Î T Beno

Kloeckner

SYMMETRIC SPACES OF HIGHER RANK DO NOT ADMIT DIFFERENTIABLE COMPACTIFICATIONS

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank 1 spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answer negatively.

Introduction

Let M be a symmetric space of nonpositive curvature, G its group of isometries and G 0 the identity component in G.

As a Riemannian manifold, M is a Hadamard space and is diffeomorphic to an open ball. Its Hadamard compactification (or geodesic compactification) is a topological gluing of M and its Hadamard boundary M(∞) such that M = M ∪ M(∞) is a closed ball. The group G acts continuously on M .

When M is of rank one, that is to say when it is negatively curved, this topological compactification admits "nice" models, carrying an invariant differentiable structure: in these models, the action of G is differentiable on M . Moreover, the restriction of this action to the boundary is a flat model for some geometry. The boundary spheres of the real, complex and quaternionic hyperbolic spaces wield the standard conformal, CR and quaternionic CR structures respectively. Concerning the octonionic hyperbolic plane, the corresponding geometry has not been studied yet, as far as we know.

It is natural to ask whether such a differentiable compactification exists when M is of higher rank. One could expect such a model to give birth to a new, luckily interesting, geometry.

In this paper, we give a negative answer to this question, and show that the obstruction comes from the spherical building at infinity. This combinatorial structure is trivial only in the Euclidean and rank one spaces. Thus there is an alternative: a symmetric space of nonpositive curvature admits either an interesting building at infinity or a differentiable compactification, not both.

1.1. Differentiable compactifications. Our goal is to extend the differentiable structure of M to the manifold with boundary M, so that we do not lose symmetry in the process. This leads to the following definition.

Definition 1. A differentiable Hadamard compactification of M is a differentiable (C 1 ) structure D on M compatible with the differentiable structure of M and such that the action of G is C 1 .

We can define a C r Hadamard compactification in the same way, where r can be finite, ∞ or ω, C ω meaning real analytic. When no precision is given, differentiable means C 1 .

By a C r action, we mean that the map G ×M → M is C r . It implies that G acts by C r diffeomorphisms and that the map G → Diff(M) is continuous in the C r topology.

This condition can be greatly relaxed thanks to the Bochner and Montgomery theorem: if G acts continuously by C r diffeomorphisms, then its action is in fact C r [START_REF] Bochner | Groups of differentiable and real or complex analytic transformations[END_REF].

For the sake of brevity we will often write "differentiable compactification" instead of "differentiable Hadamard compactification". However, a symmetric space admits other topological compactifications than the Hadamard one, e.g. Martin, Satake and Furstenberg compactifications. It would be interesting to extend our study to these, but the Hadamard compactification seems to be of utmost importance for our question. First, it is very natural, defined directly by the geometry of the space for a large class of Riemannian manifolds. Second, there is as far as we know little hope to get a manifold with boundary from the other compactifications. Either the infinity does not have the right dimension (e.g. the Poisson boundary) or the most natural differentiable structure is that of a manifold with boundary and corners (e.g. the maximal Satake compactification). A detailed account on all classical compactifications can be found in [START_REF] Borel | Compactifications of symmetric and locally symmetric spaces[END_REF] and [START_REF] Guivarc | Compactifications of symmetric spaces[END_REF].

1.2. Existence of differentiable compactifications. Let us now discuss the existence of differentiable compactifications for the three types of nonpositively curved symmetric spaces. Symmetric spaces of rank 1. It is well known that the real hyperbolic space H n admits a differentiable Hadamard compactification, given for example by the closure of Klein's ball: the central projection of the hyperboloid Q = -1 (where Q is the canonical Lorentzian metric on R n+1 ) gives an embedding of H n into RP n where the group SO 0 (1, n) of isometries of H n acts analytically. This construction can be generalized to all symmetric spaces of nonpositive curvature and rank 1.

It is worth noticing that H n admits other differentiable Hadamard compactifications. For example, the action of SO 0 (1, n) on Poincaré's ball extends analytically to the closed ball and the resulting action is not C 1 conjugate to the previous one (this can be seen by looking at asymptotic geodesics: they are tangent one to another in the closure of Poincaré's ball, not in Klein's ball.) Details are given in [START_REF] Kloeckner | On differentiable compactifications of the hyperbolic space[END_REF], where it is shown that H n admits an infinite number of nonconjugate analytic compactifications in the sense of definition 1. Euclidean spaces. If M is a Euclidean space, once again it admits a differentiable Hadamard compactification we briefly describe. Identify R n with the affine hyperplane {x 0 = 1} of R n+1 where n is the dimension of M. The projection of center 0 of M on the open upper unit half-sphere is a diffeomorphism. Pushing forward by this map we get an action of G (the affine group) on the open upper half-sphere whose continuous prolongation to the closed half-sphere is real analytic. This action is a real analytic Hadamard compactification of M = R n . Symmetric spaces of higher rank. The main result of this paper is the following.

Theorem 1. No noneuclidean symmetric space of rank k 2 admits a differentiable Hadamard compactification.

Structure of the paper. From now on, M is supposed to be a symmetric space of rank k 2.

We shall start with a simple remark about the natural projection of a fiber S x M of the unit tangent bundle of M on M(∞).

In the second section we prove that H 2 × R admits no differentiable Hadamard compactification.

Next we generalize this fact to every product

F × R k-1
where k 2 and F is a symmetric space of noncompact type of rank 1.

Finally, we prove Theorem 1.

Note that the different parts are more or less independent: the proof of Theorem 1 does not make use of preceding results. However some arguments of Section 4 will be useful, and Section 3 gives a good insight of the general phenomenon on the simplest case.

Apartments and the visual projection

2.1. Apartments. We give some basic vocabulary about the building structure of M(∞). More details can be found in [START_REF] Ballmann | Manifolds of nonpositive curvature[END_REF], Appendix 5. Our main reference for the building structure of a symmetric space is [START_REF] Patrick | Geometry of nonpositively curved manifolds[END_REF]. For details about general buildings, see [START_REF] Brown | Buildings[END_REF].

Let A be a maximal flat (i.e. a totally geodesic submanifold isometric to the Euclidean space of maximal dimension) of M, A its closure in

M and A(∞) = A ∩ M(∞) its boundary. A(∞) is called an apartment of M(∞). It is a topological submanifold.
Every point of M(∞) belongs to at least one apartment. A point is said to be regular if it belongs to exactly one apartment, otherwise it is said to be singular.

Let x be a point of M(∞). We denote by a(x) the set of all apartments containing x. If x is singular, it is said to have index 1 if a(x) is minimal with respect to inclusion among sets a(y) of singular y's.

The connected component of x in the set of points y such that a(x) = a(y) is a facet. Facets are topological submanifolds. If x is regular, we call its facet a Weyl chamber or simply a chamber; if x is singular of index 1, we call its facet a panel.

The dimension of every apartment is k -1 (where k is the rank of M). Chambers have dimension k -1, panels have dimension k -2.

Two facets are adjacent if their closures intersect. If they are adjacent and of different dimensions, one is contained in the closure of the other.

The facets form a simplicial complex on M(∞) if M is of noncompact type. If M has a Euclidean factor, some of the cells are spheres rather than simplicies.

This complex has the incidence structure of a spherical thick building, which means:

(1) each apartment is a spherical Coxeter complex (see [START_REF] Brown | Buildings[END_REF] for details), (2) for any two facets, there is an apartment containing both of them, (3) there exists at least three chambers adjacent to any given panel, (4) if there are two apartments A, A ′ containing two facets F and F ′ , then there is an isomorphism A → A ′ fixing F and F ′ pointwise. The group G acts by isomorphisms on this building: it preserves the adjacency relation and sends facets onto facets of the same dimension.

2.2. Non smoothness of the visual projection. Let x be a point of M. A unit vector v tangent to M at x defines a geodesic ray γ v , hence a point γ v (∞) of the Hadamard boundary M(∞). The map

π x : S x M → M(∞) v → γ v (∞)
is called the visual projection from the point x. For all x, the visual projection from x is a homeomorphism. It seems reasonable to expect the visual projections to be diffeomorphisms for a "good" differentiable Hadamard compactification. However, it cannot be.

Proposition 1. If M is a nonpositively curved symmetric space of higher rank, there is no differentiable structure on M(∞) such that all apartments are submanifolds.

Proof. Let γ be some geodesic ray in M that is singular of index 1. As the spherical building at infinity of M is thick, there are at least three (in fact, an infinite number of) chambers C 1 , C 2 , C 3 adjacent to P .

For each pair C i , C j (i = j) there is a flat

A ij such that C i ⊆ A ij (∞) and C j ⊆ A ij (∞). But A ij (∞) is an embedded submanifold of M(∞),
thus C i and C j have opposite tangent half spaces E i , E j at γ(∞). See figure 1. Thus we get three half subspaces

E 1 , E 2 , E 3 of T γ(∞) M(∞) such that E 1 = -E 2 , E 1 = -E 3 , E 2 = -E 3 , a contradiction.
Corollary 1. There is no differentiable structure on M(∞) such that π x is a diffeomorphism for all x ∈ M.

Proof. Suppose there is such a differentiable structure.

Let A be a maximal flat of M, x be a point of A.

Then

S x A is an embedded submanifold of S x M and π x is a diffeo- morphism. Thus A(∞) = π x (S x A) is a submanifold of M(∞). A contradiction with Proposition 1. 3. Study of H 2 × R
We summarize briefly the building structure of H 2 × R. The singular geodesics are those of the form {x} × R where x is a point of H 2 ; they are parallel (asymptotic at both ends) to one another. The maximal flats are the products γ × R where γ is a geodesic of H 2 (see figure 2).

The boundary of H 2 × R is a 2-sphere partitionned into two points and a family of nonintersecting curves joining them. The points are the end points of every singular geodesic, therefore panels of the building. The curves are the Weyl chambers, there is one of them for each point in the boundary of H 2 (see figure 3). The union of any two of them and of the two panels is an apartment. Proof. Suppose D is a differentiable Hadamard compactification of M = H 2 × R and M is endowed with D.

Let γ = {x} × R + be any singular geodesic ray of unit speed. We denote by γ(∞) the point of M(∞) defined by γ.

Since γ(∞) is fixed by all orientation-preserving isometries of H 2 , the derivatives of these isometries give a linear representation of PSL 2 (R) on T γ(∞) M . This representation is reducible as T γ(∞) M(∞) is an invariant subspace. Let ρ be the induced representation. As a representation of a simple Lie group ρ is trivial or faithful.

Let s x be the geodesic symmetry of H 2 around x. We identify s x with the isometry s x × Id of H 2 × R. For every time t ∈ R, ds x (γ(t)) has eigenvalues 1, -1, -1. By continuity, ds x (γ(∞)) must have the same eigenvalues, thus the restriction to

T γ(∞) M(∞) of ds x (γ(∞)) is -Id, that is to say ρ(s x ) = -Id.
Let y be any point of H 2 different from x. Then ρ(s y ) = -Id too. Thus ρ(s x s y ) = Id, but s x s y is a non-trivial hyperbolic transformation (it is a translation along the geodesic containing x and y). Thus ρ is neither faithful nor trivial, a contradiction.

Product of a Euclidean space by a rank 1 space

We now generalize Proposition 2 to the case when M = F × R k-1 is the product of an Euclidean space by a symmetric space F of rank 1.

However, as we will need it later, we prove something stronger.

Definition 2. A weak differentiable Hadamard compactification of M is defined as a differentiable Hadamard compactification where we replace G by its identity component G 0 .

We prove that M admits no weak differentiable Hadamard compactification. Thus, in order to generalize the argument used in the proof of Proposition 2 we need the geodesic symmetries to belong to G 0 . Of course this is false if F is of odd dimension, and we shall use another argument in this case. Proposition 3. Let F be a rank 1 symmetric space of noncompact type. If dim F is even, then the geodesic symmetries belong to G 0 . If dim F is odd, then F is a real hyperbolic space H 2m+1 . Proof. From the classification of symmetric spaces (see for example [START_REF] Helgason | Differential geometry and symmetric spaces[END_REF], Chapter IX) we know that the rank 1 symmetric spaces of noncompact type are: the real hyperbolic spaces, the complex hyperbolic spaces, the quaternionic hyperbolic spaces and an exceptionnal space, the octonionic hyperbolic plane (therefore the last assertion is clear). The identity components of their isometry groups are respectively SO 0 (1, n), SU(1, n), Sp(1, n) and F 4(-20) , which are simple Lie groups.

We shall use the following criterion. Let g = k + p be a Cartan decomposition of the Lie algebra of G 0 . Then the geodesic symmetries are in G 0 if and only if k contains a maximal abelian algebra of g (see [START_REF] Helgason | Differential geometry and symmetric spaces[END_REF], Chapter IX 3).

It is know sufficient to compare the ranks of g and k (most of them can be found in [START_REF] Knapp | Representation theory of semisimple groups[END_REF], Appendix C):

• in the real case, g = so(n, 1) is of rank ⌊ 1 2 (n+ 1)⌋ and k = so(p), of rank ⌊n/2⌋. These two ranks coincide exactly when n is even, • in the complex case, g = su(n, 1) and k = s(u(n) × u(1)) are both of rank n, • in the quaternionic case, g = sp(n, 1) and k = sp(n) × sp(1) are both of rank n + 1, • in the octonionic case, g = f 4(-20) and k = so(9) are both of rank 4. I wish to thank Fokko du Cloux, Jérôme Germoni and Bruno Sévennec for explaning this case to me.

We can now prove the following.

Proposition 4. If M = F ×R k-1
where F is a rank 1 symmetric space of noncompact type, then M admits no weak differentiable Hadamard compactification. In particular, M admits no differentiable Hadamard compactification.

Proof. Suppose there is such a differentiable structure on M . We identify an isometry g of F with the isometry g × Id of M. We denote the component of identity of the group of isometries of F by G F 0 and consider it a subgroup of G 0 . We first suppose that F is of even dimension. Let s x and s y be the geodesic symmetries around two different points x and y of F . Let d be a geodesic of R k-1 . Then γ 1 = {x} × d and γ 2 = {y} × d are asymptotic and z = γ 1 (+∞) = γ 2 (+∞) is a singular point of index 1 of M(∞).

Differentiation g → dg(z) gives us a linear representation of G F 0 in T z M . This representation is reducible since T z M(∞) is an invariant subspace. Let ρ be the representation induced on T z M(∞).

We now decompose the representation ρ. For all t ∈ R, the eigenvalues of ds x (γ 1 (t)) are 1 with multiplicity k -1 and -1 with multiplicity dim F . Thus, ρ(s x ) must have eigenvalues 1 with multiplicity k -2 and -1 with multiplicity dim F (an eigendirection transverse to the boundary must have a nonnegative eigenvalue).

We shall decompose ρ using the following lemma.

Lemma 1. The panel P of γ 1 (∞) is a differentiable submanifold of M(∞).
Proof. The panel P is pointwise fixed by all isometries of F . Thus it is pointwise fixed by s x . In a local chart, it is defined by p ∈ P ⇒ s x (p)p = 0. Since s x -Id has rank dim F and dim M(∞) = dim F +k-2, the inverse function theorem implies that P is contained in a differentiable submanifold of M(∞) of dimension k-2 (namely the set of fixed points of s x ). But, as a panel in the boundary of a rank k symmetric space, it is an open topological manifold of dimension k -2. Thus P is a differentiable submanifold of dimension k -2 of M(∞).

The tangent space T z P is an invariant subspace of ρ. Thus, this representation splits in two parts : ρ = ρ 0 ⊕ ρ 1 where ρ 0 is the trivial representation of dimension k-2 and ρ 1 is a representation of dimension dim F . Now we have a representation ρ 1 (which, as G F 0 is simple, must be faithful or trivial) with ρ 1 (s x ) = -Id. So ρ 1 cannot be trivial. But ds y (γ 2 (t)) has the same eigenvalues as ds x (γ 1 (t)), and thus ρ 1 (s y ) = -Id too. Now we have ρ 1 (s x s y ) = Id with s x s y a hyperbolic transformation, so ρ 1 cannot be faithful, a contradiction.

Suppose now that F is of odd dimension.

We have F = H 2m+1 , the real hyperbolic space. The geodesic symmetries are not in G 0 (their determinant is -1) and we shall use Proposition 1.

Let A be a maximal flat. Then A is the product of a geodesic γ of F by R k-1 . Let r ∈ G F 0 be the rotation of angle π around γ in F ; A is the set of fixed points of r, and A(∞) is the set of fixed points of r in M(∞). Since r is an involution, A(∞) is a submanifold of M(∞), a contradiction to Proposition 1.

We give an alternative proof for the odd dimension case, less elegant but more useful for the proof of Theorem 1. We can define the representation ρ 1 like in the even dimension case. Then ρ 1 is a representation of dimension dim F of G F 0 . Since, for all x ∈ M fixed by r, dr(x) has eigenvalues -1 with multiplicity 2m and 1 with multiplicity k, ρ 1 (r) has eigenvalues -1 with multiplicity 2m and 1 with multiplicity 1. Thus ρ 1 (r) = Id, hence ρ 1 is not trivial.

But G F 0 = SO 0 (2m + 1, 1) admits no non trivial representation of dimension less than 2m + 2, a contradiction.

Proof of Theorem 1

We shall now prove Theorem 1 with the same ideas that we used for the previous propositions.

Let M be a noneuclidean symmetric space of nonpositive curvature of dimension n and rank k > 1. As before, G is the group of all isometries of M, G 0 is the identity component of G and g is the Lie algebra of G.

Suppose that there exists a differentiable Hadamard compactification D of M.

We denote by α the action of G on M . We also denote by α the corresponding action of g. The first step is to find in M an embedded product F × R k-1 . Let γ be a singular geodesic of index 1. Let F γ be the union of geodesics parallel to γ (recall that parallel means that they are both positively and negatively asymptotic). Then, F γ is a totally geodesic submanifold of M isometric to a product F × R k-1 where F is a symmetric space of rank 1 (see [START_REF] Patrick | Geometry of nonpositively curved manifolds[END_REF] Section 2.11).

Let F γ be the closure of

F γ in M and F γ (∞) = F γ ∩ M(∞).
It would be interesting to prove that F γ is a submanifold of M , since we could directly use Proposition 4 to get a contradiction, but a weaker statement (namely Lemma 7) will be sufficient.

Up to a change of parametrization we can write γ(t) = (p, (t, 0, . . . , 0))

where p ∈ F , and F t = F × {(t, 0 . . . , 0)} is identified with its embedding into M. Since F , identified with F 0 , is a totally geodesic submanifold of M, the Lie algebra of the group G F of isometries of F is a subalgebra of g and the identity component G F 0 of G F is a subgroup of G 0 (that's why we needed the stronger statement in Proposition 4). Thus taking derivatives gives us a representation ρ of G F 0 on T γ(∞) M . Let k t ⊕ p t be the Cartan decomposition of g at γ(t). Since F γ is a totally geodesic submanifold we have a further decomposition p t = p t F ⊕ p t eucl ⊕ p t 0 where the terms are pairwise orthogonal (with respect to the Killing form), p t F is mapped by α onto T p F t , p t eucl is mapped onto T p R k-1 and p t 0 is mapped onto (T p F γ ) ⊥ . We define p t γ = p t F ⊕ p t eucl ; α maps p t γ onto T γ(t) F γ . Moreover, p t γ is the set of all Killing fields in p commuting with that Killing field X ∈ p t such that α(X) is the unit tangent vector of γ.

We shall split ρ in three parts in correspondence with the splitting p t = p t F ⊕ p t eucl ⊕ p t 0 . To achieve this, we use the following stability result.

Lemma 2. Let K be any compact group and (µ t ) t∈R be a continuous family of linear representations of K on some finite-dimensional real vector space V . Then for all pairs (t 1 , t 2 ) of real numbers, the representations µ t 1 and µ t 2 are conjugate.

Proof. As K is compact and V is finite-dimensional, the conjugacy class of a representation µ t is determined by its character. More precisely, the multiplicity in µ t of some irreducible representation ν is given by the scalar product of the characters of µ t and ν, hence is a continuous map. This multiplicity is an integer and is thus constant. Lemma 3. The tangent subspaces T γ(t) F t , T γ(t) R k-1 and (T γ(t) F γ ) ⊥ admit limits when t → ∞, denoted respectively by V F , V eucl and V 0 . Moreover one has T γ(∞) M = V F ⊕ V eucl ⊕ V 0 . that this system is of rank at least d = dim V 0 + dim V F at γ(∞). We can extract a subsystem of d equations that is of maximal rank at γ(∞). The inverse function theorem implies that this subsystem defines a submanifold of M(∞) of dimension k -2 = dim V eucl ∩ T γ(∞) M(∞) and containing P . But P is topologically a manifold of dimension k -2 and thus must be a differentiable submanifold of M(∞).

Since K F 0 acts trivially on P , its tangent space must be V eucl ∩ T γ(∞) M(∞).

Since G acts trivially on P , it must preserve its tangent space. We are now ready to prove the following.

Lemma 7. The subspace V F is invariant by ρ.

Proof. From previous lemmas we know that V 0 and V eucl ∩ T γ(∞) M(∞) are invariant subspaces for ρ. Since ρ is totally reducible, there exists some subspace V ′ invariant by ρ such that one has the following decomposition T γ(∞) M(∞) = V ′ ⊕ V eucl ∩ T γ(∞) M(∞) ⊕ V 0 . But V ′ must be invariant by ρ ∞ , and thus V ′ = V F and V F is invariant by ρ.

Denote by ρ 1 the representation of G F 0 induced by ρ on V F . Since the restriction of ρ 1 to K F 0 is the limit of the restriction of ρ t to T γ(t) F t , we can now use the same arguments as in the proof of Proposition 4.

If F is of even dimension, ρ 1 is neither trivial nor faithful, a contradiction.

If F is of odd dimension, ρ 1 is a nontrivial representation of dimension dim F of G F 0 , a contradiction. Theorem 1 is proved. Note that we actually get something stronger: there exists no weak differentiable Hadamard compactification of M; the obstructions to differentiability appear in the identity component of G.

Figure 1 .

 1 Figure 1. Three chambers meeting at a panel.

Figure 2 .

 2 Figure 2. A family of parallel singular geodesics lying on the same flat.

Figure 3 .

 3 Figure 3. Here we show the Weyl chambers in the boundary of H 2 × R. Any choice of two of them corresponds to a geodesic of H 2 , a flat of H 2 × R and an apartment.

Proof. Let K F 0 be the isotropy group of p in F . Taking derivatives gives linear representations ρ t of K F 0 in T γ(t) M for all t ∈ R. Then T γ(t) F t , T γ(t) R k-1 and (T γ(t) F γ ) ⊥ are invariant spaces of ρ t .

By continuity, the restriction ρ ∞ of ρ to K F 0 splits into three parts and the conclusion holds.

From Lemma 2 we deduce that the action of ρ ∞ is conjugate with that of ρ 0 . Since ρ t acts trivially on T γ(t) R k-1 , ρ ∞ acts trivially on V eucl .

We shall now prove that V F is an invariant subspace for ρ.

Lemma 4. Let O be the orbit of γ(∞) under the action of G 0 . Then

O and, for all t, the restriction of α γ(∞) to p t 0 is one-to-one and onto V 0 .

The definition of O shows that α sends p t 0 on T γ(∞) O. We want to prove that this map is one-to-one and onto.

Let H be an element of p t 0 . By definition, α(exp(H))(γ) is not parallel to γ. If α(H) γ(∞) = 0, then α(H) γ(-∞) = 0. But after conjugacy by the geodesic symmetry at γ(t) we find α(-H) γ(-∞) = 0, a contradiction. Thus the restriction of α γ(∞) to p t 0 is one-to-one. Since dim

Lemma 5. The subspace V F contains no subspace where ρ is trivial.

Let X be that vector of p 0 eucl such that γ ′ (t) = α(X) γ(t) . The linear operator d γ(∞) α(exp X) -Id acting on V 0 is of maximal rank.

Proof. The linear action ρ ∞ of K F 0 on T γ(∞) M(∞) is transitive on V F and ρ ∞ is the restriction of ρ to K F 0 thus V F contains no trivial part. To prove the second part of the lemma, we use the root space decomposition g = g 0 + g λ given by some maximal flat containing γ. We have

g λ and Ad(exp X) = e λ(X) I on g λ . Since α is onto from p 0 0 to V 0 , it is onto from λ(X) =0 g λ to V 0 . For all H ∈ g λ , we have

and thus d γ(∞) α(exp X) -Id is nondegenerate on V 0 . Lemma 6. The panel P of γ(∞) is a submanifold of M(∞) and its tangent space at γ(∞) equals V eucl ∩ T γ(∞) M(∞).

Proof. The panel P is contained in the set of the points of M(∞) left fixed by the actions of G and of exp(X). Written in local coordinates, this gives us an infinite system of equations. By Lemma 5 we know


J R Gomez-Solano

L Bellon

A Petrosyan

S Ciliberto

Steady state fluctuation relations for systems driven by an external random force

Keywords: 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion

We experimentally study the fluctuations of the work done by an external Gaussian random force on two different stochastic systems coupled to a thermal bath: a colloidal particle in an optical trap and an atomic force microscopy cantilever. We determine the corresponding probability density functions for different random forcing amplitudes ranging from a small fraction to several times the amplitude of the thermal noise. In both systems for sufficiently weak forcing amplitudes the work fluctuations satisfy the usual steady state fluctuation theorem. As the forcing amplitude drives the system far from equilibrium, deviations of the fluctuation theorem increase monotonically. The deviations can be recasted to a single master curve which only depends on the kind of stochastic external force.

Introduction. -Fluctuation relations are a very important theoretical result for the description of nonequilibrium microscopic systems since they quantify the statistical properties of fluctuating energy exchanges under rather general conditions [1]. In particular, the socalled fluctuation theorem (FT) [2,3] quantifies the asymmetry of the distribution of positive and negative fluctuations of a given time-integrated quantity (injected work, entropy production, etc.). For a system in contact with a thermostat at temperature T and driven by an external force in a non-equilibrium steady state, the FT states that the ratio of the probability of finding a positive fluctuation with respect to that of the corresponding negative value for the work W τ done by the force during a time interval τ satisfies ln

P (W τ = W ) P (W τ = -W ) → W k B T , τ c ≪ τ, (1)
where τ c is the longest characteristic relaxation time of the system. Equation (1) has been tested in several experiments such as fluidized granular media [4], a colloidal particle dragged by an optical trap [5], electrical circuits [6], mechanical harmonic oscillators [7] and a colloidal particle near the stochastic resonance [8]. New fluctuation relations have been proposed as well for the entropy production [9] or by considering modifications of the statistical properties of the thermal bath [10][11][12]. In all of these examples the force which drives the system out of equilibrium is inherently deterministic. However, it has been recently argued that the nature (deterministic or stochastic) of the forcing can play an important role in the distribution of the injected work leading to possible deviations from the relation (1) for large fluctuations (W τ / W τ > 1). Indeed, it has been found in experiments and simulations such as a Brownian particle in a Gaussian white [13] and colored [14] noise bath, turbulent thermal convection [15], wave turbulence [16], a vibrating metalic plate [17], an RC electronic circuit [18] and a gravitational wave detector [19] that the probability density functions of the work done by a stochastic force are not Gaussian but asymmetric with two exponential tails leading to violations of the FT in the form of effective temperatures or nonlinear relations between the left and the rigth hand side of eq. ( 1). It is important to remark that in the systems previously cited the steady state FT is violated because in such a case the external random force acts itself as a kind of thermal bath. One question which naturally arises is what the work fluctuation relations will become when in addition to the external random forcing a true thermalization process is allowed. In this situation there are two sources of work fluctuations: the external force and the thermal bath. As pointed out in [12,17], one is interested in the distribution of the work fluctuations done by the external random force in presence of a thermostat and the conditions under which the FT could be valid.

In the present work we address these questions in two experimental systems: a Brownian particle in an optical trap and a micro-cantilever used for atomic force microscopy (AFM). Both are in contact with a thermal bath and driven out of equilibrium by an external random force whose amplitude is tuned from a small fraction to several times the amplitude of the intrinsic thermal fluctuations exerted by the thermostat. Colloidal particle in an optical trap. -The first system we study consists on a spherical silica bead of radius r = 1 µm immersed in ultrapure water which acts as a thermal bath. The experiment is performed at a room temperature of 27 ± 0.5 • C at which the dynamic viscosity of water is η = (8.52 ∓ 0.10) × 10 -4 Pa s. The motion of the particle is confined by an optical trap which is created by tightly focusing a Nd:YAG laser beam (λ = 1064 nm) by means of a high numerical aperture objective (63×, NA = 1.4). The trap stiffness is fixed at a constant value of k = 5.4 pN/µm. The particle is kept at h ≈ 10 µm above the lower cell surface to avoid hydrodynamic interactions with the walls. Figure 1(a) sketches the configuration of the bead in the optical trap. An external random force is applied to the particle by modulating the position of the trap x 0 (t) using an acousto-optic deflector, along a fixed direction x on the plane perpendicular to the beam propagation (+z). The modulation corresponds to a Gaussian Ornstein-Uhlenbeck noise of mean x 0 (t) = 0 and covariance x 0 (s)x 0 (t) = A exp(-|t -s|/τ 0 ). The correlation time of the modulation is set to τ 0 = 25 ms whereas the value of its amplitude A is tuned to control the driving intensity. We determine the particle barycenter (x, y) by image analysis using a high speed camera at a sampling rate of 1 kHz with an accuracy better than 10 nm. See ref. [20] for more details about the experimental apparatus. The attractive force exerted by the optical trap on the bead at time t along x is given by -k(x(t)x 0 (t)). Hence, for the experimentally accessible timescales the dy- namics of the coordinate x is described by the overdamped Langevin equation

γ ẋ = -kx + ζ T + f 0 . (2)
In eq. ( 2) γ = 6πrη is the viscous drag coefficient, ζ T is a Gaussian white noise (

ζ T = 0, ζ T (s)ζ T (t) = 2k B T γδ(t -s))
which mimics the collisions of the thermal bath particles with the colloidal bead and f 0 (t) = kx 0 (t) plays the role of the external stochastic force. The standard deviation δf 0 of f 0 is chosen as the main control parameter of the system. Besides the correlation time τ 0 of f 0 there is a second characteristic timescale in the dynamics of eq. ( 2): the viscous relaxation time in the optical trap τ γ = γ/k = 3 ms < τ 0 . In order to quantify the relative strength of the external force with respect to the thermal fluctuations, we introduce a dimensionless parameter which measures the distance from equilibrium

α = x 2 x 2 eq -1, (3)
where x 2 is the variance of x in the presence of f 0 > 0 whereas x 2 eq is the corresponding variance at equilibrium (f 0 = 0). The dependence of α on δf 0 is quadratic, as shown in fig. 2(a). This quadratic dependence is a consequence of the linear response of the system to the external forcing described by the linear Langevin eq. ( 2).

The work done by the external random force on the colloidal particle (in k B T units) is Thus, by measuring simultaneously the time evolution of the barycenter position of the particle and the driving force we are able to compute directly the work injected into the system by the driving. In figs. 2(b)-(d) we show the probability density functions (PDF) of w τ for different values of τ and α. We observe that for a fixed value of α, the PDFs have asymmetric exponential tails at short integration times and they become smoother as the value of τ increases. For α = 0.20 they approach a Gaussian profile (fig. 2(b)) whereas asymmetric non-Gaussian tails remain for increasing values of α. As shown in figs. 2(c)-(d), the asymmetry of these tails becomes very pronounced for large α > 1 even for integration times as long as τ = 250 ms = 10τ 0 , where we have taken τ 0 because it is the largest correlation time of the dynamics. As pointed out in [18], the deviations of the linear relation of eq. ( 1) (with respect to w τ ) can occur for extreme values of the work fluctuations located on these tails.

w τ = 1 k B T t+τ t ẋ(t ′ )f 0 (t ′ )dt ′ . (4)
We define the asymmetry function of the PDF P as

ρ(w) = lim τ τc →∞ ln P (w τ = w) P (w τ = -w) , (5)
so that eq. ( 1) reads

ρ(w) = w. (6)
From the experimental PDFs of w τ we compute ρ(w) as the logarithm in eq. ( 5) for integration times τ = 10τ 0 . We checked that for this value the limit of eq. ( 5) has been attained. Figure 3 shows the profile of the asymmetry functions for different values of α. We notice that for sufficiently small values (α = 0.20, 0.51 < 1), the FT given by eq. ( 6) is verified by the experimental data. To our knowledge, this is the first time that the FT holds for a random force without introducing any prefactor in the linear relation of eq. ( 6). It is important to point out that any deviation from the linear relation of eq. ( 6) for extreme fluctuations is unlikely since we probed values as large as w τ / w τ ∼ 5. Indeed it is argued [12][13][14]18], that, for strongly dissipative systems driven by a random force, the deviations from FT may occur around w τ / w τ ∼ 1. Furthermore in the present case the validity of the FT for weak driving amplitudes α < 1 is consistent with the fact that for integration times τ > 25 ms, the ratio ρ(w)/w has converged to its asymptotic value 1 for all measurable w. Note that this convergence to the FT prediction is quite similar to that measured in system driven out of equilibrium by deterministic forces [6][7][8]. For instance in the case of a harmonic oscillator driven by a sinusoidal external force the asymptotic value of ρ(w)/w is reached for integration times larger than the forcing period [7].

In contrast, deviations from eq. ( 6) are expected to occur for 1 < α because the fluctuations of injected energy produced by the external random force become larger than those injected by the thermal bath. Indeed fig. 3 shows that for values above α = 1.84, eq. ( 6) is not verified any more but ρ becomes a nonlinear function of w τ . For small values of w τ it is linear with a slope which decreases as the driving amplitude increases whereas there is a crossover to a slower dependence around w τ / w τ ∼ 1, a qualitatively similar behavior to those reported in [13,[16][17][18][19]. We finish this section by emphasizing that we have clearly found that for an experimental system whose dynamics correspond to a first order Langevin equation subjected to both thermal and external noises, the FT can be satisfied or not depending on the relative strength of the external driving. The details about how this deviations arise and the convergence to generic work fluctuation relations will be given further. We first analyze the experiment on the AFM. AFM cantilever. -A second example of a system for which thermal fluctuations are non-negligible in the energy injection process at equilibrium is the dynamics of the free end of a rectangular micro-cantilever used in AFM measurements. The cantilever is a mechanical clampedfree beam, which can be bended by an external force F and is thermalized with the surrounding air. The experiment is sketched in fig. 1(b).

We use conductive cantilevers from Nanoworld (PPP-CONTPt). They exhibit a nominal rectangular geometry: 450 µm long, 50 µm wide and 2 µm thick, with a 25 nm PtIr 5 conductive layer on both sides. The deflection is measured with a home made interferometric deflection sensor [21], inspired by the original design of Schonenberger [22] with a quadrature phase detection technique [23]: the interference between the reference laser beam reflecting on the chip of the cantilever and the sensing beam on the free end of the cantilever gives a direct measurement of the deflection X. Our detection system has a very low intrinsic noise, as low as 4 pm rms in the 100 kHz bandwidth we are probing [21,24].

From the power spectrum of the deflection fluctuations of the free end at equilibrium (F = 0) we verify that the cantilever dynamics can be reasonably modeled as a stochastic harmonic oscillator with viscous dissipation [24,25]. Hence, in the presence of the external force the dynamics of the vertical coordinate X of the free end is described by the second order Langevin equation

m Ẍ + γ Ẋ = -kX + ζ T + F, (7)
where m is the effective mass, γ the viscous drag coefficient, k the stiffness associated to the elastic force on the cantilever and ζ T models the thermal fluctuations. m, γ and k can be calibrated at zero forcing using fluctuation dissipation theorem, relating the observed power spectrum of X to the harmonic oscillator model: in our experiment we measure m = 2.75 × 10 -11 kg, γ = 4.35 × 10 -8 kg/s and k = 8.05 × 10 -2 N/m. The amplitude of the equilibrium thermal fluctuations of the tip position (i.e.

x 2 eq = k B T /k ≃ 2 10 -10 m) is two orders of magnitude larger then the detection noise (i.e. 4 pm rms). The signal to noise ratio is even better when the system is driven by an external force F . The characteristic timescales of the deflection dynamics are the resonance period of the harmonic oscillator τ k = 2π (m/k) = 116 µs and the viscous relaxation time τ γ = m/γ = 632 µs, which is the longest correlation time.

When a voltage V is applied between the conductive cantilever and a metallic surface brought close to the tip (h ∼ 10 µm apart), an electrostatic interaction is created. The system behaves as a capacitor with stored energy E c =1 2 C(X)V 2 , with C the capacitance of the cantilevertip/surface system. Hence, the interaction between the cantilever and the opposite charged surface gives rise to an attractive external force F = -∂ X E c = -aV 2 on the free end, with a = ∂ X C/2. If we apply a static voltage V , the force F can be deduced from the stationary solution of eq. ( 7): kX = -aV 2 , where X is the mean measured deflection. k being already calibrated, we validate this quadratic dependence 1 of forcing in V and measure a = 1.49 × 10 -11 N/V 2 . As the electrostatic force F is only attractive, its mean value cannot be chosen to be 0. We thus generated a driving voltage V designed to create a Gaussian white noise forcing f 0 around an offset F : F = F + f 0 . The variance δf 0 of f 0 is the main control parameter of the system. In the absence of fluctuations ζ T and f 0 , eq. ( 8) has the stationary solution X = F /k. This solution corresponds to the mean position attained by the free end in the presence of the zero mean fluctuating forces. Hence, we focus on the dynamics of the fluctuations x = X -X around X which are described by the equation

mẍ + γ ẋ = -kx + ζ T + f 0 . (8)
Figure 4(a) shows the dependence between the parameter α defined in eq. ( 3) for the stochastic variable x and the control parameter δf 0 . We find that this dependence is quadatric verifying the linearity of the stochastic dynamics of the free end of the cantilever. On the other hand, the work done by the external random force during an integration time τ is computed from eq. ( 4). The corresponding PDFs are shown in figs. 4(b)-(d). Unlike the colloidal particle, the PDFs do not converge to a Gaussian distribution but to a profile with asymmetric exponential tails even for the smallest driving amplitude (α = 0.19) and for integration times as long as τ = 8τ γ , as shown in figs. 4(b)-(d). Surprisingly, when computing the asymmetry function for α = 0.19 < 1 and τ = 4τ γ the steady state FT of eq. ( 6) is perfectly verified, as shown in fig. 5. Work fluctuations as large as 2.5 times their mean value located on the exponential tails are probed and hence deviations from FT are unlikely for the same reasons discussed for the case of the Brownian particle.

In fig. 5 wee see that for 1.21 ≤ α, the deviations from eq. ( 6) appear as a nonlinear relation with a linear part for small fluctuations whose slope decreases as α increases and a crossover for larger fluctuations, qualitatively similar to the behavior observed for the colloidal particle, as shown clearly in the inset of fig. 5. In the following we discuss the properties of these deviations as the energy injection process becomes dominated by the external force. Fluctuation relations far from equilibrium. -We address now the question of how the deviations from eq. ( 6) arise as the external stochastic force drives the system far from equilibrium. As shown previously, for 1 α, the forcing amplitude is strong enough to destroy the conditions for the validity of the FT for w τ . We note that there are two well defined limit regimes depending on the driving amplitude: one occuring at small values of α for which the steady state FT is valid, and the limit α ≫ 1 for which the the role of the thermal bath must be negligible in the energy injection process, which must be completely dominated by the external stochastic force. In order to investigate whether the transition between these two regimes is abrupt or not, we proceed by noting that for the latter the stochastic force term ζ T in eqs. ( 2) and ( 8) will be negligible compared to f 0 . This implies that the resulting statistical time-integrated properties of the corresponding non-equilibrium steady state will be invariant under a normalization of the timescales and the temperature of the system. In particular, the resulting fluctuation relations for w τ must lead to a master curve for the asymmetry function in the far from equilibrium limit α ≫ 1. The information about the transition of the fluctuation relations to this regime is given by the convergence to the master curve.

We introduce the normalized work w * τ as

w * τ = τ c τ w τ 1 + α . (9)
The physical idea behind this normalization is that for α ≫ 1, the thermal bath alone works as a heat reservoir for viscous dissipation whereas its coupling with the external forcing plays the role of a non-equilibrium thermal bath at an effective temperature k x 2 /k B = (1 + α)T ≈ αT . The prefactor τ c /τ is introduced in such a way that w * τ represents the average normalized work done during the largest correlation time of the system. Accordingly, the asymmetry function must be redefined as

ρ * (w * ) = lim τ /τc→∞ τ c τ ln P (w * τ = w * ) P (w * τ = -w * ) . (10)
Figure . 6(a) shows the asymmetry function ρ * for the normalized work w * τ on the colloidal particle at large values of α for which eq. ( 6) is violated. The timescale τ c in the computation of ( 9) and ( 10) is taken as the correlation time (τ 0 = 25 ms) of the Ornstein-Uhlenbeck forcing of eq. ( 2). For comparison we also show the corresponding curves at α = 0.20, 0.51 as blue circles and red squares respectively, for which eq. ( 6) holds. The convergence to a master curve is verified, which means that for a sufficiently strong forcing the thermal bath acts only as a passive reservoir for the energy dissipation without providing any important contribution to the energy injection into the system. Evidently, the normalized asymmetry function for the values α that verify the FT lie far from the master curve. We point out that the transition to the limit α ≫ 1 is rather continuous since intermediate regimes occur, as observed for α = 1.84. In this case neither the FT is satisfied as shown previously in fig. 3 nor the master curve is attained since the strength of thermal noise is still comparable to that of the external noise.

The results for the normalized asymmetry function of the work done on the cantilever by the external force are shown in fig. 6(b). The curve corresponding to the verification of the FT for α = 0.19 is also plotted for comparison. The convergence to a master curve is also checked as the value of α increases. Indeed, when comparing our normalized experimental curves with the analytic expression carried out by [13] for the asymmetry function of the work distribution on a Brownian particle driven entirely by a Gaussian white noise ρ * (w * ) = 4w * w * < 1/3

7 4 w * + 3 2 -1 4w * w * ≥ 1/3 , (11)
we check that the assumption of the convergence of the energy injection process into the cantilever to that of a Langevin dynamics for a harmonic oscillator entirely dominated by the external noise is valid. Finite α corrections can be detected for large values of w * τ indicating that the thermal bath still influences the energy injection into the cantilever. This corrections seem to vanish as the system is driven farther from equilibrium, as observed in fig. 6 for α = 22.10.

Finally, we point out that the profile of the master curve strongly depends on the kind of stochastic force: a Gaussian Ornstein-Uhlenbeck process in the first example and a Gaussian white noise in the second one. Non-Gaussian extensions of the external random force are expected to lead to striking modification of the fluctuation relations in the limit α ≫ 1, as recently investigated for an asymmetric Poissonian shot noise [12].

Conclusions. -We have studied the FT for the work fluctuations in two experimental systems in contact with a thermal bath and driven out of equilibrium by a stochastic force. The main result of our study is that the validity of FT is controlled by the parameter α. For small α 1 we have shown that the validity of the steady state FT is a very robust result regardless the details of the intrinsic dynamics of the system (first and second order Langevin dynamics) and the statistical properties of the forcing (white and colored Gaussian noise). Indeed these specific features vanish when the integration of w τ is performed for τ much larger than the largest correlation time of the system.

In contrast for large α 1, when the randomness of the system becomes dominated by the external stochastic forcing, we have shown that FT is violated. For α ≫ 1 the results at different driving amplitudes can be set on a master curve by defining a suitable effective temperature which is a function of α. We have shown that this master curve is system dependent. * * *

Fig. 1 :

 1 Fig. 1: a) Colloidal particle in the optical trap with modulated position. b) AFM cantilever close to a metallic surface. See text for explanation.

Fig. 2 :

 2 Fig. 2: (a) Dependence of the parameter α on the standard deviation of the Gaussian exponentially correlated external force f0 acting on the colloidal particle. (b) Probability density functions of the work wτ for α = 0.20; (c) α = 3.89; and (d) α = 10.77. The symbols correspond to integration times τ = 5 ms (•), 55 ms (2), 105 ms (3), 155 ms (⊳), 205 ms(⊲) and 255 ms ( * ). The solid black lines in (b) and (c) are Gaussian fits.

Fig. 3 :

 3 Fig. 3: Asymmetry function of the PDF of the work done by the external force on the colloidal bead computed at τ = 10τ0 for different values of the parameter α: 0.20(•), 0.51(2), 1.84(3), 3.89(⊲), 6.69(⊳), 10.77( * ). The dashed line represents the prediction of the fluctuation theorem ρ(w) = w. Inset: Expanded view for α ≥ 3.89.

Fig. 4 :

 4 Fig. 4: (a) Dependence of the parameter α on the standard deviation of the Gaussian white external force f0 acting on the cantilever. (b) Probability density functions of the work wτ for α = 0.19; (c) α = 3.03; and (d) α = 18.66. The symbols correspond to integration times τ = 97 µs (•), 1.074 µs (2), 2.051 ms (3), 3.027 ms (⊳), 4.004 ms(⊲) and 4.981 ms ( * ). The black dashed lines in (b)-(d) represent the exponential fits of the corresponding tails.

Fig. 5 :

 5 Fig. 5: Asymmetry function of the probability density function of the work done by the external force on the AFM cantilever computed at τ = 4τγ for different values of the parameter α: 0.19(•), 1.21(2), 3.03(3), 6.18(⊲), 9.22(⊳), 12.77( * ), 15.46(×), 18.66(▽), 22.10(△). The dashed line corresponds to the prediction of the fluctuation theorem ρ(w) = w. Inset: Expanded view for α ≥ 9.22.

Fig. 6 :

 6 Fig. 6: (a) Asymmetry function of the PDF of the normalized work done by the Gaussian Ornstein-Uhlenbeck force on the colloidal particle for different values of the parameter α. (b) Asymmetry function of the PDF of the normalized work done by the Gaussian white force on the cantilever for different values of the parameter α. The thick solid line represents the analytical expression given by eq. (11).

The quadratic dependance is valid only after taking care to compensate for the contact potential between the tip and the sample, which gives a small correction of the order of a few tens of mV.


Luc Hillairet

EIGENVALUE VARIATIONS AND SEMICLASSICAL CONCENTRATION

Introduction

In this paper, we aim at expliciting some relations between the behaviour of analytic eigenvalue branches of a spectral problem and concentration properties of its eigenfunctions. In order to illustrate this we consider a semiclassical Schrödinger operator -h 2 ∆ + V on R d (in which ∆ is the non-positive Euclidean Laplace operator). Viewing h as an analytic parameter, this problem enters the usual perturbation theory and the eigenvalues are organised into analytic eigenbranches E j (h). It is then natural to address the behaviour of these eigenbranches when h goes to 0. This point of view is fundamentally different from the usual semiclassical perspective. Indeed, any analytic eigenbranch is expected to cross a typical energy window [E 1 , E 2 ] so that any semiclassical expansion in this window corresponds actually to different eigenbranches. Our point is that knowing how the eigenfunctions concentrate in the semiclassical regime yields some result on the behaviour of eigenbranches. For instance, we have the following theorem (see section 3). Theorem 1.1. Let E j (h) be an analytic eigenbranch of the family of operators -h 2 ∆ + V on L 2 (R d ). The following then holds.

(1) The eigenbranch E j (h) converges to some limit E 0 when h tends to 0.

(2) The limit E 0 is necessarily a critical value of the potential V.

We will also use the same kind of ideas to prove an integrated remainder estimate for an analytic variation of metrics on a smooth compact manifold X. (see Thm. 4.3 in section 4).

As we have already pointed out, the study of analytic eigenbranches and semiclassical analysis are two different worlds so that such a relation between them is a priori quite surprising. However, as it will become clear later, the formula that expresses the derivative of any analytic eigenbranch is actually a semiclassical quantity. This is the fact we will be exploiting here. This idea is also used in the two recent papers [START_REF] Hillairet | The eigenvalues of a domains with small slits[END_REF] and [START_REF] Hassell | Ergodic Billiards that are not Quantum Unique Ergodic, with an appendix[END_REF]. In [START_REF] Hillairet | The eigenvalues of a domains with small slits[END_REF] we deal with the limiting behaviour of analytic eigenbranches in a singular setting. The main ingredient of the proof consists in controlling certain quadratic forms evaluated on the eigenfunctions of the spectral problem under consideration. We claim here that such a control may actually be interpreted as a condition on the associated semiclassical measures. Such an interpretation is not necessarily helpful since rather little is known in general about semiclassical measures and the overall strategy of [START_REF] Hillairet | The eigenvalues of a domains with small slits[END_REF] relies more on direct estimates for solutions of one dimensional ODE's. In [START_REF] Hassell | Ergodic Billiards that are not Quantum Unique Ergodic, with an appendix[END_REF], a part of the proof relies on the fact that an assumption on semiclassical measures allows to prove an integrated remainder estimate of the form we will consider here. Section 4 can actually be seen as a formalization of this argument of [START_REF] Hassell | Ergodic Billiards that are not Quantum Unique Ergodic, with an appendix[END_REF].

Acknowledgments :

The ideas in this paper grew out from several discussions with Chris Judge and Andrew Hassell, I would like to thank both of them for their stimulating influence. I would also like to thank the CRM in Montréal and the organizers of the workshop "Spectrum and Dynamics" in April, 2008 and the MSRI in Berkeley where part of this work was done. Finally, I would like to thank Gilles Carron, Yves Colin de Verdière, Patrick Gérard and Andrew Hassell who made several comments on the first version of this paper that helped me improve the results and the exposition.

quantization and Semiclassical measures

In this section we will briefly recall some known facts about quantization and semiclassical measures. We will use both the classical and the semiclassical theory of pseudodifferential operators (See [START_REF] Taylor | Pseudodifferential Operators[END_REF][START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] for background on these notions)

In the case of Schrödinger operators in R d , the principal symbol of a (h-)pseudodifferential operator is a function in T * (R d ). When working on a smooth compact Riemannian manifold, we will denote by S * X the quotient of T * X\{0} by (x, ξ) ∼ (x, λξ), λ > 0. The subsets of S * X are the conical subsets of T * X\{0}. The symbol of a pseudodifferential operator A is defined by testing A again fast oscillatory functions. It is then a function on T * M \{0}. The symbol of a classical pseudodifferential operator is conically invariant and thus defines a function on S * X.

Remark 2.1. The principal symbol of a pseudodifferential operator may also be read off from the Schwartz kernel of A. Observe, however, that when performing this identification, one has to pay careful attention to the density that is used (especially when dealing with varying Riemannian metrics).

A crucial point is the ability of quantizing any reasonable function on S * X. This is done via a so-called quantization procedure. There are several possible choices, but it is useful to choose a quantization that respects positivity (see [START_REF] De Verdière | Ergodicité et fonctions propres du laplacien[END_REF][START_REF] Helffer | Ergodicité et limite semi-classique[END_REF]) 1 . Thus, for any function a on S * X there exists a pseudodifferential operator Op + (a) of order 0 whose principal symbol is a. If A is any other pseudodifferential operator with principal symbol a then Op + (a) -A is a pseudodifferential operator of order -1. Moreover if, a is non-negative, then Op + (a) is also non-negative, as an operator.

In the semiclassical setting, the homogeneity condition on a is dropped and we may quantize any smooth function on T * (X) with compact support.

Using this quantization, we define the so-called semiclassical measures. We present the construction in the semiclassical setting since in the classical one we will use a non-concentration estimate rather than semiclassical measures.

We start with a Schrödinger operator on R d , and we consider a sequence of normalized eigenfunctions u h of the equation

-h 2 ∆u h + V (x)u h = E h u h ,
such that the corresponding eigenvalues E h converge to some non-critical energy E 0 . To any compactly supported function a the map a → Op + (a)u h , u h defines a measure that converges weakly (up to extracting a subsequence). Any such limit is called a semiclassical measure. Remark 2.2. One could in fact associate semiclassical measures to any bounded sequence in L 2 . In the sequel we will consider only semiclassical measures that are associated with eigenfunctions of the Laplace operator.

Among the basic facts satisfied by any semiclassical measure, we mention only the two following properties (see [START_REF] De Verdière | Ergodicité et fonctions propres du laplacien[END_REF][START_REF] Helffer | Ergodicité et limite semi-classique[END_REF])

(1) Any semiclassical measure is a probability measure supported on the energy surface Σ E0 := {|ξ| 2 + V (x) = E 0 }. (2) Any semiclassical measure is invariant under the Hamiltonian flow associated with |ξ| 2 + V (x) on Σ E0 .

Remark 2.3. Under suitable hypotheses, we may extend the definition to smooth functions that are not necessarily compactly supported in T * R d . In particular, for a sequence of eigenfunctions with energy E h converging to E 0 , the quantity ∇u h 2 dx is uniformly bounded. This allows to prove that |ξ| 2 dµ < +∞ and then that

|ξ| 2 dµ = lim h→0 ∇u h 2 dx.
In the classical setting we can also define semiclassical measures. The main difference is that instead of a small parameter going to zero, we have to consider a sequence of eigenfunctions with eigenvalues growing to infinity and we recover a probability measure on S * X. It can be pointed out that the estimate in corollary 4.7 that we use in section 4 is a kind of non-concentration estimate. Indeed, such an estimate would give some information on any semiclassical measure when passing to the high energy limit. The assumptions on semiclassical measure in section 3 and on geometric control in section 4 are thus very similar in nature.

Schrödinger operators from the analytic viewpoint

In this section we consider the semiclassical Schrödinger equation in X = R d . The eigenvalue problem consists in looking for eigenfunctions u h and eigenvalues

E h such that (1) -h 2 ∆u h + V (x)u h = E h u h ,
in which ∆ is the (non-positive) Laplace operator associated with the Euclidean metric. Let us assume that V is smooth and confining (i.e. lim |x|→∞ V (x) = +∞) so that there is a complete orthonormal set of eigenfunctions and the corresponding eigenvalues grow to infinity. This eigenvalue problem may be put into a generalized eigenvalue problem in the sense of Kato (cf [START_REF] Kato | Perturbation theory for linear operators[END_REF]). We change notations by letting h = t thus emphasizing that we will now consider an analytic perturbation problem. An integration by parts leads to the following equivalent problem

(2) q t (u t , v) = E t • n(u, v),
In which q t is the quadratic form

q t (u) = t 2 |∇u(x)| 2 dx + V (x)|u(x)| 2 dx defined on D = H 1 (X) ∩ L 2 ((1 + V (x) 2 ) 1 2 dx)
and n is the standard Riemannian quadratic form on L 2 (X). Observe that although q t is real-analytic for t ∈ R, D is the form domain of q t only for t = 0. Consequently, usual analytic perturbation theory tells us that the eigenvalues organize into real-analytic eigenbranches for t ∈ (0, 1]. A standard question in this setting is to address the limiting behaviour of these eigenbranches when t goes down to 0. Theorem 3.1. Let E t be an eigenvalue branch of [START_REF] Burq | Geometric control in the presence of a black box[END_REF], then E t converges to a limit E 0 when t goes to 0. Moreover, this limit E 0 is a critical value of V.

Proof. Let E t be an eigenvalue branch and u t the corresponding normalized eigenvector branch. The derivative of E t may be computed by differentiating eq. ( 2) (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]), this yields

(3) Ėt = 2t X |∇u t | 2 (x)dx.
This expression is obviously non-negative so that E t is a non-decreasing function of t. Since, for any t we have E t ≥ min X V (x), E t has a limit E 0 when t goes to 0. The condition on concentration of semiclassical measures then enters in the following lemma. Lemma 3.2. Assume that for any semiclassical measure µ at the energy E 0 ,

(C) ΣE 0 |ξ| 2 dµ(x) > 0.
Then, for any eigenbranch

(E t , u t ) such that lim E t = E 0 , (4) ∃c > 0, lim inf t→0 t 2 • X |∇u t (x)| 2 dx ≥ c.
Assuming this lemma, we finish the proof of the theorem. If we suppose, for a contradiction, that E 0 satisfies condition (C), then using (3), and (4) we get

Ėt ≥ 2c t ,
and this is a contradiction since E t has to converge when t goes down to 0. So condition (C) is not fulfilled and there exists a semiclassical measure at energy E 0 such that

ΣE 0 |ξ 2 |dµ = 0.
This implies that the support of µ is contained in the level set {ξ = 0} ⊂ Σ E0 .

Since dµ has to be invariant by the hamiltonian flow

ẋ = 2ξ ξ = -∇V (x)
this forces E 0 to be a critical value of V .

It remains to prove lemma 3.2.

Proof of Lemma 3.2. As it is standard in this kind of settings (see [START_REF] Hassell | Eigenfunction concentration in polygonal billiards[END_REF] or [START_REF] Burq | Geometric control in the presence of a black box[END_REF]), we actually work in the reverse direction, assuming that (4) is not true. We may thus find a sequence t n going to 0 and corresponding eigenfunctions and eigenvalues E tn and u tn , such that

t 2 n X |∇u tn (x)| 2 dx ≤ 1 n .
We now change notations again and let t n = h since the rest of the argument is of semiclassical nature (and, as usual in this setting h actually stands for h n ). We thus have

(5) 0 ≤ h 2 X |∇u h (x)| 2 dx ≤ ε(h).
where ε(h) is some function going to 0 with h. The sequence u h is L2 normalized so that we may extract subsequences and find associated semiclassical measures at energy E 0 since lim E h = E 0 . Let dµ be one of these semiclassical measures, by definition (see remark 2.3 above)

(6) lim h→0 h 2 • X |∇u h (x)| 2 dx = ΣE 0 |ξ 2 |dµ,
where the limit is understood along the subsequence defining µ. Putting ( 5) and ( 6) together yields

ΣE 0 |ξ 2 |dµ = 0,
thus finishing the proof of the lemma.

Remark 3.3. Since the energy surface is defined by |ξ| 2 + V (x) = E 0 , we can replace everywhere

ΣE 0 |ξ 2 |dµ by ΣE 0 (E 0 -V (x))dµ.
In dimension 1 this result can be refined using that the spectrum of a onedimensional Schrödinger operator is known to be simple. If we assume that the potential V has only non-degenerate minima, then the spectrum near the bottom of the energy is known (see [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]) and there is an infinite number of eigenvalues close to the minimum. Since the eigenvalue branches cannot cross, the non-degenerate minimum is then only possible limit. 2

Integrated remainder estimates

In this section, we consider a smooth compact manifold X of dimension d ≥ 2 and a real-analytic family of Riemannian metrics (g τ ) τ ∈[-1,1] . We denote by ∆ τ the Laplace operator associated with g τ and by E j (τ ) the associated analytic branches of eigenvalues. We will also denote by •, • τ the scalar product on L 2 (X, g τ ). We now define the usual following counting functions :

N (τ, E) = ♯{j | E j (τ ) ≤ E}, R M (τ, E) = ♯{j | E j (τ ) ∈ (E -M, E + M ] } = N (τ, E + M ) -N (τ, E -M ).
Remark 4.1. Observe that since (g τ ) is analytic in τ , the scalar products •, • τ define equivalent norms. The same is true at the level of H 1 : the norms • H 1 (X,gτ ) defined by :

u 2 H 1 (X,gτ ) = M g τ (∇ τ u, ∇ τ u)dvol gτ + u 2 τ ,
are all equivalent.

The following lemma expresses the variation of E t using a family of pseudodifferential operators. It also serves as a definition of the family A τ that will be used in the rest of the paper.

Lemma 4.2.

There exists a real-analytic family of symbols a τ , and, for any τ 0 , there exists a constant C such that, for any normalized eigenbranch (E n (τ ), u n (τ )) we have

∀|τ | ≤ τ 0 , Ėn (τ ) E n (τ ) -Op + (a τ )u n (τ ), u n (τ ) τ ≤ CE n (τ ) -1 2 .
We will denote by A τ = Op + (a τ ).

Proof. First, by differentiating the eigenvalue equation

E n (τ ) = ∆ τ u n (τ ), u n (τ ) τ ,
we find a second-order differential operator ∆τ such that Ėn (τ ) = ∆τ u n (τ ), u n (τ ) τ .

(Observe that since the branch is normalized and using the eigenvalue equation we have that ∆ u, u = E u, u = 0). Denote by a τ the principal symbol of ∆τ (∆ τ + 1) -1 , and let

A τ = Op + (a τ ). By definition R τ = ∆τ (∆ τ + 1) -1 -A τ is
an analytic family of pseudodifferential operators of order -1. In particular, R τ is uniformly bounded from L 2 into H 1 . This yields the bound Ėn (τ )

E n (τ ) + 1 -Op + (a τ )u n (τ ), u n (τ ) τ ≤ CE n (τ ) -1 2 .
We thus get Ėn (τ )

E n (τ ) -1 + E -1 n Op + (a τ )u n (τ ), u n (τ ) τ ≤ CE n (τ ) -1 2 1 + E -1 n ,
from which the claimed bound follows since Op + (a τ )u n (τ ), u n (τ ) τ is uniformly bounded.

The main result of this section is then the following theorem (the notion of geometric control is defined in definition 4.4 in the following section). Theorem 4.3. Suppose that a 0 is non-negative and that there exists ε > 0 such that the subset {(x, ξ) ∈ S * X , a 0 (x, ξ) > ε} geometrically controls (S * X, g 0 ), then for any fixed M , there exists τ 0 , E 0 and K such that

(7) ∀ E > E 0 , τ0 -τ0 R M (t, E) dt ≤ K E d 2 -1 .
The universal remainder in Weyl's law gives that for a fixed t, R M (t, E) = O(E d-1

2 ) as E goes to infinity. The theorem thus says that averaging with respect to the perturbation greatly improves this estimate.

The rest of the section is devoted to the proof of this theorem. The central part of the argument is actually interesting by itself and consists in proving a somewhat explicit uniform control estimate.

4.1.

A uniform control estimate. We begin by recalling the notion of geometric control observing that since the velocity is constant along a geodesic, the geodesic flow of a Riemannian metric is well-defined on S * X. Definition 4.4. We say that an open subset U of S * X geometrically controls (S * X, g) if for any (x, ξ) ∈ S * X, there exists T ∈ R and (x 0 , ξ 0 ) ∈ U such that (x, ξ) = Φ T (x 0 , ξ 0 ) where Φ . (., .) denotes the geodesic flow of g.

It is well-known that geometric control implies a non-concentration estimate for eigenfunctions. More precisely, if U is an open subset of X such that U = S * U geometrically controls (S * X, g) then, there exists some positive c and an energy E 0 such that (8)

U |u n | 2 dvol g ≥ c X |u n | 2 dvol g ,
for any eigenfunction u n of the Laplace operator associated with g such that E n ≥ E 0 . (Actually the restriction on E for this estimate may be released using the principle of unique continuation for solutions of second order elliptic PDE's see remark 4.6 below) Such a control estimate can be proved by a contradiction argument relying on known properties of semiclassical measures ([6], [START_REF] Burq | Geometric control in the presence of a black box[END_REF] for the scheme of such an argument). Such an approach can probably be adapted to get a control estimate that is uniform in τ. We propose here a slightly different proof that also allows us to get some uniform control on c. Proposition 4.5. Let U be an open subset of S * X that geometrically controls (S * X, g 0 ). There exists τ 0 ∈ R, E 0 ∈ R, a zeroth-order pseudodifferential operator Π on X and c > 0 such that :

(1) The symbol π of Π is supported in U and, 0 ≤ π(x, ξ) ≤ 1 on S * X.

(2) For any |τ | ≤ τ 0 and any u eigenfunction of ∆ τ with energy greater than E 0 then the following non-concentration estimate holds :

(9) Πu τ ≥ c u τ .
Remark 4.6. Estimate (8) may be extended to any eigenfunction, without any restriction on the energy. Such an extension requires that one knows that an eigenfunction of a Laplace operator cannot vanish on an open subset (thus proving the control estimate for any finite number of eigenfunctions). The principle of unique continuation gives this property for eigenfunctions of the Laplace operator. It is not clear to the author if this non-vanishing property holds in the cotangent bundle. It can also be observed that, strictly speaking, the formulation with the microlocal cutoff only has a meaning in the high energy limit since the microlocal cutoff is defined modulo smoothing operators.

Proof. Let (x, ξ) ∈ S * X, the geometric control assumption implies that there exists some T (= T (x, ξ)) and (x 0 , ξ 0 ) ∈ U such that Φ T 0 (x 0 , ξ 0 ) = (x, ξ). Using the continuity properties of solutions of an O.D.E. both with respect to initial conditions and with respect to parameters, we can find -some τ max (x, ξ), -two conical neighbourhoods of (x, ξ) :

V = V (x,ξ) and W = W (x,ξ) , -a conical neighbourhood Ũ = Ũ(x,ξ) of (x 0 , ξ 0 ) such that V is compactly included in W and Φ -T τ (W) ⊂ Ũ ⋐ U for any |τ | ≤ τ max (x, ξ). Since S * X is compact, we can find a finite collection ((x i , ξ i )) 1≤i≤N such that S * X = N i=1 V (xi,ξi)
. We now use the index i to denote any quantity formerly indexed by (or attached to) (x i , ξ i ). Denote by τ 0 the minimum of all τ max (x i , ξ i ). We can then find a smooth partition of unity π i such that each π i is identically 1 in V i and identically 0 outside W i . Note that, by construction,

∀ |τ | ≤ τ 0 , N i=1 Φ -Ti τ (W i ) ⊂ N i=1 Ũi ⋐ U.
We denote by π 0 a smooth function which is identically 1 in N i=1 Ũi and 0 outside U. Denote by Π i = Op + (π i ), any by U τ (T ) = exp iT √ ∆ τ , we claim that for any i > 0, and for any |τ | ≤ τ 0 , the operator

R i = Π i U τ (T i ) (Id -Π 0 ) is smoothing.
Indeed, its wave-front set is included (y, η, x, ξ) | (y, η) ∈ supp.(π i ), (x, ξ) ∈ supp.(1π 0 ), (y, η) = Φ Ti τ (x, ξ) , which is empty by construction. In particular, each R i,τ is bounded from H -1 (X) into L 2 (X). Moreover, a careful analysis of the construction of the Hadamard parametrix (as it is for instance presented in [START_REF] Bérard | On the wave equation on a compact Riemannian manifold without conjugate points[END_REF]) shows that there exists a uniform constant C such that :

(10) ∀ |τ | ≤ τ 0 , ∀w ∈ H -1 (X), R i w L 2 (X,g0) ≤ C u H -1 (X,g0) ,
(see the appendix for a brief sketch of proof). Since the (π i ) i>1 are a partition of unity, i>1 Π i = Id. We thus have (11)

u 2 0 = N i=1 Π i u, u 0 .
Using that u is an eigenfunction of ∆ τ , we have

| Π i u, u 0 | = | Π i U τ (T i )u, u 0 | ≤ | Π i U τ (T i )Π 0 u, u 0 | + | R i u, u 0 | .
We use Cauchy-Schwarz inequality on both terms and the following facts on the norm of the different operators : -Π i is bounded from L 2 (X) to itself, -U τ (T ) is an isometry from L 2 (X), -and (10). This yields :

| Π i u, u 0 | ≤ c i Π 0 u 0 + C u H -1 (X,g0) • u 0 .
Using that the H -1 (X, g τ ) norms are all equivalent and that u 2 H -1 (X,gτ ) = (1 + E) -1 u 2 τ when u is an eigenfunction of ∆ τ , we get

| Π i u, u 0 | ≤ c i Π 0 u + C(1 + E) -1 2 u 0 • u 0 , for a constant C independent of τ ∈ [-τ 0 , τ 0 ].
Summing all these inequalities and denoting by c the maximum of the c i we obtain:

u 0 ≤ (N c) Π 0 u + CN (1 + E) -1 2 u 0 .
This yields the estimate in the proposition with Π = Π 0 and c -1 = 2cN as soon as CN (1 + E) -1 2 is less than 1 2 . This proposition has the following useful corollary.

Corollary 4.7. Let a τ be an analytically dependent family of non-negative symbols of order 0 such that {a 0 (x, ξ) > ε} geometrically controls (S * X, g 0 ) then there exist positive κ, τ 0 and E 0 such that, for any |τ | ≤ τ 0 and any u eigenfunction of ∆ τ with energy greater than E 0 then

A τ u, u 0 ≥ κ u 2
τ , where A τ denotes Op + (a τ ).

Remark 4.8. In the following proof, we will use several times the fact that, for any symbols of order 0, Op + (bc) -Op + (b) • Op + (c) is a pseudodifferential operator of order -1. We will denote by R any operator arising from this kind of operation. Observe that if b and c depend analytically on τ , then R also depends analytically on τ .

Proof. Let τ 0 , Ẽ0 and Π be given by applying Proposition 4.5 to U = {a 0 (x, ξ) > ε} .

If needed, we restrict τ 0 so that, for any |τ | ≤ τ 0 , a τ is bounded below by ε 2 on U. Observe that

A τ u, u 0 = A τ Π 2 u, u 0 + A τ (Id -Π 2 )u, u 0 = Op + (a τ π 2 )u, u 0 + Op + (a τ (1 -π 2 )u, u 0 + R τ u, u 0 ≥ Op + (a τ π 2 )u, u 0 + R τ u, u 0 ,
where the last inequality holds because π(x, ξ) ≤ 1 and we have chosen a positive quantization. Since the support of π is included in U we have

∀(x, ξ) ∈ S * X, a τ (x, ξ)π 2 (x, ξ) ≥ ε 2 π 2 (x, ξ), which is quantized into Op + (a τ π 2 )u, u 0 ≥ ε 2 Op + (π 2 )u, u 0 ≥ ε 2 Πu 2 0 + R τ u, u 0
, with another remainder operator R τ . Putting these two inequalities together, we obtain

(12) Au, u 0 ≥ ε 2 Πu 2 0 + R τ u, u 0 .
Since R is of order -1 and the norms on H -1 associated with (g τ ) |τ |≤τ0 are uniformly equivalent, there exists some uniform C such that

| R τ u, u 0 | ≤ CE -1 2 u τ . Using Proposition 4.5, for |τ | ≤ τ 0 and E ≥ E 0 , we have Πu 0 ≥ c 1 Πu τ ≥ c 2 u τ so that we get (13) A τ u, u ≥ c ε 2 -CE -1 2 • u 2 τ .
The corollary then holds with κ = cε 4 as soon as E is large enough so that CE -1 2 ≤ cε 4 .

4.2.

Proof of Theorem 4.3. We consider normalized eigenbranches and we start from the formula for the derivative of the eigenfunctions : (in the following equation, everything except the constant C depends on τ )

d dτ ln(E) ≥ Au, u -CE -1 2 .
Under the assumptions of the theorem, we may use proposition 4.5 and its corollary. This provides us with κ > 0, τ 0 and E 0 such that

∀ |τ | ≤ τ 0 , ∀ E(τ ) ≥ E 0 , d dτ ln(E) ≥ κ. (14)
The rest of the argument is merely reproduced from [START_REF] Hassell | Ergodic Billiards that are not Quantum Unique Ergodic, with an appendix[END_REF].

Denote by R

M (E) = τ0 -τ0 R M (t, E) dt. A straightforward manipulation of the integral yields : (15) R M (E) = j Leb{τ | E j (τ ) ∈ (E -M, E + M ] },
where Leb denotes the Lebesgue measure. We consider now E such that E -M ≥ E 0 . According to (14), any eigenbranch that crosses the energy E -M has to do it in an increasing manner. This implies that any eigenbranch for which the set

I j := {t | E j (τ ) ∈ (E -M, E + M ] } is not empty satisfies E j (-τ 0 ) ≤ E + M and E j (τ 0 ) ≥ E -M.
In particular, the sum in (15) has only a finite number of positive terms, and moreover, this number is bounded by N -τ0 (E + M ) which is bounded by CE d 2 according to Weyl's law. Estimate (14) also implies that I j is an interval whose extremities we denote by τ - j and τ + j 3

. Integrating (14) we obtain

κ(τ + j -τ - j ) ≤ ln E j (τ + j ) E j (τ - j ) ≤ ln E + M E -M ≤ C E , 3
We do not specify whether I j is open or closed at each of its extremity since this fact depends on whether τ ± j coincide or not with ±τ 0 , and it is irrelevent when computing the Lebesgue measure.

as soon as E is large enough (indepently of j). This bound each term of the sum in (15), since the number of terms is also bounded explicitly using Weyl's law, we obtain

R M (E) ≤ C • E d 2 -1 .
Remark 4.9. The fact that a 0 is non-negative implies in particularly that the derivative of vol gτ X is increasing. In particular, the variation cannot be volumepreserving.

Remark 4.10. Corollary (4.7) implies that for any semiclassical measure associated with ∆ τ , the following lower bound holds :

adµ ≥ κ.
If we replace the control assumption in Thm.(4.3) by the following (weaker) assumption :

∀ τ, ∀ µ semiclassical measure associated with ∆ τ , adµ > 0, then, following the same lines of proof, we would get a 'almost-everywhere' result in the spirit of [START_REF] Hassell | Ergodic Billiards that are not Quantum Unique Ergodic, with an appendix[END_REF].

Remark 4.11. This result can probably be extended to manifolds with boundary with the precaution that, in this case we would have to control the possible escape of the mass to the boundary.

Examples :

(1) A natural variation to look at is to simply take g τ = exp(τ )g 0 . In this case the control assumption is trivially satisfied so that we get a uniform integrated remainder. However, this is not a new result since looking at a fixed energy interval and dilating the volume is equivalent to fixing the volume and dilating the energy interval so that the integrated remainder is actually a regularized remainder in Weyl's law. And it is known that integrating the remainder in Weyl's law with respect to the energy yields a much better estimate than the universal remainder term. (2) For a more general conformal variation on a surface (d = 2) we have g τ :=

(1 + τ f (x)) 2 g 0 , the eigenvalue problem can be put in the form

X ∇ 0 u∇ 0 φ dvol g0 = E X uφ(1 + τ f (x)) 2 dvol g0 .
This gives a 0 (x, ξ) = 2f (x). Choosing f non-negative and supported in a control region yields examples where the theorem applies. (3) On the contrary, the variation considered in [START_REF] Hassell | Ergodic Billiards that are not Quantum Unique Ergodic, with an appendix[END_REF] on a Donnelly surface doesn't satisfy the control estimate because of the cylinder of periodics orbits and of periodic geodesic in the hyperbolic wings.

I would like to thank San Vũ Ngo . c and Frédéric Faure for pointing out that, in the onedimensional case, the simplicity of the spectrum allows to improve the result.

Appendix : Uniform Hadamard Parametrix

The aim of this appendix is to give a (rather sketchy) justification of the uniform estimate [START_REF] Taylor | Pseudodifferential Operators[END_REF].

Let X be a smooth manifold without boundary and g τ a real-analytic family of riemannian metrics.

Denote by U τ (s) the propagator of the half-wave equation associated with ∆ τ :

and denote by k τ (s, x, y) its Schwartz kernel. This kernel k τ (s, x, y) is obtained by restricting the Schwartz kernel of cos(s

We start from the Hadamard parametrix construction as explained for instance in [START_REF] Bérard | On the wave equation on a compact Riemannian manifold without conjugate points[END_REF]. Concretely, for s less than some small ε and x, y such that d τ (x, y) ≤ 2ε, we make the ansatz kN (τ, s, x, y)

in which the u k are smooth functions of x, y and T α+k (z) are oscillatory distributions of R with wave-front in {(z = 0, ζ > 0)}. The coefficients u k satisfy some transport equations from which it is straightforward that they also depend smoothly on τ . We fix a smooth cutoof function ρ := ρ(d τ (x, y)) that cuts off in d τ (x, y) ≤ 2ε and we let k N = kN × ρ A computation then yields that k N solves the half-wave equation up to a remainder term r N (τ, s, x, y) which is C l(N ) where l(N ) tends to infinity with N. Applying Duhamel's principle, we get k(τ, s, x, y)k N (τ, s, x, y) = s 0 k(τ, ss ′ , x, w)r N (τ, s ′ , w, y)ds ′ dw.

Here again, a detailed analysis shows that, for s ≤ ε and for any N, we finally have k(τ, s, x, y) = k N (τ, s, x, y) + R N (τ, s, x, y), in which everything is smooth in τ, and R N is C l(N ) . Since k N has a fixed order as a distribution of (s, x, y), applying it to a C l function gives a C l-β with a loss β that is independant of N.

We now fix some T , and we use the semigroup properties of U to write

with a fixed small δ. We now plug into this expression the decomposition

Expanding everything gives a large sum in which each term is a composition of K N factors with R N factors. We address the regularity of each term. Each term has regularity Npβ where p is the number of K N factors. Thus any term except the one with only K N factors may be made as regular as wanted by choosing N large enough. Moreover, everything is regular in τ.

It remains to address the term with only K N factors. Each K N is explicit and this composition is adressed using stationary phase. It is somewhat tedious in the general case because for instance of conjugate points. However it is much simpler when we have an initial and final microlocal cutoff that destroys everything in the wave-front set. Indeed a uniform non-stationary phase then applies that yields a uniform C m bound for any m.