0, 0) on it is given by the matrix:

  e -α 0 0 0 1 0 0 -α 1   .
The group G is thus isomorphic to a semi-direct product R R 3 .

Observe that H is a subgroup of G, corresponding to γ = 0. We have the following Liouville Theorem for (G, H + ): Proof: Given an open subset U ⊂ Σ, denote z X (U ) the Lie algebra of all local conformal vector fields defined on U commuting with X. For each x 0 ∈ Σ, there exists a small neighborhood U containing x 0 , an open subset V ⊂ H + , and a diffeomorphism ψ : U → V such that ψ * (z X (U )) = h| V and ψ * (X) = C. The corollary then follows from Lemma 6.3. ♦ Lemma 6.5. The (G, H + )-structure on Σ is complete.

Proof: Let Σ be the universal cover of Σ and δ : Σ → H + a developing map, with associated holonomy morphism ρ : π 1 (Σ) → G. By construction, δ * ( X) = C, where X is the lift of X to Σ. In particular, the relation δ

• ϕ t X = ϕ t C • δ shows that δ( Σ) is a union of lines y ≡ c, for c ∈ R. Thus δ( Σ) is a horizontal strip in H + .
Next observe that G preserves the degenerate metric h 0 = dy 2 /y 2 on H + , from which Σ inherits a degenerate metric h. On a 2-fold cover of Σ, there is W ∈ X (Σ) transverse to X and satisfying h(W, W ) = 1. For x ∈ Σ, the trajectory δ(ϕ t W .x) is a curve in H + with velocity of constant h 0 -length 1, defined on R since W is complete on Σ. This curve must cross every horizontal line in H + ; we conclude that δ( Σ) = H + . In fact, the open set Ω = {ϕ t X ϕ s W .x | s, t ∈ R} ⊂ Σ is mapped diffeomorphically by δ onto H + . The boundary ∂Ω is empty. Indeed, if x ∈ ∂Ω, then there is x ∈ Ω satisfying δ(x ) = δ(x) = y. Disjoint neighborhoods of both x and x in Σ map diffeomorphically under δ to a neighborhood of y. On the other hand, both neighborhoods in Σ intersect Ω, which contradicts injectivity of δ on Ω. We conclude that Ω = Σ, and completeness follows. ♦

The developing map δ identifies π 1 (Σ) with a discrete subgroup of G, and by proposition 6.1 (3), this discrete group contains an index 2 subgroup Λ isomorphic to Z 2 . There must be in Λ an element γ 0 = (α 0 , β 0 , γ 0 , τ 0 ) with α 0 = 0, otherwise Λ would preserve the lines y ≡ c in H + , and could not act cocompactly. Now it is readily checked that the centralizer L of γ 0 in G comprises elements of G of the form α, β 0 e α 0 -1 (e α -1), γ 0 α 0 α, τ , α, τ ∈ R. Hence L is isomorphic to R 2 , and {ϕ t C } acts by translations on the torus L/Λ. Because {ϕ t C } is compact in L/Λ, this would make {ϕ t X } relatively compact in Diff(Σ). Then there are

t k → ∞ and x ∈ Σ such that D x ( ϕ t k X Σ ) and D x ( ϕ t k X Σ ) -1
are bounded. Since Σ has codimension one, this implies boundedness of D x ϕ t k X and (D x ϕ t k X ) -1 . Then there is a holonomy sequence for {ϕ t k X } at x contained in P + , which implies conformal flatness by theorem 4.12. We have reached the desired contradiction.

Case z X is 2-dimensional

The case in which z X is 2-dimensional, necessarily isomorphic to R 2 , is the most involved. In subsection 7.1 below, we show that z X generates a global action of S 1 × R on M . The remaining subsections follow the main ideas of section 6 of [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF]-also the most difficult part of the proof in that paper-to arrive at a contradiction. The contradiction is with the standing assumptions that {ϕ t X } is essential and (M, g) is not conformally flat. From these assumptions we can immediately deduce the following facts about z Xorbits: Lemma 7.1.

(1) There is no z X -orbit of dimension 0. (2) There exists a closed z X -orbit Σ of dimension 1 Proof: Corollary 4.11 rules out any z X -orbit of dimension 0. Because M is compact, there are recurrent points for {ϕ t X }. At such points, the isotropy is nontrivial by proposition 3.4, and the z X -orbit is of dimension 1. It is closed because of theorem 3.3 (2) and point (1). ♦ 7.1. Global conformal action on M . The result of this subsection is that, possibly after passing to a finite cover of M , the vector fields in z X are globally defined, necessarily complete, generating a conformal action of a cylinder Z X . Proposition 7.2. After possibly replacing M by a finite cover, every local conformal vector field on M commuting with X extends to a conformal vector field defined on all of M . The resulting subalgebra of X conf (M ), isomorphic to z X , integrates to a subgroup group Z X ≤ Conf(M ) isomorphic to S 1 × R, acting locally freely on an open, dense subset of M .

Proof: Let M be the universal cover of M , with group of deck transformations Γ < Conf( M ). Denote the lifts of X and z X to M also by X and z X . By Amores' theorem [START_REF] Amores | Vector fields of a finite type G-structure[END_REF] (see section 3.2), the lifts of z X form a globally defined subalgebra of X conf ( M ). Let Γ 0 be the kernel of the holonomy representation of Γ on z X , and set M = M /Γ 0 . We will again denote by z X the corresponding subalgebra of X conf ( M ). The manifold M is a quotient of M by a group Γ ∼ = Γ/Γ 0 , and the holonomy representation of Γ on z X is faithful.

Lift the closed orbit Σ to M , and let Σ be a connected component of this lift. It is the z X -orbit of a point x0 in M .

Lemma 7.3. The manifold Σ ⊂ M is a circle.

Proof: Assume, for the sake of contradiction, that Σ is diffeomorphic to R. In this case, the stabilizer of Σ in Γ is a cyclic subgroup Γ Σ = γ ∼ = Z. First suppose X(x 0 ) = 0, which implies X is nonvanishing on Σ. Then there is T 0 = 0 such that γϕ T 0 X ∈ Is loc X (x 0 ). This group has finitely-many components by theorem 3.2, so a power

(γϕ T 0 X ) = γ ϕ T 0 X = ϕ 1 Z on a neighborhood U of x0
, where Z is a generator of Is X (x 0 ), and we assume

ϕ t Z is defined on U for all t ∈ [0, 1]. Let Y = Z -T 0 X. For t ∈ [0, 1] and x ∈ U , the composition ϕ -tT 0 X • ϕ t Z (x) is well-defined. Because X and Z commute, ϕ t Y (x) = ϕ -tT 0 X • ϕ t Z (x)
, and now

ϕ 1 Y U = γ k U .
Then (γ k ) * Y = Y on U , which implies by analyticity that (γ k ) * Y = Y on M . Because Γ fixes X and Y is independent of X, the element γ k = 1 would centralize z X , contradicting faithfulness of Γ on z X .

Next suppose X(x 0 ) = 0, which implies X ≡ 0 on Σ, and let Z ∈ z X \RX. Such Z is nonvanishing by corollary 4.11. There is T 0 = 0 such that ϕ T 0 Z (x 0 ) = γ(x 0 ), and ϕ t Z is defined on a neighborhood U of x0 for all t ∈ [0, T 0 ]. Then γ -1 ϕ T 0 Z ∈ Is loc X (x 0 ) on U . Again by theorem 3.2, the latter group contains {ϕ s X } as a finite-index subgroup. Thus, for some k ∈ Z and

S 0 ∈ R, (γ -1 • ϕ T 0 Z ) k = ϕ S 0 X on a neighborhood of x0 . Because [X, Z] = 0, the differential D x0 ϕ S 0 X (Z(x 0 )) = Z(x 0 ). Next D x0 (γ -1 • ϕ T 0 Z )(Z(x 0 )) = ±Z(x 0 ) = (γ -1 * Z)(x 0 ) Then γ * (Z) = ±Z + βX for some β ∈ R. If γ * (Z) were congruent to -Z modulo RX, then, given that ϕ -T 0 Z γ(x 0 ) = x0 , we would have γ -1 (x 0 ) = γ -1 ϕ -T 0 Z γ(x 0 ) = ϕ T 0 Z (x 0 ) = γ(x 0
) which is absurd, since Γ acts freely and γ has infinite order. Thus γ * (Z) = Z + βX. Now there is S 1 such that

(γ -1 • ϕ T 0 Z ) k = γ -k ϕ S 1 X ϕ T 0 Z ∈ Is loc X (x 0 ) Thus, for some S 2 , γ k = ϕ S 2 X ϕ T 0 Z on U . Let Y = S 2 X + T 0 Z. As above, the flow along Y is well-defined on U for t ∈ [0, 1],
and

ϕ 1 Y U = γ k U .
This leads again to (γ k ) * Y = Y on M , yielding a contradiction with the faithfulness of the action of Γ on z X . ♦ Next we show that z X globalizes on a neighborhood of the closed orbit Σ.

Lemma 7.4. There exists Y ∈ z X , and Ñ an open neighborhood of Σ, such that for all y ∈ Ñ , the orbit {ϕ t Y .y} is defined on R, included in Ñ , and 1-periodic.

Proof: If X(x 0 ) = 0, then {ϕ t X .x 0 } is periodic. Because Is loc X (x 0 ) has finitely-many components, there are

T 0 > 0, S 0 ∈ R such that ϕ T 0 X • ϕ S 0 Z
is trivial on a neighborhood of x0 , where Z generates Is X (x 0 ). Note that, for any S, the flow {ϕ s Z } is defined for s ∈ [0, S] in some neighborhood of x0 , and it commutes with {ϕ t X }. Now Y = T 0 X + S 0 Z ∈ z X generates a periodic flow {ϕ rT 0 X • ϕ rS 0 Z }, defined for r ∈ R in a neighborhood U of x0 . We can then put Ñ := t∈R ϕ t Y (U ), which is indeed an open neighborhood of Σ satisfying the conclusions of the lemma.

If X vanishes on Σ, then by corollary 4.11, any Z ∈ z X \RX does not vanish on Σ. Then Z = Z| Σ is nonvanishing and complete because Σ is compact by lemma 7.3. Let T 0 > 0 be such that ϕ T 0 Z . x0 = x0 . The flow along Z is defined on [0, T 0 ] in a neighborhood Ñ of Σ. By theorem 3.2, {ϕ s X } has finite index in Is loc X (x 0 ) and is noncompact because it is an essential flow. After replacing T 0 by a finite integer multiple, and shrinking Ñ if necessary, ϕ T 0 Z ϕ S 0 X will be trivial in restriction to Ñ , for some S 0 ∈ R. Because X is complete and [Z, X] = 0, the flow along Y = T 0 Z + S 0 X restricted to Ñ is periodic with period 1 and complete. Again, Ñ := t∈R ϕ t Y (U ) is an open neighborhood of Σ satisfying the conclusions of the lemma. ♦

Let Ω = t∈R ϕ t X . Ñ , an open set; let Y be as in lemma 7.4. For every y ∈ Ω, the pseudo-orbit {ϕ t Y .y} is defined for every t ∈ R, included in Ω, and 1-periodic. Any Z ∈ z X equals r 0 X + s 0 Y , for some r 0 and s 0 , so for every y ∈ M , we have ϕ t Z .y = ϕ tr 0 X • ϕ ts 0 Y .y, for all t such that the expression is defined. For y ∈ Ω, in particular, ϕ t Z .y is defined for every t ∈ R and lies in Ω. All vector fields of z X | Ω are thus complete, defining an action of a 2dimensional abelian Lie group Z X on Ω. For the time being, we consider Z X as a subgroup of Conf(Ω). Since the flow {ϕ t Y } is cyclic on Ω, we infer that Z X is isomorphic to a cylinder S 1 × R or a torus T 2 . The latter possibility would mean that {ϕ t X } is compact in Conf(Ω), hence in Conf( M ), and finally in Conf(M ), contradicting the essentiality hypothesis. Now assume Z X is a cylinder. There exists a homomorphism ρ : Γ → Aut(Z X ) integrating the representation Γ → Aut(z X ). Indeed, for γ ∈ Γ and Z ∈ z X the local flows {γϕ s Z γ -1 } and {ϕ s γ * Z } coincide on Ω; in particular, γϕ 1 Z γ -1 is well-defined on Ω and belongs to Z X . Because ρ(Γ ) fixes ϕ t X for all t, it follows that |ρ(Γ )| ≤ 2. Because ρ is faithful, |Γ | ≤ 2. Then M is compact, and all vector fields of z X are complete. Thus z X integrates to the action of a group Z X ∼ = S 1 × R, as claimed in the proposition.

We replace M by M in the sequel. To complete the proof of the proposition, it remains to check that the action of Z X is locally free on a dense open subset of M . If not, there would be a nonempty open subset U on which all Z X -orbits have dimension 1. The identity component of the isotropy group Is loc X (x 0 ) at a point x 0 ∈ U would fix a nonzero vector v tangent to the orbit and act trivially on the quotient T x 0 M/Rv. Basic linear algebra shows that the differential of the isotropy at x 0 , identified with a subgroup of CO(1, 2), must be trivial in this case. On the other hand, theorem 4.9 says, given our assumption on (M, g), that the isotropy must be linearizable, yielding a contradiction. ♦ Now we replace M by the finite covering given by the above proposition. The remainder of this section follows the arguments of section 6 of [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF], with some simplifications available in dimension three. We obtain directly from the Z X -action a foliation by degenerate surfaces on an open, dense subset Ω ⊂ M in section 7.2. The leaves are projections of leaves of an integrable distribution in a reduction of the Cartan bundle over Ω (section 7.3). Then we appeal to [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF] to better understand how 2-dimensional orbits can accumulate on closed orbits (section 7.4). A contradiction ultimately results from the fact that only finitely many 1-dimensional orbits can attract 2-dimensional orbits, while the 2-dimensional orbits should accumulate on uncountably many distinct 1-dimensional orbits (section 7.5). 7.2. Description of the Z X -orbits.

7.2.1.

All orbits are degenerate. Proposition 7.5. There are two types of Z X -orbits in M , both of which occur:

(1) Circular lightlike orbits, with linear, unipotent isotropy. These form an analytic subset Σ of M .

(2) Cylindrical orbits on which the metric is degenerate. Each of these contains an orbit of type (1) in its closure. These fill an open, dense subset Ω f of M .

Proof: By corollary 4.10, there are no Z X -fixed points; moreover, all 1dimensional orbits are closed-otherwise, there would be a fixed point in the closure by theorem 3.3 [START_REF] Bader | Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds[END_REF]. The closed, lightlike orbits are precisely the zero-set of the analytic function, given by a choice of metric g in the conformal class and Y ∈ z X \RX by

ϕ(x) = g x (X, X) 2 + g x (Y, Y ) 2 + g x (X, Y ) 2
Since we assume that (M, g) is not conformally flat, the elements of the isotropy algebra can be assumed linearizable by theorem 4.9. In a 1-dimensional, lightlike orbit, the Z X -isotropy fixes a lightlike tangent vector. Such isotropy easily seen to be balanced or unipotent. In the first case, the isotropy is stable and leads to conformal flatness by proposition 4.8. We have proved all the claimed properties of orbits of type [START_REF] Amores | Vector fields of a finite type G-structure[END_REF].

Let Ω f be the set on which Z X acts locally freely; it is open and dense by proposition 7.2. If there were a closed, 2-dimensional orbit, then {ϕ t X } would have a recurrent point on this orbit, contradicting proposition 3.4. Thus every 2-dimensional orbit is not closed, and, by theorem 3.3 (2), contains a closed 1-dimensional orbit in its closure. Now we focus on the linear part of a holonomy sequence {p k } for an unbounded sequence {h k } ⊂ Z X at x ∈ Ω f . By proposition 4.4, the subspace z X (x) ⊂ T x M is approximately stable for {D x h k }, because Z X centralizes z X . This means that ω x(z X ) belongs modulo p to (g/p) AS (p k ), for any x ∈ π -1 (x). Assuming p k is in A P + -form, the presence of a 2-dimensional approximately stable subspace in g/p makes it contracting, balanced, mixed, or of bounded distortion, as in definition 4.2.

Let Ω ⊆ Ω f be the open subset where the Cotton tensor is nonzero; by our standing assumption, it is also dense. For x ∈ Ω, a holonomy sequence {p k } as above is of bounded distortion or mixed type by proposition 4.7. For these types, (g/p) AS (p k ) is a degenerate plane. Thus all orbits in Ω are degenerate. On the other hand, if a point x ∈ Ω f has Riemannian or Lorentzian orbit, then so do the points in a neighborhood of x. Since Ω is open and dense, we conclude that all orbits of Ω f -that is, all 2-dimensional orbits-are degenerate.

We rule out nondegenerate orbits of dimension 1 in lemma 7.6 directly below, which completes the proof. ♦ Lemma 7.6. There are no 1-dimensional spacelike or timelike orbits.

The proof of this lemma makes use of the exponential map of the Cartan connection, defined in subsection 4.3.2. The exponential map will appear frequently in the remainder of this section.

Proof. If a point x has a timelike orbit, then ϕ is negative in a neighborhood of x. This neighborhood intersects Ω f , where ϕ is nonnegative, a contradiction. Now suppose that x has a 1-dimensional spacelike orbit, and let x ∈ π -1 (x) be such that the isotropy image ( ÎX ) x is linear, contained in G 0 = CO(1, 2). It fixes a spacelike vector, which makes it conjugate in G 0 to a 1-parameter diagonal group

{ ĥt : (x 1 , x 2 , x 3 ) → (e t x 1 , x 2 , e -t x 3 )} Let Ẑ ∈ co(1,
2) be the generator of { ĥt }, corresponding under ι x to a generator Z ∈ Is X (x). Note that the corresponding linear vector field Ẑ on R 1,2 is timelike along the line R(E 1 + E 3 ) (with the metric I of section 4.1).

Let γ(t) = exp x(t(E 1 + E 3 )) and γ = π • γ. The lift γ determines a metric along γ in the conformal class [g] by

u, v γ(t) = I ω (-1) γ(t) û, ω (-1) γ(t) v
where ω (-1) is the component of ω on g -1 ∼ = R 1,2 , and û, v are any lifts of u, v to T γ(t) M . We will approximate Z, Z γ(t) by computing

d dt 0 ω (-1) γ(t) (Z) = ( Ê1 + Ê3 ) x.ω (-1) (Z)
where Ê1 + Ê3 is the vector field on M satisfying ω( Ê1 + Ê3 )

≡ E 1 + E 3 .
Using that L Z ω( Ê1 + Ê3 ) ≡ 0, the Cartan curvature gives

0 = Ω x( Ê1 + Ê3 , Z) = ( Ê1 + Ê3 ) x.ω(Z) + E 1 + E 3 , Ẑ Thus ( Ê1 + Ê3 ) x.ω (-1) (Z) = Ẑ, E 1 + E 3 (-1) = E 1 -E 3 Now ω (-1) γ(t) (Z) = t(E 1 -E 3 + R(t)), lim t→0 R(t) = 0 and Z, Z γ(t) = I (t(E 1 -E 3 + R(t)), t(E 1 -E 3 + R(t))) = t 2 (-2 + Q(t)) lim t→0 Q(t) = 0
For sufficiently small t, this inner product is negative along γ. This would mean that Z ∈ z X is timelike on a nonempty open set, contradicting that all Z X -orbits in the open, dense set Ω f are degenerate.

Corollary 7.7. Let {h k } ⊂ Z X be an unbounded sequence, and suppose that h k .x → y for x ∈ Ω and y ∈ Σ. Then any holonomy sequence for {h k } at x is of mixed type.

Proof. We established during the proof of proposition 7.5 above that holonomy sequences at points of Ω are of bounded distortion or mixed type. By proposition 7.5, there is Z ∈ z X with Z(x) spacelike. It is also approximately stable for {D x h k }. If {h k } has bounded distortion at x, then lim k D x h k (Z) is a nonzero spacelike vector tangent to the orbit of y. This would contradict the result from proposition 7.5 above that the orbit of y is one-dimensional and lightlike.

7.3. Z X -orbits in the Cartan bundle. In this section we construct a Z Xinvariant reduction of M Ω and show that Z X -orbits there are tangent to a special distribution defined by the Cartan connection. These properties will be key to controlling the accumulation of 2-dimensional orbits in Ω on 1-dimensional orbits in ∂Ω.

7.3.1.

Adjoint approximately stable spaces. As in [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF], we will call {p k } an ACL holonomy sequence at x ∈ Ω f if it is in A P + -form; it corresponds to h k .x → y for y ∈ Σ; and the isotropy in Z X with respect to ŷ = lim h k .x k .p -1 k is linear. Every ACL holonomy sequence at x ∈ Ω has (g/p) AS (p k ) = E ⊥ 1 , because its linear component is of mixed type by corollary 7.7.

We will next describe g AS (p k ), which also reflects the nonlinear part of {p k }. We use the G 0 -invariant decomposition g

= g -1 ⊕ g 0 ⊕ g 1 with g -1 ∼ = R 1,2 to identify E ⊥
1 with a subspace of g.

Proposition 7.8 ([31] Prop 6.5). For {p k } an ACL holonomy sequence at

x ∈ Ω, E ⊥ 1 ⊂ g AS (p k ) Proof: Write {p k = d k τ k }. By [31, Lem 6.4], {ξ k = ln τ k } is contained in the line RE t 1 I ⊂ p + ∼ = R 1,2 * . Moreover, for D k = ln d k , the sequence {e β(D k ) ξ k } is bounded. Now, it follows that for {p k } as above and v ∈ E ⊥ 1 , Ad(p k ).v = Ad(d k )(v + [ξ k , v]) = Ad(d k ).v + e β(D k ) [ξ k , v],
noting that [ξ k , v] is in the root space g β and [ξ k , [ξ k , v]] = 0. This expression is bounded, so the desired inclusion follows. ♦

7.3.2. Reduction of M over Ω. Write Ωf = M Ω f
. Recall that Z X acts locally freely in Ω f with degenerate orbits, by proposition 7.5. Given x ∈ Ω f , the orthogonal n(x) to the orbit of x is a lightlike line, tangent to the orbit. We define a reduction of Ωf given by the conformal frames in which this orthogonal is the line [E 1 ], as follows. Denote N the null cone of the Lorentzian inner product I in R 1,2 , and by P(N ) its projectivization. Let

η : Ωf → P(N ) x → ω (-1) x ( n(x))
where n(x) is any lift of n(x) to T x M , and ω (-1) denotes the component on g -1 ∼ = R 1,2 . This map is well-defined, and in fact analytic, in Ω f . The

level set of [E 1 ] is a reduction R ⊂ Ωf to Q 0 P + , where Q 0 < G 0 is the stabilizer of [E 1 ]
. The Z X -action preserves orbits, so it preserves the orthogonals, and it leaves ω invariant; thus Z X preserves R . Now we restrict to Ω = M Ω . Let q 1 be the annihilator of

E 1 in p + ∼ = R 1,2 * , with corresponding connected subgroup Q 1 < P + . Let Q = Q 0 Q 1 < P . Define R = {x ∈ Ω : ω -1 x (E ⊥ 1 ) ⊂ T xR }
This construction and the following proposition are very similar to [31, Sec

.

Proposition 7.9. The set R is a Z X -invariant reduction of Ω to Q.

Proof: Let x ∈ Ω. By proposition 7.5, there is

h k → ∞ in Z X such that h k .
x → y ∈ Σ, and the isotropy at y is linear and unipotent. There is thus an ACL holonomy sequence {p k } for {h k } with respect to x ∈ π -1 (x). By corollary 7.7 it is of mixed type. As observed in the proof of proposition 7.5, ω x(z X ) belongs modulo p to (g/p) AS (p k ), which equals E ⊥ 1 . For x = π(x), the projection z X (x) is the tangent space to the orbit of x. Thus the orthogonal n(x) corresponds under ω to [E 1 ], and x ∈ R . Whenever there is an ACL holonomy sequence of mixed type with respect to x ∈ Ω, then x ∈ R . By proposition 7.8, E ⊥ 1 ⊂ g AS (p k ). By proposition 4.5, for all X ∈ E ⊥ 1 , for s sufficiently small, {p k } is also a holonomy sequence at γ(s) = exp x(sX ), and this point is in Ω. Thus γ(x) is in R , for s sufficiently small, which implies ω -1

x (X) ∈ T xR . Then x ∈ R; moreover, every π-fiber of Ω intersects R. To verify that R is a reduction of R to Q, we will express it as the level set of a smooth-actually, analytic-map on R . As R is a reduction of Ω, there is, at each x ∈ R , a two-dimensional subspace of ω(T xR ) projecting modulo p to E ⊥ 1 , varying smoothly with x. This subspace can be expressed as the graph of a linear homomorphism E ⊥ 1 → p, unique up to addition of a homomorphism E ⊥ 1 → q 0 p + , corresponding to addition of vertical vectors tangent to R . Then we compose with the projection to p/p + ∼ = g 0 to define Φ : R → Hom(E ⊥ 1 , g 0 /q 0 ) Observe that x ∈ R belongs to R if and only if Φ(x) = 0.

The map Φ is Q 0 P + -equivariant, where, for ϕ ∈ Hom(E ⊥ 1 , g 0 /q 0 ), (g

• τ )(ϕ) = ( Ad g| g 0 /q 0 ) • ϕ • ( Ad g -1 E ⊥ 1 ) + (Ad τ -Id)| E ⊥ 1
For τ ∈ P + , the image Ad τ (E ⊥ 1 + q 0 ) ≡ E ⊥ 1 + q 0 mod p + if and only if τ ∈ Q 1 . The affine Q 0 P + -action on Hom(E ⊥ 1 , g 0 /q 0 ) factors through Aff(R). The orbit of 0 is 1-dimensional, with stabilizer Q.

Because every π-fiber of Ω intersects R, which is in turn contained in R , the image of the latter under Φ is contained in the orbit of 0. Now R, the inverse image of 0, is a smooth Q-reduction of M over Ω. It is Z X -invariant and analytic because R and ω are. ♦

The geometric interpretation of R is as the conformal normalized 2-frames at points x ∈ Ω in which the orbits are totally geodesic (infinitesimally at x).

The fact from the proof of proposition 7.9 that {p k } is also a holonomy sequence at γ(s) = exp x(sX ) for all X ∈ E ⊥ 1 gives that γ(s) ∈ R, for all s such that γ(s) = π • γ(s) ∈ Ω. It follows that ω -1

x (E ⊥ 1 ) ⊂ T xR.

7.3.3. Foliation of R. Let D = ω -1 (E ⊥ 1 + q), an analytic distribution on M . The restriction to R is tangent to R because, from the previous paragraph, ω -1 (E ⊥ 1 ) ⊂ T R, and R is a principal Q-bundle. When M is 3-dimensional, we can prove integrability of D in R without using the Cartan curvature.

Lemma 7.10. The distribution D R is integrable. The projection on M of the leaves of this distribution coincide with Z X -orbits in Ω.

Proof:

The key fact here is that the Z X -orbits in R are tangent to D. Indeed, as noted in the proofs of propositions 7.5 and 7.9, ω x(z X ) is congruent modulo p to E ⊥ 1 if there is a mixed ACL holonomy sequence with respect to x, and more generally, if

x ∈ R. Because R is Z X -invariant, z X (x) ⊂ T xR. Thus z X (x) ⊂ Dx for all x ∈ R.
The (Z X × Q)-orbits in R are integral leaves for D, projecting to the twodimensional, degenerate Z X -orbits in Ω. ♦ 7.4. Accumulation of 2-dimensional orbits on 1-dimensional orbits. Before applying the results of the previous section, we focus on the geometry around 1-dimensional Z X -orbits. 7.4.1. Plaques at 1-dimensional orbits. We define a distinguished degenerate surface around each 1-dimensional orbit. The following proposition is an aggregate of propositions 6.11 and 6.12 (see also remark 6.14) of [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF].

Proposition 7.11. Let y ∈ M have closed, isotropic Z X -orbit and suppose that Is X (y) is linear and unipotent with respect to ŷ ∈ π -1 (y). Then:

(1) The point ŷ belongs to the closure R.

(2) Let γ : [0, 1] → M a continuous path, smooth on [0, 1), such that γ([0, 1)) ⊂ Ω and γ(1) = y. Then there exists a continuous lift γ : [0, 1] → M , smooth on [0, 1), such that γ([0, 1)) ⊂ R, and γ(1) = ŷ. (3) There is a neighborhood U of 0 in E ⊥ 1 + q such that exp ŷ(U ) is an integral submanifold of D.

The third point of the proposition follows rather easily from the first, because by analyticity, integrability of D on R extends to the closure. Definition 7.12. Let exp ŷ(U ) be as in proposition 7.11. For U small enough, π(exp ŷ(U )) is a degenerate 2-dimensional submanifold of M . It will be called a plaque at y, and denoted P y .

Observe that if ŷ and ŷ are two points of π -1 (y) such that Is X (y) is linear and unipotent with respect to ŷ and ŷ , then ŷ = ŷ.q, where q ∈ Q (actually q belongs to the subgroup

Q 0 Q 1 ⊂ Q, where Q 1 is the 1-dimensional subgroup of Q 1 normalized by Q 0 ). In particular Ad(q -1 )(E ⊥ 1 + q) = E ⊥ 1 + q.
The relation exp ŷ (Ad(q -1 ).U) = exp ŷ(U ).q is a consequence of the second axiom for ω part (2), and implies that the projections on M of exp ŷ (Ad(q -1 ).U) and exp ŷ(U ) are the same. Thus all plaques at y have the same germ, in the sense that if P y and P y are two of them, then P y ∩ P y is open in P y and P y . Proposition 7.13 (see [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF] Rem 6.17). Let ∆ be a 1-dimensional Z X -orbit. Let P y be a plaque at y ∈ ∆. If O is a 2-dimensional Z X -orbit of Ω containing y in its closure, then P y ∩ O has nonempty interior in P y .

Proof: By theorem 3.3 (1), the orbit O is a semi-analytic set. We will use the following result, known as the "curve selecting lemma." Lemma 7.14 (see [START_REF] Lojasiewicz | Ensembles semi-analytiques[END_REF] Sec 19, Prop 2). Let S be a semi-analytic subset of M , and y ∈ S. Assuming y is not an isolated point of S, there exists an analytic arc γ : [0, 1) → S extending continuously to 1, with γ(1) = y.

This lemma applied to S = O provides a continuous path γ : [0, 1] → M , which is smooth on [0, 1), and satisfies γ([0, 1)) ⊂ O ⊂ Ω, as well as γ(1) = y. Let ŷ ∈ π -1 (y) with linear, unipotent holonomy. Pproposition 7.11 (2) yields a continuous lift γ of γ, such that γ is smooth on [0, 1), γ([0, 1)) ⊂ R, and γ(1) = ŷ.

Because γ([0, 1)) ⊂ O, the image γ((0, 1)) lies in an integral leaf of D (this leaf is the preimage of O in R). Let U be a neighborhood of 0 in g sufficiently small that for every t ∈ [0, 1], the set U is mapped diffeomorphically by exp γ(t) onto its image. Set V = U ∩ (E ⊥ 1 + q) and L t = exp γ(t) (V) for t ∈ [0, 1]. A suitable choice of U ensures that V is connected and that π(L t ) is a hypersurface for all t. Lemma 7.15. For t close enough to 1, L t contains ŷ.

Proof: For t 0 close enough to 1, the image U = exp γ(t 0 ) (U) contains ŷ. Let (X 1 , . . . , X m ) be a basis of E ⊥ 1 + q, and X1 , . . . , Xm the corresponding vector fields on M with ω( Xi ) ≡ X i ∀i. Write

γ (t) = m i=1 a i (t) Xi (γ(t)) ∀t ∈ (t 0 -, 1)
where a i : (t 0 -, 1) → R are some smooth functions. The ODE

β (t) = m i=1 a i (t) Xi (γ(t))
is defined on the open set exp γ(t 0 ) (U), but also on the submanifold L t 0 . Uniqueness of solutions with the intial condition β(t 0 ) = γ(t 0 ) implies that γ([t 0 , 1)) ⊂ L t 0 . Then ŷ = γ(1) ∈ L t 0 because L t 0 is closed in exp γ(t 0 ) (U). ♦

A plaque P y at y is, by definition, the projection of exp ŷ(W ), where W is a neighborhood of 0 in E ⊥ 1 + q. Now exp ŷ(W ) and L t 0 are two integral leaves of D having a common point ŷ. It follows that the intersection exp ŷ(W )∩L t 0 is open in both exp ŷ(W ) and L t 0 . Projecting to M gives an open subset of P y ∩ O. ♦ Corollary 7.16. Let ∆ be a 1-dimensional orbit of Z X . The set of 2dimensional Z X -orbits in Ω containing ∆ in their closure is at most countable. In particular, the set of 1-dimensional Z X -orbits which are accumulated by 2-dimensional orbits is uncountable. A collection of pairwise disjoint nonempty open subsets of P y is at most countable, so I y is also countable. ♦ 7.5. Stratification of the set of 1-dimensional orbits, conclusion. Now we denote by Σ the complement of Ω. By proposition 7.5, Σ is a nowhere dense, analytic subset of M , comprising the points with 1-dimensional, closed, lightlike Z X -orbits orbits. For k ∈ N, we denote by Σ (k) the set of smooth points of dimension k in Σ-namely, the points contained in some k-dimensional analytic submanifold of M contained in Σ. Observe that Σ (k) = ∅ for every k ≥ 3, since Σ is nowhere dense.

For a semi-analytic set Σ, the dimension of Σ equals the maximal integer k, denoted k max , such that Σ (k) = ∅. The set of singular points of Σ, denoted Σ sing , is the complement Σ\Σ (kmax) . By [START_REF] Lojasiewicz | Ensembles semi-analytiques[END_REF]Thm 4] (see also [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF]Thm 7.2] and [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF]Rem 7.3]), Σ sing is a closed, semi-analytic subset of Σ, of dimension less than dim Σ.

Lemma 7.17. The complement of Σ (2) in Σ is a finite union of 1-dimensional Z X -orbits, possibly empty. integral leaf of D, and so α is contained in P y for t near t 0 . Varying α over paths through y in Σ (2) gives that a neighborhood of y in Σ (2) is contained in a plaque P y . Because both sets are smooth surfaces near y, we can shrink P y so that it is contained in Σ (2) . ♦ Proposition 7.18 together with proposition 7.13 shows that no 1-dimensional orbit of Σ (2) can be accumulated by a 2-dimensional orbit, because the latter is contained in Ω. By lemma 7.17, the complement of Σ (2) in Σ is a finite union of 1-dimensional orbits. We infer that only finitely many 1dimensional orbits can be accumulated by 2-dimensional orbits. This is a contradiction to corollary 7.16.

Case z X is 1-dimensional

The last case, in which z X = RX, is shown in this section to also lead to conformal flatness. The 1-dimensional orbits of Z X = {ϕ t X } cannot be closed by theorem 3.2, so there must be fixed points by theorem 3.3. This last part of the proof is very similar to section 5, treating global fixed points, in [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF], with slight simplification here stemming from Z X being 1-dimensional.

We first recall the model space, the 3-dimensional Einstein universe Ein 1,2 , which was introduced at the beginning of section 3. This space can be realized as the projectivization of the lightcone for the quadratic form

Q 2,3 (x) = 2x 0 x 4 + 2x 1 x 3 + x 2 2
on R 5 with basis E 0 , . . . , E 4 . It is a compact projective variety and inherits a Lorentzian conformal structure from Q 2,3 . We will denote o = [E 0 ] ∈ Ein 1,2 and take P to be the stabilizer of o in PO [START_REF] Bader | Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds[END_REF][START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF].

The following proposition will conclude the proof of theorem 1.2. Proposition 8.1. Suppose that z X = RX. Then (M, g) is conformally flat.

The proof occupies the remainder of this section. 8.1. Unipotent or hyperbolic isotropy. Let x 0 be a Z X -fixed point. By theorem 4.9, M is conformally flat or X is linearizable at x 0 . Assuming the latter, let x0 ∈ π -1 (x 0 ) with ω x0 (X) = X ∈ g 0 . Recall from theorem 3.2 that ( ÎX ) x is algebraic, which means that it is closed under Jordan decomposition (see, eg, [START_REF] Morris | Ratner's theorems on unipotent flows[END_REF]Thm 4.3.3]). Thus X is R-semisimple, nilpotent, or elliptic. The elliptic case would contradict noncompactness of Z X , since the isotropy monomorphism ι x : Is loc (x) → P (see section 3.2) is proper. 8.2. Hyperbolic isotropy implies conformal flatness. Suppose X is semisimple, so it generates an R-split 1-parameter subgroup of G 0 . If this subgroup were balanced or contracting, then (M, g) would be conformally flat by proposition 4.8. Up to conjugation in G 0 or replacing X with -X, we may assume e t X : (x, y, z) → (e t(b-a) x, e -a y, e t(-a-b) z) with 0 ≤ a < b

Recall the exponential map of ω from subsection 4.3.2. Let α = exp x0 (sE 1 ), and let α = π • α. The set α + = {α(s) : s > 0} is a Z X -orbit. Let x 1 be an accumulation point of α(s k ) for a sequence s k → ∞. It is shown in Section 5.2 of [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF] that the isotropy of X at x 1 is also hyperbolic, with a and b exchanged. We recall the main points of that argument here.

α + = {x 0 } ∪ α + ∪ {x 1 } and x 1 is a Z X -fixed point.
The proof is based on the reasonable topological properties of the orbit closure α + guaranteed by semi-analyticity as in theorem 3.3. A corollary is that α + admits a smooth parametrization by a finite interval.

To compute the isotropy of {ϕ t X } at x 1 , we compare with the flow {e t X } along the lightlike geodesic of Ein 1,2 corresponding to α. The path α has a development in PO(2, 3), which is the 1-parameter subgroup generated by E 1 , viewed as an element of g -1 ⊂ g. The projection of this path to PO(2, 3)/P ∼ = Ein 1,2 is the curve e sE 1 .o. In the homogeneous coordinates on RP 4 , this path is [1 : s : 0 : 0 : 0]; it lies in the projectivization of the totally isotropic plane P 0 = span{e 0 , e 1 } ⊂ R 2,3 . Take R ∈ g to be the generator of a rotation in the plane P 0 . Note that {e θR .o} coincides with α for 0 ≤ θ < π/2. Define β(θ) = exp x0 (θR) and β = π • β. There is a path θ in P such that

β(θ) = α(s). θ s = tan θ, 0 ≤ θ < π/2 Lemma 8.3 ([31] Lemma 5.6). There is x1 ∈ π -1 (x 1 ) equal to lim θ→π/2 β(θ).
There is a smooth lift β of β to M , related to β by

β(θ) = β(θ).p θ 0 ≤ θ < π/2
for a path p θ in P defined on [0, π/2). It is shown that ω P (p θ ) satisfies a linear, first-order, inhomogeneous ODE with coefficients all in terms of β and its derivatives. It follows that the ω P -derivative of p θ is bounded on [0, π/2).

As P is complete with respect to the left-invariant framing determined by ω P , the limit of p θ as θ → π/2 exists, thus defining the corresponding limit of β(θ).

Lemma 8.4 ([31] Lemma 5.7). The isotropy of ϕ t X with respect to x1 is

g t =        e tb e ta 1 e -ta e -tb        ∈ G 0 < PO(2, 3)
This value is obtained from the limit as θ → π/2 of the holonomy of {ϕ t X } at β(θ). A shortcut to computing this limit is given by computing in PO [START_REF] Bader | Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds[END_REF][START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF] the conjugate e -(π/2)•R e t X e (π/2)•R = g t . Now the isotropy ( ÎX ) x1 = {g t } is contracting, so (M, g) is conformally flat by proposition 4.8. 8.3. All unipotent isotropy not possible. Now suppose that the isotropy of X at x0 is a nilpotent element X ∈ g 0 with respect to x0 ∈ π -1 (x 0 ). Here we briefly recall the proof of [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF]Prop 5.8].

Take U ⊂ g -1 such that π • exp x0 is a diffeomorphism of U onto its image, which we denote U . For v = 0 in U, let y 0 = exp x0 (v). The connected component of y 0 in Z X .y 0 ∩ U equals π • exp x0 (( ÎX ) x0 .v ∩ U)
and is bounded away from x 0 . By theorem 3.3, Z X .y 0 is locally connected and contains Z X .y 0 as a relatively open subset. Therefore, x 0 / ∈ Z X .y 0 for any y 0 = x 0 in U .

Choose v not fixed by the isotropy. For y 0 as above, Z X .y 0 contains a point x 1 = y 0 with closed Z X -orbit by theorem 3.3. This closed orbit is necessarily a fixed point by theorem 3.2. We can assume, by the results above for hyperbolic isotropy, that the isotropy of Z X is again unipotent at x 1 . Now for y 1 ∈ Z X .y 0 sufficiently close to x 1 , we have x 1 ∈ Z X .y 1 . On the other hand, the argument in the previous paragraph gives that Z X .y 1 is bounded away from x 1 -contradiction. 9. Appendix: Proof of theorem 4.12 This appendix is devoted to the proof of theorem 4.12. We will actually prove a more general statement, for smooth manifolds and sequences of local conformal transformations. It is clear that theorem 4.12 follows directly from theorem 9.1 9.1. Conformal geodesic segments. The strategy to prove theorem 9.1 is to exhibit dynamical properties of the sequence {f k } which force conformal flatness on an open subset. The dynamical behavior of {f k } around a point x is understood via the action of its holonomy sequences on conformal geodesics in the model space, introduced below. The 3-dimensional Minkowski space will be taken to have the quadratic form q(x) := 2x 1 x 3 + x 2 2 , and will be denoted by R 1,2 . A conformal immersion j o : R 1,2 → Ein 1,2 is given in homogeneous coordinates on RP 4 by the formula j o : x → 1 : x 1 : x 2 : x 3 : -q(x) 2 , mapping the origin in R 1,2 to o ∈ Ein 1,2 . For us a conformal geodesic segment of Ein

p k =   1 t k v * k - t 2 k q(v k ) 2 0 I 3 -t k v k 0 0 1   ∈ P + < O(2, 3).
Here v k is a sequence of R 1,2 satisfying v k = 1, and t k ≥ 0. The expression v * k stands for v t k I, where I is as in section 4.1. Observe that {p k }, hence {t k }, is unbounded, because {f k } is unbounded. After taking a subsequence, we assume that t k → ∞, and that there is a vector v = lim v k .

Recall the conformal immersion j o : R 1,2 → Ein 1,2 from section 9.1 above. Let x k → x ∈ R 1,2 . For u ∈ [0, 1], and k ∈ N, define β k (u) := j o (ux k ), and β(u) := j o (ux). Observe that each β k and β are conformal geodesic segments emanating from o. Define x k (u) := (j o ) -1 (p k .β k (u)). From the matrix expression of p k , and the formula for j o , (1)

x k (u) = 1 + t k u x k , v k + t 2 k u 2 q(v k )q(x k ) 4 -1 • ux k + t k u 2 q(x k ) 2 v k .
After possibly taking a further subsequence of {p k }, we may assume that t k |q(v k )| converges in [0, ∞]. There are two subcases:

First case: t k |q(v k )| → ∞. In this case, q(v k ) is nonzero for k large enough, so that after perhaps taking a subsequence of {p k }, we may assume that the sign of q(v k ) is constant. Choose = ±1 so that q(v k ) ≥ 0 for all sufficiently large k. Choose x such that (a) q(x) is positive. (b) x, v is positive.

Let u ∈ (0, 1] and write:

1+t k u x k , v k + t 2 k u 2 q(v k )q(x k ) 4 = u 2 t 2 k q(v k ) q(x k ) 4 + x k , v k ut k q(v k ) + 1 u 2 t 2 k q(v k )
.

Under the assumption t k |q(v k )| → ∞,

lim k→∞ q(x k ) 4 + x k , v k ut k q(v k ) + 1 u 2 t 2 k q(v k ) = q(x) 4 ,
so that for k big enough,

q(x k ) 4 + x k , v k ut k q(v k ) + 1 u 2 t 2 k q(v k ) ≥ q(x k ) 8 .
If t k u ≥ 1, we can infer from (1) and the previous inequalities that

(2)

x k (u) ≤ 8 t k q(v k )q(x k ) x k + |q(x k )| 2 .
Observe that (2) also holds trivially if u = 0.

Suppose t k u ≤ 1, and note that conditions (a) and (b) on x imply that for k big enough,

1 + t k u x k , v k + t 2 k u 2 q(v k )q(x k ) 4 ≥ 1.
Then for k large enough,

(3)

x k (u) ≤ u x k + u 2 t k |q(x k )| 2 ≤ 1 t k x k + |q(x k )| 2
From (3), we infer: for any x k → x. In particular, {p k } is stable at j o (x) = β(1) in the sense of [START_REF]Dégénerescence locale des transformations conformes pseudo-riemanniennes[END_REF]Def. 4.1]. It was proved in [START_REF]Dégénerescence locale des transformations conformes pseudo-riemanniennes[END_REF] Lemma 4.3, that if {p k } has this stability property at some point z, it admits a pointwise stable holonomy sequence at z. Lemma 9.4 is thus proved in this case.

Second case: lim k→∞ t k q(v k ) = a ∈ R. This time, take x ∈ R 1,2 such that:

(a) q(x) = 0. (b) x, v > 0 Let u ∈ (0, 1], and write

1 + t k u x k , v k + t 2 k u 2 q(v k )q(x k ) 4 = 1 + t k u x k , v k + u 4 t k q(v k )q(x k ) .
Because x k , v k + u 4 t k q(v k )q(x k ) tends to x, v as k → ∞,

1 + t k u x k , v k + t 2 k u 2 q(v k )q(x k ) 4 ≥ 1 2 t k u x, v
for k large enough.

We infer from (1) that:

(5)

x k (u) ≤ 2 x, v x k t k + |q(x k )| 2 .
This inequality holds trivially for u = 0, giving lim k→∞ sup u∈[0,1]

x k (u) = 0.

Taking x k ≡ x gives lim k→∞ p k ([β]) = o. Moreover, if {x k } is any sequence converging to x, inequality [START_REF] Čap | Essential Killing fields of parabolic geometries[END_REF] shows that lim k→∞ p k (j o (x k )) = o. As in the first case, this implies that {p k } admits a pointwise, stable holonomy sequence at j o (x) = β(1), and Lemma 9.4 is proved. ♦

  by the scalar d -3 extends the representation on U to G 0 . 4.2. Cartan curvature. Let (M, M , ω) be the canonical Cartan geometry modeled on Ein 1,2 associated to (M, [g]), from section 3. Recall the notation G = PO(2, 3).

Corollary 4 . 6 .

 46 [START_REF]Dégénerescence locale des transformations conformes pseudo-riemanniennes[END_REF] Lem 4.3, 4.6] If {f k } has a stable holonomy sequence {p k } at x, then {p k } is also a holonomy sequence for {f k } on a neighborhood of x.

4. 4 .

 4 Stability implies conformal flatness. We record some immediate consequences of the properties outlined above. Proposition 4.7. If there is a balanced or a contracting holonomy sequence at x ∈ M , then the Cotton-York tensor vanishes at x. Proof. Let {p k = d k τ k } be a holonomy sequence at x, with {d k } ⊂ A satisfying the balanced or contracting condition in definition 4.2. The Prepresentation on the Cotton module U factors through the projection to G 0 , which has weights 3α + wβ, w = -2, -1, 0, 1, 2. Now p k acts by d k , for which (3α + wβ)(ln d k ) → ∞, for all possible w. Thus U AS (p k ) = 0. The conclusion follows from proposition 4.4.

( 3 )

 3 , such that [X, Y ] = [X, Z] = 0, and [Y, Z] = X. Since X and Y commute and span a 2-dimensional space at each point of Ω, there exist local coordinates (x, y, z) in which X = ∂ ∂x and Y = ∂ ∂y . Because the z X -orbits are 2-dimensional, Z is of the form λ ∂ ∂x + µ ∂ ∂y for some functions λ and µ.

  ∂y , and X = ∂ ∂x which are conformal for the metric g = ϕ * g.Again, let p = (p 1 , p 2 , p 3 ) be a point in our coordinate chart. The vector field U = Z -p 2 X -p 3 Y vanishes at p and is a conformal for g .

Lemma 6 . 3 .♦Corollary 6 . 4 .

 6364 Let U and V be two connected open subsets of H + , and f : U → V a diffeomorphism such that f * C = C and f * (h) = h. Then f is the restriction of a unique element of G. Proof: Because f preserves C, it is of the form f (x, y) = (x + η(y), ψ(y)). Now f also preserves h, hence f * acts as an automorphism of h. In particular f * B = bB, with b = 0. We get ψ(y) = 1 b y, showing that b > 0. Set b = e α . Next, f * (A) has the form A + cB + dC, from which we deduce η(y) = βy + γ ln(y) + τ, β, γ, τ ∈ R The surface Σ is endowed with a (G, H + )-structure.

Proof:

  Given y ∈ ∆, the set of 2-dimensional Z X -orbits O ⊂ Ω with ∆ ⊂ O coincides with the set of orbits O with y ∈ O. Denote this set of orbits I y . Let P y be a plaque containing y. For every O ∈ I y , the interior of O ∩P y is nonempty by proposition 7.13. Choose U O a connected component of this interior. If O = O are two distinct orbits in I y , then U O ∩ U O = ∅.

Lemma 8 . 2 ([ 31 ]

 8231 Lemma 5.4).

Theorem 9 . 1 .

 91 Let (M, g) be a smooth, compact, 3-dimensional Lorentzian manifold. Let {f k } ⊂ Conf loc (M, [g]) be an unbounded sequence defined on a common neighborhood U of x ∈ M . If {f k } admits a holonomy sequence at x contained in P + , then there exists a nonempty open subset U ⊂ M which is conformally flat.

Proposition 9 . 2 Corollary 9 . 3 .

 9293 (see[START_REF]Local dynamics of conformal vector fields[END_REF] Prop 6.3). Let (M, g) be a smooth Lorentzian manifold. Let {f k } < Conf(M, [g]) with holonomy sequence {p k } along xk → x in M . Assume that there exists a conformal geodesic segment β : [0, 1] → Ein 1,n-1 emanating from o such that lim k→∞ p k .[β] = o. Then any pointwise holonomy sequence of {p k } at β(1) admits a subsequence which is a holonomy sequence for {f k } with respect to some converging sequence ŷk → ŷ in M . This proposition, together with proposition 4.8, brings us closer to theorem 9.1, through the following corollary: Let (M, g) be a smooth, 3-dimensional, Lorentzian manifold. Let {p k } be a holonomy sequence for {f k } along xk → x in M . Suppose there exists a conformal geodesic segment β : [0, 1] → Ein

  x k ≡ x gives lim k→∞ p k ([β]) = o. Moreover, (4) shows that lim k→∞ p k (j o (x k )) = o

  1 with Poincaré recurrence to produce nontrivial local isotropy for isometric actions. Recall that a recurrent point for an unbounded subgroup H < Conf(M, [g]) is x ∈ M with h k .x → x for some unbounded sequence {h k } ⊂ H.

	Proposition 3.4. (compare [20, Prop. 5.1], [19, Prop. 3.3]) Let (M, g) be
	a compact, real-analytic pseudo-Riemannian manifold, and let {ϕ t X } be a
	noncompact conformal flow. At each recurrent point x for {ϕ t X }, the local isotropy Is loc X (x) has noncompact identity component.

  9, Sec 7.4], [5, Def 2.10] for a non-approximate version, and [16, Sec 4.4] for a related notion of stability and stable foliations). Definition 4.3. Let V be a P -module, and let {p k } be a sequence in P . The approximately stable set for {p k

  1,2 emanating from o will be a curve α : [0, 1] → Ein 1,2 of the form s → p.j o (sw), where p ∈ P and w ∈ R 1,2 . 9.2. Local dynamics via conformal geodesic segments. Let {p k } be a holonomy sequence for {f k } ⊂ Conf loc (M ) at x ∈ M . It is an unbounded sequence of conformal transformations of Ein 1,2 fixing o, which in turn admits holonomy sequences at other points of Ein 1,2 . The following proposition, borrowed from[START_REF]Local dynamics of conformal vector fields[END_REF], explains how conformal geodesic segments relate holonomy sequences of {f k } and {p k }.

  1,2 emanating from o such that lim k→∞ p k .[β] = o. If {p k } admits a pointwise holonomy sequence at β(1) which is stable, then a nonempty open subset U ⊂ M is conformally flat. Recall definition 4.2 for stable holonomy sequences. 9.3. Lemma ensuring stable holonomy sequence. Theorem 9.1 is a direct consequence of Corollary 9.3 and Lemma 9.4 below, which is the main technical result of this section. Lemma 9.4. Let {p k } be a sequence of P + . After passing to a subsequence, there exists a conformal geodesic segment β : [0, 1] → Ein 1,2 emanating from o such that lim k→∞ p k .[β] = o, and such that {p k } admits a pointwise holonomy sequence at β(1) which is stable. Proof: Denote the Euclidean norm on R 3 by • . Write

Proof: If dim Σ = 1, then Σ (2) = ∅, and Σ sing is actually empty. Indeed, if Σ sing = ∅, it is a closed semi-analytic set of dimension 0, namely, a finite number of points. These are all Z X -fixed points because Σ sing is Z Xinvariant. But corollary 4.11 rules out any Z X -fixed points. Thus Σ is a compact, Z X -invariant, analytic submanifold of M , which means it is a finite union of circular Z X -orbits.

If dim Σ = 2, then Σ sing is the complement of Σ (2) in Σ. If nonempty, it has dimension 1 or 0. Let Σ sing be the singular set of Σ sing . If Σ sing = ∅, then dim Σ sing = 0, and, as above, it is a nonempty set of Z X -fixed points, which leads to a contradiction. Therefore Σ sing = ∅, which means Σ sing is a closed, 1-dimensional, Z X -invariant, analytic submanifold of M , again a finite union of circular Z X -orbits. ♦ Here is the key proposition leading to a contradiction. Proposition 7.18 (see [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF] Sec 6.4). Let y ∈ Σ (2) . Then there exists a distinguished plaque Σ containing y and contained in Σ (2) .

Proof: We recall here the main points of the argument from [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF] that Σ is covered by finitely-many plaques, with some simplifications for dimension 3. Denote by u + the unipotent subalgebra of g 0 annihilating E 1 ; it is 1dimensional in our case. Let α : (-, ) → Σ be a differentiable path through y, which we can assume is contained in Σ (2) . For each t, there is a point α(t) ∈ π -1 (α(t)) with ω α(t) (z X ) ⊂ RE 1 modulo p and with u + ⊂ ω α(t) (z X ), by proposition 7.5. In fact, there is a differentiable lift α of α satisfying these conditions. Note that by proposition 7.11 [START_REF] Amores | Vector fields of a finite type G-structure[END_REF], the image of α is contained in R.

Because u + ⊂ ω α(t) (z X ) for all t, the latter subspace, which is 2-dimensional, is abelian, a standard fact that can be deduced from part (2) of the second axiom for ω. Together with a generator U of u + , it is spanned by an element of the form E 1 + A + ξ with A ∈ a and ξ ∈ q 1 , the centralizer of u + in p + , which is contained in q 1 . Note that centralizing u + means A ∈ ker β. An argument similar to that in the proof of lemma 7.6 shows that if α(A) = 0, then an element of z X would be timelike somewhere (see also [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF]Lem 6.13]), contradiciting proposition 7.5. Therefore A = 0, and ω α(t) (z X ) is spanned by U and E 1 + ξ for some ξ ∈ q 1 . On the other hand, if A t 0 = ω(α (t 0 )) were transverse to E ⊥ 1 + q, then calculations with the formula

(see the end of [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF]Sec. 6.4]) would give that ω α(t) (z X ) is not contained in the subspace spanned by U and E 1 + ξ for t close to t 0 . From this contradiction, we conclude that α (t) ∈ D for all t. It follows that α is contained in an