Subelliptic wave equations are never observable

It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic ray meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $\Delta=-\sum_{i=1}^m X_i^*X_i$ on a manifold $M$, and any measurable subset $\omega\subset M$ such that $M\backslash \omega$ contains in its interior a point $q$ with $[X_i,X_j](q)\notin \text{Span}(X_1,\ldots,X_m)$ for some $1\leq i,j\leq m$, we show that for any $T_0>0$, the wave equation with subelliptic Laplacian $\Delta$ is not observable on $\omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the associated sub-Riemannian distance) spending a long time in $M\backslash \omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.

1. Introduction 1.1. Setting. Let n ∈ ‫ގ‬ * and let M be a smooth connected compact manifold of dimension n with a nonempty boundary ∂ M. Let µ be a smooth volume on M. We consider m ⩾ 1 smooth vector fields X 1 , . . . , X m on M which are not necessarily independent, and we assume that the following Hörmander condition [1967] holds: The vector fields X 1 , . . . , X m and their iterated brackets [X i , X j ], [X i , [X j , X k ]], etc.
span the tangent space T q M at every point q ∈ M.
We consider the sub-Laplacian defined by where the star designates the transpose in L 2 (M, µ) and the divergence with respect to µ is defined by L X µ = (div µ X )µ, where L X stands for the Lie derivative. Then is hypoelliptic; see [Hörmander 1967, Theorem 1.1]. We consider with Dirichlet boundary conditions and the domain D( ) which is the completion in L 2 (M, µ) of the set of all u ∈ C ∞ c (M) for the norm ∥(Id− )u∥ L 2 . We also consider the operator (− ) 1/2 with domain D((− ) 1/2 ) which is the completion in L 2 (M, µ) of the set of all u ∈ C ∞ c (M) for the norm ∥(Id − ) 1/2 u∥ L 2 .

Main result.
Our main result is the following.
Theorem 2. Let T 0 > 0 and let ω ⊂ M be a measurable subset. We assume that there exist 1 ⩽ i, j ⩽ m and q in the interior of M\ω such that [X i , X j ](q) / ∈ Span(X 1 (q), . . . , X m (q)). Then the subelliptic wave equation (1) is not exactly observable on ω in time T 0 .
Consequently, using a duality argument (see Section 4.2), we obtain that exact controllability also does not hold in any finite time.
Corollary 4. Let T 0 > 0 and let ω ⊂ M be a measurable subset. We assume that there exist 1 ⩽ i, j ⩽ m and q in the interior of M\ω such that [X i , X j ](q) / ∈ Span(X 1 (q), . . . , X m (q)). Then the subelliptic wave equation (1) is not exactly controllable on ω in time T 0 .
In what follows, we denote by D the set of all vector fields that can be decomposed as linear combinations with smooth coefficients of the X i : D = Span(X 1 , . . . , X m ) ⊂ T M.
D is called the distribution associated to the vector fields X 1 , . . . , X m . For q ∈ M, we denote by D q ⊂ T q M the distribution D taken at point q.
The assumptions of Theorem 2 are satisfied as soon as the interior U of M \ ω is nonempty and D has constant rank < n in U . Indeed, under these conditions, we can argue by contradiction: assume that for any q ∈ U and any 1 ⩽ i, j ⩽ m, it holds [X i , X j ](q) ∈ Span(X 1 (q), . . . , X m (q)) = D q . Then we have [D, D] ⊂ D in U , i.e., D is involutive. By Frobenius's theorem, D is then completely integrable, which contradicts Hörmander's condition.
The following examples show that the assumptions of Theorem 2 are also satisfied in some nonconstantrank cases: Example 5. In the Baouendi-Grushin case, for which X 1 = ∂ x 1 and X 2 = x 1 ∂ x 2 are vector fields on (−1, 1) x 1 × ‫ޔ‬ x 2 , where ‫ޔ‬ = ‫,ޚ/ޒ‬ the corresponding sub-Laplacian = X 2 1 + X 2 2 (here, µ = d x 1 d x 2 for simplicity) is elliptic outside of the singular submanifold S = {x 1 =0}. Therefore, the corresponding subelliptic wave equation is observable on any open subset containing S (with some finite minimal time of observability, see [Bardos et al. 1992]), but according to Theorem 2, it is not observable in any finite time on any subset ω such that the interior of M \ ω has a nonempty intersection with S.
Then, we have [X 1 , X 2 ] = 2x 1 ∂ x 3 . The only points at which this bracket belongs to the distribution Span(X 1 , X 2 ) are the points for which x 1 = 0. Since this set of points has empty interior, the assumptions of Theorem 2 are satisfied as soon as M \ ω has nonempty interior.
Remark 7. The assumption of compactness on M is not necessary; we may remove it and just require that the subelliptic wave equation (1) in M is well-posed. It is for example the case if M is complete for the sub-Riemannian distance induced by X 1 , . . . , X m since is then essentially self-adjoint [Strichartz 1986].
Remark 8. Theorem 2 remains true if M has no boundary. In this case, (1) is well-posed in a space slightly smaller than (2): a condition of null average has to be added since nonzero constant functions on M are solutions of (1); see Section 1.5. The observability inequality of Theorem 2 remains true in this space of solutions; anticipating the proof, we notice that the spiraling normal geodesics of Proposition 17 still exist (since their construction is purely local), and we subtract from the initial datum u k 0 of the localized solutions constructed in Proposition 16 their spatial average M u k 0 dµ.
Remark 9. Thanks to abstract results (see for example [Miller 2012]), Theorem 2 remains true when the subelliptic wave equation (1) is replaced by the subelliptic half-wave equation ∂ t u + i √ − u = 0 with Dirichlet boundary conditions. 1.3. Ideas of the proof. In the sequel, we define a normal geodesic 1 to be the projection on M of a bicharacteristic (parametrized by time) for the principal symbol of the wave equation (1). We will give a more detailed definition in Section 1.4.
The proof of Theorem 2 mainly requires two ingredients: (1) There exist solutions of the free subelliptic wave equation (1) whose energy concentrates along any given normal geodesic.
(2) There exist normal geodesics which "spiral" around curves transverse to D, and which therefore remain arbitrarily close to their starting point on arbitrarily large time intervals.
Combining the two above facts, the proof of Theorem 2 is straightforward (see Section 4.1). Note that the first point follows from the general theory of propagation of complex Lagrangian spaces, while the second point is the main novelty of this paper. Since our construction is purely local (meaning that it does not "feel" the boundary and only relies on the local structure of the vector fields), we can focus on the case where there is a In the sequel, we assume it is the case.
Let us give an example of vector fields where the spiraling normal geodesics used in the proof of Theorem 2 are particularly simple. We consider the three-dimensional manifold with boundary M 1 = (−1, 1) x 1 × ‫ޔ‬ x 2 × ‫ޔ‬ x 3 , where ‫ޔ‬ = ‫ޚ/ޒ‬ ≈ (−1, 1) is the one-dimensional torus. We endow M 1 with the vector fields X 1 = ∂ x 1 and X 2 = ∂ x 2 − x 1 ∂ x 3 . This is the Heisenberg manifold with boundary. We endow M 1 with an arbitrary smooth volume µ. The normal geodesics we consider are given by They spiral around the x 3 -axis x 1 = x 2 = 0.
Here, one should think of ε as a small parameter. In the sequel, we denote by x ε the normal geodesic with parameter ε.
Clearly, given any T 0 > 0, for ε sufficiently small, we have x ε (t) ∈ V for every t ∈ (0, T 0 ). Our objective is to construct solutions u k of the subelliptic wave equation (1) such that ∥(u k 0 , u k 1 )∥ H×L 2 = 1 and the energy of u k (t, · ) concentrates outside of an open set V t containing x ε (t), i.e., tends to 0 as k → +∞ uniformly with respect to t ∈ (0, T 0 ). As a consequence, the observability inequality (4) fails.
The construction of solutions of the free wave equation whose energy concentrates on geodesics is classical in the elliptic (or Riemannian) case; these are the so-called Gaussian beams, for which a construction can be found for example in [Ralston 1982]. Here, we adapt this construction to our subelliptic (sub-Riemannian) setting, which does not raise any problem since the normal geodesics we consider stay in the elliptic part of the operator . It may also be directly justified with the theory of propagation of complex Lagrangian spaces (see Section 2).
In the case of general vector fields X 1 , . . . , X m , the existence of spiraling normal geodesics also has to be justified. For that purpose, we first approximate X 1 , . . . , X m by their nilpotent approximations, and we then prove that, for these approximations, such a family of spiraling normal geodesics exists, as in the Heisenberg case.
1.4. Normal geodesics. In this section, we explain in more details what normal geodesics are. As said before, they are natural extensions of Riemannian geodesics since they are projections of bicharacteristics.
We denote by S m phg (T * ((0, T ) × M)) the set of polyhomogeneous symbols of order m with compact support and by m phg ((0, T ) × M) the set of associated polyhomogeneous pseudodifferential operators of order m whose distribution kernel has compact support in (0, T ) × M (see Appendix A).
We set P = ∂ 2 tt − ∈ 2 phg ((0, T ) × M), whose principal symbol is with τ the dual variable of t and g * the principal symbol of − . For ξ ∈ T * M, we have (see Appendix A) Here, given any smooth vector field X on M, we denote by h X the Hamiltonian function (momentum map) on T * M associated with X defined in local (x, ξ )-coordinates by h X (x, ξ ) = ξ(X (x)).
In T * ‫ޒ(‬ × M), the Hamiltonian vector field ⃗ p 2 associated with p 2 is given by ⃗ { · , · } denotes the Poisson bracket (see Appendix A). Since ⃗ p 2 p 2 = 0, we get that p 2 is constant along the integral curves of ⃗ p 2 . Thus, the characteristic set C( p 2 ) = { p 2 =0} is preserved by the flow of ⃗ p 2 . Null-bicharacteristics are then defined as the maximal integral curves of ⃗ p 2 which live in C( p 2 ). In other words, the null-bicharacteristics are the maximal solutions of This definition needs to be adapted when the null-bicharacteristic meets the boundary ∂ M, but in the sequel, we only consider solutions of (7) on time intervals where x(t) does not reach ∂ M.
In the sequel, we take τ = − 1 2 , which gives g * (x(s), ξ(s)) = 1 4 . This also implies that t (s) = s + t 0 and, taking t as a time parameter, we are led to solve In other words, the t-variable parametrizes null-bicharacteristics in a way that they are traveled at speed 1.
Remark 10. In the subelliptic setting, the cosphere bundle S * M can be decomposed as S * M = U * M ∪S , where U * M = g * = 1 4 is a cylinder bundle, = {g * =0} is the characteristic cone and S is the sphere bundle of ; see [Colin de Verdière et al. 2018, Section 1].
We denote by φ t : S * M → S * M the (normal) geodesic flow defined by φ t (x 0 , ξ 0 ) = (x(t), ξ(t)), where (x(t), ξ(t)) is a solution of the system given by the first two lines of (8) and initial conditions (x 0 , ξ 0 ). Note that any point in S is a fixed point of φ t and that the other normal geodesics are traveled at speed 1 since we took g * = 1 4 in U * M (see Remark 10). The curves x(t) which solve (8) are geodesics (i.e., local minimizers) for a sub-Riemannian metric g; see [Montgomery 2002, Theorem 1.14].
1.5. Observability in some regions of phase-space. We have explained in Section 1.3 that the existence of solutions of the subelliptic wave equation (1) concentrated on spiraling normal geodesics is an obstruction to observability in Theorem 2. Our goal in this section is to state a result ensuring observability if one "removes" in some sense these normal geodesics.
For this result, we focus on a version of the Heisenberg manifold described in Section 1.3 which has no boundary. This technical assumption avoids us using boundary microlocal defect measures in the proof, which, in this sub-Riemannian setting, are difficult to handle. As a counterpart, we need to consider solutions of the wave equation with null initial average, in order to get well-posedness.
We consider the Heisenberg group G, that is, ‫ޒ‬ 3 with the composition law .
‫ޚ‬ is a co-compact subgroup of G, the left quotient M H = \G is a compact three-dimensional manifold and, moreover, X 1 and X 2 are well-defined as vector fields on the quotient. We call M H endowed with the vector fields X 1 and X 2 the "Heisenberg manifold without boundary". Finally, we define the Heisenberg Laplacian H = X 2 1 + X 2 2 on M H . Since [X 1 , X 2 ] = −∂ x 3 , it is a hypoelliptic operator. We endow M H with an arbitrary smooth volume µ.
As said above, normal geodesics corresponding to a large momentum ξ 3 are precisely the ones used to contradict observability in Theorem 2. We expect to be able to establish observability if we consider only solutions of (1) whose ξ 3 (in a certain sense) is not too large. This is the purpose of our second main result. Set Note that since ξ 3 is constant along null-bicharacteristics, V ε and its complement V c ε are invariant under the bicharacteristic equations (10).
In the next statement, we define a horizontal strip to be the periodization under the action of the co-compact subgroup of a set of the form where I is a strict open subinterval of [0, 2π ).
Theorem 11. Let B ⊂ M H be an open subset and suppose that B is sufficiently small, so that ω = M H \B contains a horizontal strip. Let a ∈ S 0 phg (T * M H ), a ⩾ 0, such that, denoting by j : and in particular a does not depend on time. There exists κ > 0 such that, for any ε > 0 and any T ⩾ κε −1 , it holds for some C = C(ε, T ) > 0 and for any solution u ∈ C 0 ‫;ޒ(‬ D((− H ) 1/2 )) ∩ C 1 ‫;ޒ(‬ L 2 0 ) of (9). The term ∥(u 0 , u 1 )∥ 2 L 2 ×H ′ 0 in the right-hand side of (11) cannot be removed; i.e., our statement only consists of a weak observability inequality. Indeed, the usual way to remove such terms is to use a unique continuation argument for eigenfunctions ϕ of , but here it does not work since Op(a)ϕ = 0 does not imply in general that ϕ ≡ 0 in the whole manifold, even if the support of a contains j (T * ω) for some nonempty open set ω: in some sense, there is no "pseudodifferential unique continuation argument".
1.6. Comments on the existing literature.
Elliptic and subelliptic waves. The exact controllability/observability of the elliptic wave equation is known to be almost equivalent to the so-called geometric control condition (GCC) (see [Bardos et al. 1992]) that any geodesic enters the control set ω within time T. In some sense, our main result is that GCC is not satisfied in the subelliptic setting, as soon as M\ω contains in its interior a point x at which is "truly subelliptic". For the elliptic wave equation, in many geometrical situations, there exists a minimal time T 0 > 0 such that observability holds only for T ⩾ T 0 : when there exists a geodesic γ : (0, T 0 ) → M traveled at speed 1 which does not meetω, one constructs a sequence of initial data (u k 0 , u k 1 ) k∈‫ގ‬ * of the wave equation whose associated microlocal defect measure is concentrated on (x 0 , ξ 0 ) ∈ S * M taken to be the initial conditions for the null-bicharacteristic projecting onto γ . Then, the associated sequence of solutions (u k ) k∈‫ގ‬ * of the wave equation has an associated microlocal defect measure ν which is invariant under the geodesic flow: ⃗ pν = 0, where ⃗ p is the Hamiltonian flow associated to the principal symbol p of the wave operator. In particular, denoting by π : T * M → M the canonical projection, π * ν gives no mass to ω since γ is contained in M \ω, and this proves that observability cannot hold.
In the subelliptic setting, the invariance property ⃗ pν = 0 does not give any information on ν on the characteristic manifold , since ⃗ p = −2τ ∂ t + ⃗ g * vanishes on . This is related to the lack of information on propagation of singularities in this characteristic manifold; see the main theorem of [Lascar 1982]. If one instead tries to use the propagation of the microlocal defect measure for subelliptic half-wave equations, one is immediately confronted with the fact that √ − is not a pseudodifferential operator near . This is why, in this paper, we used only the elliptic part of the symbol g * (or, equivalently, the strictly hyperbolic part of p 2 ), where the propagation properties can be established, and then the problem is reduced to proving geometric results on normal geodesics.
Subelliptic Schrödinger equations. The recent article [Burq and Sun 2019] deals with the same observability problem, but for subelliptic Schrödinger equations: namely, the authors consider the Baouendi-Grushin Schrödinger equation x + x 2 ∂ 2 y is the Baouendi-Grushin Laplacian. Given a control set of the form ω = (−1, 1) x × ω y , where ω y is an open subset of ‫,ޔ‬ the authors prove the existence of a minimal time of control L(ω) related to the maximal height of a horizontal strip contained in M G \ω. The intuition is that there are solutions of the Baouendi-Grushin Schrödinger equation which travel along the degenerate line x = 0 at a finite speed; in some sense, along this line, the Schrödinger equation behaves like a classical (half)-wave equation. What we want here is to explain in a few words why there is a minimal time of observability for the Schrödinger equation, while the wave equation is never observable in finite time as shown by Theorem 2.
The plane ‫ޒ‬ 2 x,y endowed with the vector fields ∂ x and x∂ y also admits normal geodesics similar to the 1-parameter family q ε , namely, for ε > 0, These normal geodesics, denoted by γ ε , also "spiral" around the line x = 0 more and more quickly as ε → 0, and so we might expect to construct solutions of the Baouendi-Grushin Schrödinger equation with energy concentrated along γ ε , which would contradict observability when ε → 0 as above for the Heisenberg wave equation.
However, we can convince ourselves that it is not possible to construct such solutions: in some sense, the dispersion phenomena of the Schrödinger equation exactly compensate for the lengthening of the normal geodesics γ ε as ε → 0 and explain that even these Gaussian beams may be observed in ω from a certain minimal time L(ω) > 0 which is uniform in ε.
To put this argument into a more formal form, we consider the solutions of the bicharacteristic equations for the Baouendi-Grushin Schrödinger equation i∂ t u − G u = 0 given by It follows from the hypoellipticity of G (see [Burq and Sun 2019, Section 3] for a proof) that Therefore ε 2 |ξ y | ≳ 1, and hence |y(t)| ≳ t, independently from ε and ξ y . This heuristic gives the intuition that a minimal time L(ω) is required to detect all solutions of the Baouendi-Grushin Schrödinger equation from ω, but that for T 0 > L(ω), no solution is localized enough to stay in M\ω during the time interval (0, T 0 ). Roughly speaking, the frequencies of order ξ y travel at speed ∼ ξ y , which is typical for a dispersion phenomenon. This picture is very different from the one for the wave equation (which we consider in this paper) for which no dispersion occurs.
With similar ideas, in [Letrouit and Sun 2021], the interplay between the subellipticity effects measured by the nonholonomic order of the distribution D (see Section 3.1) and the strength of dispersion of Schrödinger-type equations was investigated. More precisely, for γ = ∂ 2 x +|x| 2γ ∂ 2 y on M = (−1, 1) x ‫ޔ×‬ y , and for s ∈ ‫,ގ‬ the observability properties of the Schrödinger-type equation (i∂ t − (− γ ) s )u = 0 were shown to depend on the value κ = 2s/(γ + 1). In particular it is proved that, for κ < 1, observability fails for any time, which is consistent with the present result, and that for κ = 1, observability holds only for sufficiently large times, which is consistent with the result of A slightly different problem is the approximate controllability of hypoelliptic PDEs, which was studied in [Laurent and Léautaud 2022] for hypoelliptic wave and heat equations. Approximate controllability is weaker than exact controllability, and it amounts to proving "quantitative" unique continuation results for hypoelliptic operators. For the hypoelliptic wave equation, it is proved in [Laurent and Léautaud 2022] that for T > 2 sup x∈M (dist(x, ω)) (here, dist is the sub-Riemannian distance), the observation of the solution on (0, T ) × ω determines the initial data, and therefore the whole solution.
1.7. Organization of the paper. In Section 2, we construct exact solutions of the subelliptic wave equation (1) concentrating on any given normal geodesic. First, in Section 2.1, we show that, given any normal geodesic t → x(t) which does not hit ∂ M in the time interval (0, T ), it is possible to construct a sequence (v k ) k∈‫ގ‬ of approximate solutions of (1) whose energy concentrates along t → x(t) during the time interval (0, T ) as k → +∞. By "approximate", we mean here that ∂ 2 tt v k − v k is small, but not necessarily exactly equal to 0. In Section 2.1, we provide a first proof for this construction using the classical propagation of complex Lagrangian spaces. Another proof using a Gaussian beam approach is provided in Appendix B. Then, in Section 2.2, using this sequence (v k ) k∈‫ގ‬ , we explain how to construct a sequence (u k ) k∈‫ގ‬ of exact solutions of (∂ 2 tt − )u = 0 in M with the same concentration property along the normal geodesic t → x(t).
In Section 3, we prove the existence of normal geodesics which spiral in M, spending an arbitrarily large time in M\ω. These normal geodesics generalize the example described in Section 1.3 for the Heisenberg manifold with boundary. The proof proceeds in two steps: first, we show that it is sufficient to prove the result in the so-called "nilpotent case" (Section 3.2), and then we prove it in the nilpotent case (Section 3.3).
In Section 4.1, we use the results of Sections 2 and 3 to conclude the proof of Theorem 2. In Section 4.2, we deduce Corollary 4 by a duality argument. Finally, in Section 4.3, we prove Theorem 11.
2. Gaussian beams along normal geodesics 2.1. Construction of sequences of approximate solutions. We consider a solution (x(t), ξ(t)) t∈[0,T ] of (8) on M. We shall describe the construction of solutions of The following proposition, which is inspired by [Ralston 1982;Macià and Zuazua 2002], shows that it is possible, at least for approximate solutions of (12).
x)e ikψ(t,x) the following properties hold: • v k is an approximate solution of (12), meaning that • The energy of v k is bounded below with respect to k and t ∈ [0, T ]: • The energy of v k is small off x(t): For any t ∈ [0, T ], we fix V t an open subset of M for the initial topology of M, containing x(t), so that the mapping t → V t is continuous (V t is chosen sufficiently small so that this makes sense in a chart). Then Remark 13. The construction of approximate solutions such as the ones provided by Proposition 12 is usually done for strictly hyperbolic operators, that is, operators with a principal symbol p m of order m such that the polynomial f (s) = p m (t, q, s, ξ ) has m distinct real roots when ξ ̸ = 0; see for example [Ralston 1982]. The operator ∂ 2 tt − is not strictly hyperbolic because g * is degenerate, but our proof shows that the same construction may be adapted without difficulty to this operator along normal bicharacteristics. This is due to the fact that along normal bicharacteristics, ∂ 2 tt − is indeed strictly hyperbolic (or equivalently, is elliptic). It was already noted by [Ralston 1982] that the construction of Gaussian beams could be done for more general operators than strictly hyperbolic ones, and that the differences between the strictly hyperbolic case and more general cases arise while dealing with propagation of singularities. Also, in [Hörmander 1985, Chapter 24.2], it was noticed that "since only microlocal properties of p 2 are important, it is easy to see that hyperbolicity may be replaced by ∇ ξ p 2 ̸ = 0." Hereafter we provide two proofs of Proposition 12. The first proof is short and is actually quite straightforward for readers acquainted with the theory of propagation of complex Lagrangian spaces, once one has noticed that the solutions of (8) which we consider live in the elliptic part of the principal symbol of − . For the sake of completeness, and because this also has its own interest, we provide in Appendix B a second proof, longer but more elementary and accessible without any knowledge of complex Lagrangian spaces; it relies on the construction of Gaussian beams in the subelliptic context. The two proofs follow parallel paths, and indeed, the computations which are only sketched in the first proof are written in full detail in the second proof, given in Appendix B.
First proof of Proposition 12. The construction of Gaussian beams, or more generally of a WKB approximation, is related to the transport of complex Lagrangian spaces along bicharacteristics, as reported for example in [Hörmander 1985, Chapter 24.2; Ivrii 2019, Volume I, Part I, Chapter 1.2]. Our proof follows the lines of [Hörmander 1985, pages 426-428].
A usual way to solve (at least approximately) evolution equations of the form where P is a hyperbolic second-order differential operator with real principal symbol and C ∞ coefficients, is to search for oscillatory solutions In this expression as in the rest of the proof, we suppress the time variable t. Thus, we use x = (x 0 , x 1 , . . . , x n ), where x 0 = t in the earlier notation, and we set x ′ = (x 1 , . . . , x n ). Similarly, we take the notation ξ = (ξ 0 , ξ 1 , . . . , ξ n ), where ξ 0 = τ previously, and ξ ′ = (ξ 1 , . . . , ξ n ). The bicharacteristics are parametrized by s as in (7), and without loss of generality, we only consider bicharacteristics with x(0) = 0 at s = 0, which implies in particular x 0 (s) = s because of our choice τ 2 (s) = g * (x(s), ξ(s)) = 1 4 . Taking charts of M, we can assume M ⊂ ‫ޒ‬ n . The precise argument for reducing to this case is written at the end of Appendix B. Also, in the sequel, P = ∂ 2 tt − . Plugging the ansatz (17) into (16), we get with and L is a transport operator given by For v k to be an approximate solution of P, we are first led to cancel the higher-order term in (18), i.e., which we solve for initial conditions (i.e., we fix such a ψ 0 , and then we solve (20) for ψ). Indeed, it will be sufficient for our purpose for (20) to be satisfied at second order along the curve x(s); i.e., D α x f (x(s)) = 0 for any |α| ⩽ 2 and any s. For that, we first notice that the choice ∇ψ(x(s)) = ξ(s) ensures that (20) holds at orders 0 and 1 along the curve s → x(s) (see Appendix B for detailed computations). Now, we explain how to choose D 2 ψ(x(s)) adequately in order for (20) to hold at order 2.
However, if the phase ψ 0 is complex, quadratic, and satisfies the condition Im(D 2 ψ 0 ) > 0, where D 2 ψ 0 denotes the Hessian, no blow-up happens, and the solution is global-in-time. Let us explain why. Indeed, 0 = {(x ′ , ∇ψ 0 (x ′ ))} then lives in the complexification of the tangent space T * M, which may be thought of as ‫ރ‬ 2(n+1) . We take coordinates (y, η) on T * ‫ޒ‬ n+1 or T * ‫ރ‬ n+1 and we consider the symplectic forms defined by σ = dy j ∧ dη j and σ ‫ރ‬ = dy j ∧ dη j .
Because of the condition Im(D 2 ψ 0 ) > 0, 0 is called a "strictly positive Lagrangian space" (see [Hörmander 1985, Definition 21.5.5]), meaning that iσ ‫ރ‬ (v, v) > 0 for v in the tangent space to 0 . For any s, the symplectic forms σ and σ ‫ރ‬ are preserved by (s, · ), meaning that (s, · ) * σ = σ and (s, · ) * σ ‫ރ‬ = σ ‫ރ‬ ; therefore σ = 0 on the tangent space to s , and iσ ‫ރ‬ (v, v) > 0 for v tangent to s . It precisely means that s is also a strictly positive Lagrangian space. Then, by [Hörmander 1985, Proposition 21.5.9], we know that there exists ψ(s, · ) complex and quadratic with Im(D 2 ψ(s, · )) > 0 such that s = {(x ′ , ∇ x ′ ψ(s, x ′ ))} (to apply [Hörmander 1985, Proposition 21.5.9], recall that, for In other words, the key point in using complex phases is that strictly positive Lagrangian spaces are parametrized by complex quadratic phases ϕ with Im(D 2 ϕ) > 0, whereas real Lagrangian spaces were not parametrized by real phases (see explanations above). This parametrization is a diffeomorphism from the Grassmannian of strictly positive Lagrangian spaces to the space of complex quadratic phases with ϕ with Im(D 2 ϕ) > 0. Hence, the phase for s ∈ [0, T ] and y ′ ∈ ‫ޒ‬ n is smooth and for this choice (20) is satisfied at second order along s → x(s) (the rest R(x ′ , ξ ′ ) plays no role since it vanishes in a neighborhood of s → x(s)).
Then, we note that A 2 vanishes along the bicharacteristic if and only if La 0 (x(s)) = 0 (see also [Hörmander 1985, equation (24.2.9)]). According to (19), this turns out to be a linear transport equation on a 0 (x(s)), with leading coefficient ∇ ξ p 2 (x(s), ξ(s)) different from 0. Given a ̸ = 0 at (t = 0, x ′ = x ′ (0)), this transport equation has a solution a 0 (x(s)) with initial datum a, and, by Cauchy uniqueness, a 0 (x(s)) ̸ = 0 for any s. We can choose a 0 in a smooth (and arbitrary) way outside the bicharacteristic. We choose it to vanish outside a small neighborhood of this bicharacteristic, so that no boundary effect happens.
With these choices of ψ and a 0 , the bound (13) then follows from the following result whose proof is given in [Ralston 1982, Lemma 2.8].
Lemma 14. Let c(x) be a function on ‫ޒ‬ n+1 which vanishes at order S − 1 on a curve for some S ⩾ 1. Suppose that Supp c ∩ {|x 0 | ⩽ T } is compact and that Im ψ(x) ⩾ ad(x) 2 on this set for some constant a > 0, where d(x) denotes the distance from the point x ∈ ‫ޒ‬ d+1 to the curve . Then there exists a constant C such that Let us now sketch the end of the proof, which is given in Appendix B in full detail. We apply Lemma 14 to S = 3, c = A 1 and to S = 1, c = A 2 , and we get which implies (13). The bounds (14) and (15) follow from the facts that Im(D 2 ψ(s, · )) > 0 and v k (x) = k n/4−1 a 0 (x)e ikψ(x) . □ Remark 15. An interesting question would be to understand the delocalization properties of the Gaussian beams constructed along normal geodesics in Proposition 12. Compared with the usual Riemannian case done for example in [Ralston 1982], there is a new phenomenon in the sub-Riemannian case since the normal geodesic x(t) (or, more precisely, its lift to S * M) may approach the characteristic manifold = {g * =0}, which is the set of directions in which is not elliptic. In finite time T as in our case, the lift of the normal geodesic remains far from , but it may happen as T → +∞ that it goes closer and closer to . The question is then to understand the link between the delocalization properties of the Gaussian beams constructed along such a normal geodesic, and notably the interplay between the time T and the semiclassical parameter 1 k .

Construction of sequences of exact solutions in M.
In this section, using the approximate solutions of Section 2.1, we construct exact solutions of (12) whose energy concentrates along a given normal geodesic of M which does not meet the boundary ∂ M during the time interval [0, T ].
Suppose (v k ) k∈‫ގ‬ is constructed along x(t) as in Proposition 12 and u k is the solution of the Cauchy Then: • The energy of u k is bounded below with respect to k and t ∈ [0, T ]: there exists A > 0 such that, for all t ∈ [0, T ], lim inf k→+∞ E(u k (t, · )) ⩾ A.
• The energy of u k is small off x(t): For any t ∈ [0, T ], we fix V t an open subset of M for the initial topology of M, containing x(t), so that the mapping t → V t is continuous (V t is chosen sufficiently small so that this makes sense in a chart). Then Proof of Proposition 16. Set h k = (∂ 2 tt − )(θ v k ). We consider w k the solution of the Cauchy problem   Differentiating E(w k (t, · )) and using Gronwall's lemma, we get the energy inequality Therefore, using (13), we get sup t∈[0,T ] E(w k (t, · )) ⩽ Ck −1 . Since u k = θ v k − w k , we obtain that for every t ∈ [0, T ], where the last equality comes from the fact that θ and its derivatives are bounded and ∥v k ∥ L 2 ⩽ Ck −1 when k → +∞. Using (14), we conclude that (22) holds.

Existence of spiraling normal geodesics
The goal of this section is to prove the following proposition, which is the second building block of the proof of Theorem 2, after the construction of localized solutions of the subelliptic wave equation (1) done in Section 2. We say that X 1 , . . . , X m satisfies the property (P) at q ∈ M if the following holds: (P) For any open neighborhood V of q, for any T 0 > 0, there exists a nonstationary normal geodesic t → x(t), traveled at speed 1, such that x(t) ∈ V for any t ∈ [0, T 0 ].
Proposition 17. At any point q ∈ M such that there exist 1 ⩽ i, j ⩽ m with [X i , X j ](q) / ∈ D q , property (P) holds.
In Section 3.1, we define the so-called nilpotent approximations X q 1 , . . . , X q m at a point q ∈ M, which are first-order approximations of X 1 , . . . , X m at q ∈ M such that the associated Lie algebra Lie( X q 1 , . . . , X q m ) is nilpotent. Roughly, we have X q i ≈ X i (q), but low-order terms of X i (q) are not taken into account for defining X q i , so that the high-order brackets of the X q i vanish (which is not generally the case for the X i ). These nilpotent approximations are good local approximations of the vector fields X 1 , . . . , X m , and their study is much simpler.
The proof of Proposition 17 splits into two steps: first, we show that it is sufficient to prove the result in the nilpotent case (Section 3.2), then we handle this simpler case (Section 3.3).
3.1. Nilpotent approximation. In this section, we recall the construction of the nilpotent approximations The integer r (q) is called the nonholonomic order of D at q, and it is equal to 2 everywhere in the Heisenberg manifold for example. Note that it depends on q; see Example 5 in Section 1.2 (the Baouendi-Grushin example). For 0 ⩽ i ⩽ r (q), we set n i (q) = dim D i q , and the sequence (n i (q)) 0⩽i⩽r (q) is called the growth vector at point q. We set Q(q) = r (q) i=1 i(n i (q) − n i−1 (q)), which is generically the Hausdorff dimension of the metric space given by the sub-Riemannian distance on M; see [Mitchell 1985]. Finally, we define the nondecreasing sequence of weights w i (q) for 1 ⩽ i ⩽ n as follows. Given any 1 ⩽ i ⩽ n, there exists a unique 1 ⩽ j ⩽ n such that n j−1 (q) + 1 ⩽ i ⩽ n j (q). We set w i (q) = j. For example, for any q in the Heisenberg manifold, w 1 (q) = w 2 (q) = 1 and w 3 (q) = 2; indeed, the coordinates x 1 and x 2 have "weight 1", while the coordinate x 3 has "weight 2" since ∂ x 3 requires a bracket to be generated.
Regular and singular points. We say that q ∈ M is regular if the growth vector (n i (q ′ )) 0⩽i⩽r (q ′ ) at q ′ is constant for q ′ in a neighborhood of q. Otherwise, q is said to be singular. If any point q ∈ M is regular, we say that the structure is equiregular. For example, the Heisenberg manifold is equiregular, but not the Baouendi-Grushin example.
The nonholonomic order of a smooth germ of vector field X at q, denoted by ord q (X ), is the real number defined by ord q (X ) = sup{σ ∈ ‫ޒ‬ : ord q (X f ) ⩾ σ + ord q ( f ) for all f ∈ C ∞ (q)}.
For example, it holds ord q ([X, Y ]) ⩾ ord q (X ) + ord q (Y ) and ord q ( f X ) ⩾ ord q ( f ) + ord q (X ). As a consequence, every X which has the property that X (q ′ ) ∈ D i q ′ for any q ′ in a neighborhood of q is of nonholonomic order ⩾ −i.
Privileged coordinates. Locally around q ∈ M, it is possible to define a set of so-called "privileged coordinates" of M; see [Bellaïche 1996].
A family (Z 1 , . . . , Z n ) of n vector fields is said to be adapted to the sub-Riemannian flag at q if it is a frame of T q M at q and if Z i (q) ∈ D w i (q) q for any i ∈ {1, . . . , n}. In other words, for any i ∈ {1, . . . , r (q)}, the vectors Z 1 , . . . , Z n i (q) at q span D i q . A system of privileged coordinates at q is a system of local coordinates (x 1 , . . . , x n ) such that In particular, for privileged coordinates, we have ∂ x i ∈ D w i (q) q \D w i (q)−1 q at q, meaning that privileged coordinates are adapted to the flag.
Example: exponential coordinates of the second kind. Choose an adapted frame (Z 1 , . . . , Z n ) at q. It is proved in [Jean 2014, Appendix B] that the inverse of the local diffeomorphism (x 1 , . . . , x n ) → exp(x 1 Z 1 ) • · · · • exp(x n Z n )(q) defines privileged coordinates at q, called exponential coordinates of the second kind.
Dilations. We consider a chart of privileged coordinates at q given by a smooth mapping ψ q : U → ‫ޒ‬ n , where U is a neighborhood of q in M, with ψ q (q) = 0. For every ε ∈ ‫,}0{\ޒ‬ we consider the dilation δ ε : ‫ޒ‬ n → ‫ޒ‬ n defined by δ ε (x) = (ε w i (q) x 1 , . . . , ε w n (q) x n ) for every x = (x 1 , . . . , x n ). A dilation δ ε acts also on functions and vector fields on ‫ޒ‬ n by pull-back: In particular, for any vector field X of nonholonomic order k, it holds δ * ε X = ε −k X .
Coming back to the vector fields X 1 , . . . , X m , we can write the Taylor expansion Since X i ∈ D, its nonholonomic order is necessarily −1; hence it holds w(α) ⩾ w j (q) − 1 if a α, j ̸ = 0. Therefore, we may write X i as a formal series where X (s) i is a homogeneous vector field of degree s, meaning that Then X q i is homogeneous of degree −1 with respect to dilations, i.e., δ * ε X q i = ε −1 X q i for any ε ̸ = 0. Each X q i may be seen as a vector field on ‫ޒ‬ n thanks to the coordinates (x 1 , . . . , x n ). Moreover, in the C ∞ topology; all derivatives uniformly converge on compact subsets. For ε > 0 small enough we have where R ε i depends smoothly on ε for the C ∞ topology; see also [Agrachev et al. 2020, Lemma 10.58]. An important property is that ( X The nilpotent approximation of X 1 , . . . , X m at q is then defined as M q ≃ ‫ޒ‬ n endowed with the vector fields X q 1 , . . . , X q m . It is important to note that the nilpotent approximation depends on the initial choice of privileged coordinates. For an explicit example of computation of nilpotent approximation; see [Jean 2014, Example 2.8].
3.2. Reduction to the nilpotent case. In this section, we show the following.
Lemma 18. Let X 1 , . . . , X m be smooth vector fields on M satisfying Hörmander's condition, and let q ∈ M. If the property (P) holds at point 0 ∈ ‫ޒ‬ n for the nilpotent approximation X q 1 , . . . , X q m , then the property (P) holds at point q for X 1 , . . . , X m .
Note that the above lemma is true for any nilpotent approximation X q 1 , . . . , X q m at q, i.e., for any choice of privileged coordinates (see Section 3.1).
Proof of Lemma 18. We use the notation h Z for the momentum map associated with the vector field Z (see Section 1.4). We use the notation of Section 3.1, in particular the coordinate chart ψ q .
We set Y i = (ψ q ) * X i and X ε i = εδ * ε Y i which is a vector field on ‫ޒ‬ n . Recall that where R ε i depends smoothly on ε for the C ∞ topology. Therefore, using the homogeneity of X q i , we get, for any ε > 0, The vector field (δ ε ) * R ε i (x) does not depend on ε and has a size which tends uniformly to 0 as x → 0 ∈ M q ≃ ‫ޒ‬ n . Recall that the Hamiltonian H associated to the vector fields X q i is given by

Similarly, we set
We note that (27) gives where ⃗ is a smooth vector field on T * ‫ޒ‬ n such that when ∥x∥ → 0 (independently of ξ ), where π : T * ‫ޒ‬ n → ‫ޒ‬ n is the canonical projection. This last point comes from the smooth dependence of R ε i on ε for the C ∞ topology (uniform convergence of all derivatives on compact subsets of ‫ޒ‬ n ).
Given the projection of an integral curve c( · ) of ⃗ H , we denote byĉ( · ) the projection of the integral curve of ⃗ H with same initial covector. Combining (28) and (29), and using Gronwall's lemma, we obtain the following result: Therefore, if the property (P) holds at 0 ∈ ‫ޒ‬ n for X q 1 , . . . , X q m , then it holds also at 0 ∈ ‫ޒ‬ n for the vector fields Y 1 , . . . , Y m .
Using that X i = ψ * q Y i , we can pull back the result to M and obtain that the property (P) holds at point q for X 1 , . . . , X m , which concludes the proof of Proposition 17. □ Thanks to Lemma 18, it is sufficient to prove the property (P) under the additional assumption that M ⊂ ‫ޒ‬ n and Lie(X 1 , . . . , X m ) is nilpotent.
In all that follows, we assume that this is the case.

3.3.
End of the proof of Proposition 17. Let us finish the proof of Proposition 17. Our ideas are inspired by [Agrachev and Gauthier 2001, Section 6].
First step: reduction to the constant Goh matrix case. We consider an adapted frame Y 1 , . . . , Y n at q. We take exponential coordinates of the second kind at q; we consider the inverse ψ q of the diffeomorphism (x 1 , . . . , x n ) → exp(x 1 Y 1 ) · · · exp(x n Y n )(q).
Then we write the Taylor expansion (26) of X 1 , . . . , X m in these coordinates. Thanks to Lemma 18, we can assume that all terms in these Taylor expansions have nonholonomic order −1. We denote by ξ i the dual variable of x i . We use the notation n 1 , n 2 , . . . introduced in Section 3.1, and we make a strong use of (25).
Proof. We write where the a i j are homogeneous polynomials. We have Let k ⩾ n 2 +1, which means that x k has nonholonomic order ⩾ 3. If a i j (x) depends on x k , then necessarily i ⩾ n 3 + 1, since a i j (x)∂ x i has nonholonomic order −1. Thus, writing explicitlyξ k = −∂g * /∂ x k thanks to (31), there is in front of each term a factor ξ i for some i which is in particular ⩾ n 2 + 1. By Cauchy uniqueness, we deduce that ξ k ≡ 0 for any k ⩾ n 2 + 1. Now, let k ⩾ n 1 + 1, which means that x k has nonholonomic order ⩾ 2. If a i j (x) depends on x k , then necessarily i ⩾ n 2 +1, since a i j (x)∂ x i has nonholonomic order −1. Thus, writing explicitlyξ k = −∂g * /∂ x k thanks to (31), there is in front of each term a factor ξ i for some i which is ⩾ n 2 + 1. It is null by the previous conclusion; henceξ k ≡ 0. □ The previous claim will help us to reduce the complexity of the vector fields X i once again (after the first reduction provided by Lemma 18). Let us consider, for any 1 ⩽ j ⩽ m, the vector field where the sum is taken only up to n 2 . We also consider the reduced Hamiltonian on T * M Claim 2. If X red 1 , . . . , X red m satisfy property (P) at q, then X 1 , . . . , X m satisfy property (P) at q. Proof. Let us assume that X red 1 , . . . , X red m satisfy property (P) at q. Let T 0 > 0 and let (x red,ε (0), ξ red,ε (0)) be initial data for the Hamiltonian system associated to g * red which yield speed-1 normal geodesics (x red,ε (t), ξ red,ε (t)) such that x red,ε (t) → q uniformly over (0, T 0 ) as ε → 0.
We can assume without loss of generality that ξ red,ε i (0) = 0 for any i ⩾ n 2 + 1, since these momenta (preserved under the reduced Hamiltonian evolution) do not change the projection x red,ε (t) of the normal geodesic. We consider (x ε (0), ξ ε (0)) = (x red,ε (0), ξ red,ε (0)) as initial data for the (nonreduced) Hamiltonian evolution associated to g * . Then we notice that ξ ε k ≡ 0 for k ⩾ n 2 + 1 thanks to Claim 1. It follows that when i ⩽ n 2 , we have x ε i (t) = x red,ε i (t); i.e., the coordinate x i is the same for the reduced and the nonreduced Hamiltonian evolution.
Finally, we take k such that n 2 + 1 ⩽ k ⩽ n 3 . Since g * is given by (31), we havė But a k j has necessarily nonholonomic order 2 since ∂ x k has nonholonomic order −3. Thus, a k j (x) is a nonconstant homogeneous polynomial in x 1 , . . . , x n 2 . Since x ε 1 , . . . , x ε n 2 converge to q uniformly over (0, T 0 ) as ε → 0, it is also the case of x ε k according to (33), noticing that n i=1 a i j (x ε )ξ ε i ⩽ (g * ) 1 2 = 1 2 for any j. In other words, x ε n 2 +1 , . . . , x ε n 3 also converge to q uniformly over (0, T 0 ) as ε → 0. We can repeat this argument successively for k ∈ {n 3 + 1, . . . , n 4 }, k ∈ {n 4 + 1, . . . , n 5 }, etc., and we finally obtain the result: for any 1 ⩽ k ⩽ n, x ε k converges to q uniformly over (0, T 0 ) as ε → 0. □ Thanks to the previous claim, we are now reduced to proving Proposition 17 for the vector fields X red 1 , . . . , X red m . In order to keep notation as simple as possible, we simplify to X 1 , . . . , X m ; i.e., we drop the upper notation "red". Also, without loss of generality we assume that q = 0.
If we choose our normal geodesics so that x(0) = 0, then x i ≡ 0 for any i ⩾ n 2 + 1 thanks to (32). In other words, we forget the coordinates x n 2 +1 , . . . , x n in the sequel, since they all vanish. 2 Second step: conclusion of the proof. Now, we write the normal extremal system in its "control" form. We refer the reader to [Agrachev et al. 2020, Chapter 4]. We havė where the u i are the controls, explicitly given by since (x(t), ξ(t)) = e t ⃗ g * (0, ξ 0 ). Thanks to (32), we rewrite (34) aṡ where F = (a i j ), which has size n 2 × m, and u = t (u 1 , . . . , u m ). Differentiating (35), we have the complementary equationu where G is the Goh matrix (it differs from the usual Goh matrix by a factor −2 due to the absence of factor 1 2 in the Hamiltonian g * in our notation).
Let us prove that G(t) is constant in t. Fix 1 ⩽ j, j ′ ⩽ m. We notice that in (32), a i j is a constant (independent of x) as soon as 1 ⩽ i ⩽ n 1 since ∂ x i has weight −1. This implies [X j , X j ′ ] is spanned by the vector fields ∂ x n 1 +1 , ∂ x n 1 +2 , . . . , ∂ x n 2 .
Putting this into the relation {h X j , h X j ′ } = h [X j ,X j ′ ] , and using that the dual variables ξ k for n 1 +1 ⩽ k ⩽ n 2 are preserved under the Hamiltonian evolution (due to Claim 1), we get that G(t) ≡ G is constant in t.
We know that G ̸ = 0 and that G is antisymmetric. The whole control space ‫ޒ‬ m is the direct sum of the image of G and the kernel of G, and G is nondegenerate on its image. We take u 0 in an invariant plane of G; in other words its projection on the kernel of G vanishes (see Remark 19). We denote by G the restriction of G to this invariant plane. We also assume that u 0 , decomposed as u 0 = (u 01 , . . . , u 0m ) ∈ ‫ޒ‬ m , satisfies m i=1 u 2 0i = 1 4 . Then u(t) = e t G u 0 and since e t G is an orthogonal matrix, we have ∥e t G u 0 ∥ = ∥u 0 ∥. We have by integration by parts Let us now choose the initial data of our family of normal geodesics (indexed by ε). The starting point x ε (0) = 0 is the same for any ε; we only have to specify the initial covectors ξ ε = ξ ε (0) ∈ T * 0 ‫ޒ‬ m . For any i = 1, . . . , m, we impose that It follows that g * (x(0), ξ ε (0)) = m i=1 u 2 0i = 1 4 for any ε > 0. Now, we notice that Span(X 1 , . . . , X m ) is in direct sum with the Span of the [X i , X j ] for i, j running over 1, . . . , m (this follows from (37)). Fixing G 0 ̸ = 0 an antisymmetric matrix and G 0 its restriction to an invariant plane, we can specify, simultaneously to (39), that Then x ε (t) is given by (38) applied with G = ε −1 G 0 , which brings a factor ε in front of (38). Recall finally that the coefficients a i j which compose F have nonholonomic order 0 or 1; thus they are degree-1 (or constant) homogeneous polynomials in x 1 , . . . , x n 1 . Thus d ds (F(x(s))) is a linear combination ofẋ i (s) which we can rewrite thanks to (36) as a combination with bounded coefficients since m i=1 u 2 i = 1 4 of the x i (s). Hence, applying the Gronwall lemma in (38), we get ∥x ε (t)∥ ⩽ Cε, which concludes the proof.
Remark 19. Let us explain why we choose u 0 to be in an invariant plane of G. If the projection of u 0 to the kernel of G is nonzero then the primitive of the exponential of e (t/ε)G 0 u 0 contains a linear term that does not depend on ε. Then the corresponding trajectory follows a singular curve; see [Agrachev et al. 2020, Chapter 4] for a definition. This means we find normal geodesics which spiral around a singular curve and do not remain close to their initial point over (0, T 0 ), although their initial covector is "high in the cylinder bundle U * M". For example, for the Hamiltonian ξ 2 1 + (ξ 2 + x 2 1 ξ 3 ) 2 associated to the "Martinet" vector fields X 1 = ∂ x 1 , X 2 = ∂ x 2 + x 2 1 ∂ x 3 in ‫ޒ‬ 3 , there exist normal geodesics which spiral around the singular curve (t, 0, 0).
Remark 20. The normal geodesics constructed above lose their optimality quickly, in the sense that their first conjugate point and their cut-point are close to q.
⊂ V, and fix also T 0 > 0. As already explained in Section 1.3, to conclude the proof of Theorem 2, we use Proposition 16 applied to the particular normal geodesics constructed in Proposition 17.
By Proposition 17, we know that there exists a normal geodesic t → x(t) such that x(t) ∈ V ′ for any t ∈ (0, T 0 ). It is the projection of a bicharacteristic (x(t), ξ(t)) and since it is nonstationary and travels at speed 1, it holds g * (x(t), ξ(t)) = 1 4 . We denote by (u k ) k∈‫ގ‬ a sequence of solutions of (12) as in Proposition 16 whose energy at time t concentrates on x(t) for t ∈ (0, T 0 ). Because of (22), we know that ∥(u k (0), ∂ t u k (0))∥ H×L 2 ⩾ c > 0 uniformly in k. Therefore, in order to establish Theorem 2, it is sufficient to show that Since x(t) ∈ V ′ for any t ∈ (0, T 0 ), we get that for V t chosen sufficiently small for any t ∈ (0, T 0 ), the inclusion V t ⊂ V holds (see Proposition 16 for the definition of V t ). Combining this last remark with (23), we get (40), which concludes the proof of Theorem 2.

4.2.
Proof of Corollary 4. We endow the topological dual H(M) ′ with the norm The following proposition is standard; see, e.g., [Tucsnak and Weiss 2009;Le Rousseau et al. 2017].

4.3.
Proof of Theorem 11. We consider the space of functions for any t ∈ [0, T ], and we denote by H T its completion for the norm ∥ · ∥ H T induced by the scalar product We consider also the topological dual H ′ 0 of the space H 0 (see Section 1.5).
We prove the last injection. Let u ∈ H T . Writing u(t, · ) = ∞ k=1 a k (t)ϕ k ( · ) (note that there is no 0-mode since u(t, · ) has null average), we see that and thus H T embeds continuously into L 2 ((0, T ) × M H ). Then, using a classical subelliptic estimate (see [Hörmander 1967;Rothschild and Stein 1976, Theorem 17]), we know that there exists C > 0 such that Together with the previous estimate, we obtain that, for any u ∈ H T , ∥u∥ Proof of Theorem 11. In this proof, we use the notation P = ∂ 2 tt − H . For the sake of a contradiction, suppose that there exists a sequence (u k ) k∈‫ގ‬ of solutions of the wave equation such that ∥(u k 0 , u k 1 )∥ H×L 2 = 1 for any k ∈ ‫ގ‬ and as k → +∞. Following the strategy of [Tartar 1990;Gérard 1991], our goal is to associate a defect measure to the sequence (u k ) k∈‫ގ‬ . Since the functional spaces involved in our result are unusual, we give the argument in detail. First, up to extraction of a subsequence which we omit, (u k 0 , u k 1 ) converges weakly in H 0 × L 2 (M H ) and, using the first convergence in (43) and the compact embedding we get that (u k 0 , u k 1 ) ⇀ 0 in H 0 × L 2 0 . Using the continuity of the solution with respect to the initial data, we obtain that u k ⇀ 0 weakly in H T . Using Lemma 22, we obtain u k → 0 strongly in L 2 ((0, T ) × M H ).
and u k → 0 strongly in L 2 ((0, T )×M H ), the first of the two lines in (44) converges to 0 as k → +∞. Moreover, the last line is bounded uniformly in k since B ∈ 0 phg ((0, T ) × M H ). Hence (Bu k , u k ) H T is uniformly bounded. By a standard diagonal extraction argument (see [Gérard 1991] for example), there exists a subsequence, which we still denote by (u k ) k∈‫ގ‬ such that (Bu k , u k ) converges for any B of principal symbol b in a countable dense subset of C ∞ c ((0, T ) × M H ). Moreover, the limit only depends on the principal symbol b, and not on the full symbol.
Hence, we can conclude using the expression for x 3 (whose derivative is roughly (4|ξ 3 |) −1 ) and the fact that ω = M H \B contains a horizontal strip. Note that if ξ 3 = 0, the expressions of x 1 (t), x 2 (t), x 3 (t) are much simpler and we can conclude similarly. Hence, together with (49), the propagation property (48) implies that ν ≡ 0. It follows that ∥u k ∥ H T → 0. By conservation of energy, it is a contradiction with the normalization ∥(u k 0 , u k 1 )∥ H×L 2 = 1. Hence, (11) holds. □

Appendix A: Pseudodifferential calculus
We denote by an open set of a d-dimensional manifold (typically d = n or d = n + 1 with the notation of this paper) equipped with a smooth volume µ. We denote by q the variable in , typically q = x or q = (t, x) with our notation.
Let ω 0 = dp ∧ dq be the canonical symplectic form on T * written in canonical coordinates (q, p). The Hamiltonian vector field ⃗ f of a function f ∈ C ∞ (T * ) is defined by the relation In the coordinates (q, p), it reads In these coordinates, the Poisson bracket is which is also equal to ⃗ f g and −⃗ g f . Let π : T * → be the canonical projection. We recall briefly some facts concerning pseudodifferential calculus, following [Hörmander 1985, Chapter 18].
We denote by S m hom (T * ) the set of homogeneous symbols of degree m with compact support in . We also write S m phg (T * ) for the set of polyhomogeneous symbols of degree m with compact support in . Hence, a ∈ S m phg (T * ) if a ∈ C ∞ (T * ), π(Supp(a)) is a compact of , and there exist a j ∈ S m− j hom (T * ) such that, for all N ∈ ‫,ގ‬ a − N j=0 a j ∈ S m−N −1 phg (T * ). We denote by m phg (T * ) the space of polyhomogeneous pseudodifferential operators of order m on , with a compactly supported kernel in × . For A ∈ m phg ( ), we denote by σ p (A) ∈ S m phg (T * ) the principal symbol of A. The subprincipal symbol is characterized by the action of pseudodifferential operators on oscillating functions: if A ∈ m phg ( ) and f (q) = b(q)e ik S(q) with b, S smooth and real-valued, then

A quantization is a continuous linear mapping
Op : S m phg (T * ) → m phg ( ) satisfying σ p (Op(a)) = a. An example of quantization is obtained by using partitions of unity and, locally, the Weyl quantization, which is given in local coordinates by We have the following properties: (1) If A ∈ l phg ( ) and B ∈ m phg ( ), then (2) If X is a vector field on and X * is its formal adjoint in L 2 ( , µ), then X * X ∈ 2 phg ( ), σ p (X * X ) = h 2 X and σ sub (X * X ) = 0.
But y(0) cannot be proportional toẋ(0); otherwise, using (56), we would get that y(s 0 ) is proportional toẋ(s 0 ). Hence, the right-hand side in (58) is > 0, which implies that Im(M(s 0 )) is positive definite on the orthogonal complement toẋ(s 0 ). Therefore, we found a choice for the second-order derivatives of ψ along which meets all our conditions. For x = (t, x ′ ) ∈ ‫ޒ‬ × ‫ޒ‬ n and s such that t = t (s), we set and f vanishes at order 2 along for this choice of ψ.
To sum up, as in the Riemannian (or "strictly hyperbolic") case handled in [Ralston 1982], the key observation is that the invariance of σ and σ ‫ރ‬ prevents the solutions of (54) with positive imaginary part on the orthogonal complement ofẋ(0) from blowing up.
Analysis of A 2 (x). We note that A 2 vanishes along if and only if La 0 (x(s)) = 0. According to (51), this turns out to be a linear transport equation on a 0 (x(s)). Moreover, the coefficient of the first-order term, namely ∇ ξ p 2 (x(s), ξ(s)), is different from 0. Therefore, given a 0 ̸ = 0 at (t =0, x = x(0)), this transport equation has a solution a 0 (x(s)) with initial datum a 0 , and, by Cauchy uniqueness, a 0 (x(s)) ̸ = 0 for any s. Note that we have prescribed a 0 only along , and we may choose a 0 in a smooth (and arbitrary) way outside . We choose it to vanish outside a small neighborhood of .
Proof of (15). We observe that since Im(M(s)) is positive definite (uniformly in s) on the orthogonal complement ofẋ(s), there exist C, α ′ > 0 such that, for any t ∈ [0, T ], for any x ′ ∈ M, |∂ t v k (t (s), x ′ )| and |X j v k (t (s), x ′ )| are both bounded above by Ck n/4 e −α ′ kd(x ′ ,x ′ (s)) 2 . Therefore (1), (61) where, in the last line, we used the fact that |dµ/dℓ n | ⩽ C in a fixed compact subset of M (since µ is a smooth volume), and the o(1) comes from the eventual blowup of µ at the boundary of M. Now, M ⊂ ‫ޒ‬ n , and there exists r > 0 such that B d (x(s), r ) ⊂ V t (s) for any s such that t (s) ∈ (0, T ), where d( · , · ) still denotes the Euclidean distance in ‫ޒ‬ n . Therefore, we bound above the integral in (61) by Ck n/2 ‫ޒ‬ n \B d (x(s),r ) e −2α ′ kd(x ′ ,x ′ (s)) 2 dℓ n (x ′ ).
Extension of the result to any manifold M. In the case of a general manifold M, not necessarily included in ‫ޒ‬ n , we use charts together with the above construction. We cover M by a set of charts (U α , ϕ α ), where (U α ) is a family of open sets of M covering M and ϕ α : U α → ‫ޒ‬ n is an homeomorphism U α onto an open subset of ‫ޒ‬ n . Take a solution (x(t), ξ(t)) t∈[0,T ] of (8). It visits a finite number of charts in the order U α 1 , U α 2 , . . . , and we choose the charts and a 0 so that v k (t, · ) is supported in a unique chart at each time t. The above construction shows how to construct a 0 and ψ as long as x(t) remains in the same chart. For any l ⩾ 1, we choose t l so that x(t l ) ∈ U α l ∩ U α l+1 and a 0 (t l , · ) is supported in U α l ∩ U α l+1 . Since there is a (local) solution v k for any choice of initial a 0 (t l , x(t l )) and Im(∂ 2 ψ/(∂ x i ∂ x j ))(t l , x(t l )) in Proposition 12, we see that v k may be continued from the chart U α l to the chart U α l+1 . This continuation is smooth since the two solutions coincide as long as a 0 (t, · ) is supported in U α l ∩ U α l+1 . Patching all solutions on the time intervals [t l , t l+1 ] together, it yields a global-in-time solution v k , as desired.