Ugo Boscain
email: ugo.boscain@cmap.polytechnique.fr

Camille Laurent
email: camille.laurent@cmap.polytechnique.fr

The Laplace-Beltrami operator in almost-Riemannian Geometry

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated.

In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities.

This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet case. 1

Introduction

A 2-dimensional Almost Riemannian Structure (2-ARS for short) is a generalized Riemannian structure that can be defined locally by a pair of smooth vector fields on a 2-dimensional manifold M , satisfying the Hörmander condition. These vector fields play the role of an orthonormal frame.

Let us denote by (q) the linear span of the two vector fields at a point q. Where (q) is 2-dimensional, the corresponding metric is Riemannian. Where (q) is 1-dimensional, the corresponding Riemannian metric is not well defined, but thanks to the Hörmander condition one can still define the Carnot-Caratheodory distance between two points, which happens to be finite and continuous.

2-ARSs were introduced in the context of hypoelliptic operators [START_REF] Franchi | Une métrique associée à une classe d'opérateurs elliptiques dégénérés[END_REF][START_REF] Grušin | A certain class of hypoelliptic operators[END_REF], they appeared in problems of population transfer in quantum systems [START_REF] Boscain | Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy[END_REF][START_REF] Boscain | Resonance of minimizers for n-level quantum systems with an arbitrary cost[END_REF][START_REF] Boscain | Optimal control in laser-induced population transfer for two-and three-level quantum systems[END_REF], and have applications to orbital transfer in space mechanics [START_REF] Bonnard | Singular Metrics on the Two-Sphere in Space Mechanics[END_REF][START_REF] Bonnard | Conjugate and cut loci of a two-sphere of revolution with application to optimal control[END_REF]. 2-ARSs are a particular case of rank-varying sub-Riemannian structures (see for instance [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry[END_REF][START_REF] Jean | Uniform estimation of sub-Riemannian balls[END_REF][START_REF] Vendittelli | Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities[END_REF]).

Generically (i.e for an open and dense subset of the set of all 2-ARSs, in a suitable topology), the singular set Z, where (q) has dimension 1, is a 1-dimensional embedded submanifold and there are three types of points: Riemannian points, Grushin points where (q) is 1-dimensional and dim( (q)+ [ , ](q)) = 2 and tangency points where dim( (q)+ [ , ](q)) = 1 and the missing direction is obtained with one more bracket. One can easily show that at Grushin points (q) is transversal to Z. Generically, at tangency points (q) is tangent to Z and tangency points are isolated. Normal forms at Riemannian, Grushin and tangency points were established in [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF] and are described in Figure 1.

2-ARSs present very interesting phenomena. For instance, geodesics can pass through the singular set, with no singularities even if all Riemannian quantities (as for instance the metric, the Riemannian area, the curvature) explode while approaching Z. Moreover the presence of a singular set permits the conjugate locus to be nonempty even if the Gaussian curvature is always negative, where it is defined (see [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF]). See also [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF][START_REF] Agrachev | Two-dimensional almost-Riemannian structures with tangency points[END_REF][START_REF] Boscain | Lipschitz Classification of Two-Dimensional Almost-Riemannian Distances on Compact Oriented Surfaces[END_REF][START_REF] Boscain | High-order angles in almost-Riemannian geometry[END_REF] for Gauss-Bonnet-type formulas and for a classification of 2-ARSs from the point of view of Lipschitz equivalence.

In this paper we study the Laplace-Beltrami operator in a 2-ARS. The main point is that the first order terms explode as a consequence of the explosion of the area on the singular set Z. In particular we are interested in its self-adjointness. This is crucial to understand the evolution of a quantum particle and of the heat flow in a 2-ARS.

On M \ Z, this operator is defined as the divergence of the gradient. The almost-Riemannian gradient can be defined with no difficulty since it involves only the inverse metric which is well defined (by continuity) even on the singular set. More precisely, on a Riemannian manifold (M, g), ∇ is the unique operator from C ∞ (M ) to Vec(M ) satisfying g q (∇ φ(q), v) = dφ q (v), ∀ q ∈ M, v ∈ T q M . In the two-dimensional case, if in an open set Ω ⊂ M an orthonormal frame for g is given by two vector fields X 1 and X 2 , then we have on Ω,

∇(φ) = X 1 (φ)X 1 + X 2 (φ)X 2 , (1)
where by X i (φ) we mean the Lie derivative of φ in the direction of X i (i = 1, 2). This last formula can be used to define the gradient even where X 1 and X 1 are not linearly independent. In this paper, we use formula [START_REF] Agrachev | Introduction to Riemannian and sub-Riemannian geometry[END_REF] to define the gradient of a function on a 2-ARS. However the divergence of a vector field requires a notion of area and in a 2-ARS the natural area (i.e. the Riemannian one) explodes while q approaches the singular set. Let us recall that the divergence of a vector field X on a Riemannian manifold (M, g) is the unique The local representations established in [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF]. Tangency points are the most difficult to handle due to the fact that the asymptotic of the distance is different from the two sides of the singular set (see [START_REF] Bonnard | The Sphere and the Cut Locus at a Tangency Point in Two-Dimensional Almost-Riemannian Geometry[END_REF][START_REF] Boscain | A normal form for generic 2-dimensional almostriemannian structures at a tangency point[END_REF]).

function satisfying divXdω = L X dω where dω is the Riemannian volume form. In coordinates divX = 1 ω i,j ∂ i ωX j , where ω = √ detg. To see how the Riemannian quantities (and in particular dω) explode while approaching the singular set Z, choose a local orthonormal frame for the 2-ARS of the form X 1 = (1, 0) and X 2 = (0, f (x, y)), where f is some smooth function. This is always possible, see for instance [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF]. On M \ Z, for the metric g, the area element dω, the curvature K and the gradient of a smooth function one has:

g(x, y) = 1 0 0 1 f (x,y) 2 , dω(x, y) = 1 |f (x, y)| dx dy, K(x, y) = f (x, y)∂ 2 x f (x, y) -2∂ x f (x, y) 2 f (x, y) 2 , ∇(φ) = ∂ x (φ)∂ x + f 2 ∂ y (φ)∂ y .
and for a vector field

Y = (Y 1 , Y 2 ) one has div(Y ) = |f | ∂ x 1 |f | Y 1 + ∂ y 1 |f | Y 2 = ∂ x Y 1 + ∂ y Y 2 - ∂ x f f Y 1 - ∂ y f f Y 2 (2)
which is not well defined on Z even on vector fields which are linear combinations of X 1 and X 2 .

In particular it is not well defined on X 1 . As a consequence the Laplace Beltrami operator presents some singularities in the first order terms. (The choice of the area does not affect the principal symbol of the operator). More precisely, one has

∆(φ) = ∂ 2 x φ + f 2 ∂ 2 y φ - ∂ x f f ∂ x φ + f (∂ y f )(∂ y φ). (3)
The simplest case is the well known Grushin metric, which is the 2-ARS on R 2 for which an orthonormal basis is given by,

X 1 = 1 0 , X 2 = 0 x . (4)
Here the singular set Z is the y axis and on R 2 \ Z the Riemannian metric, the Riemannian area and the Gaussian curvature are given respectively by:

g = 1 0 0 1 x 2 , dω = 1 |x| dx dy, K = - 2 x 2 . (5)
It follows that the Laplace Beltrami operator is given by

∆φ := div(∇(φ)) = ∂ 2 x + x 2 ∂ 2 y - 1 x ∂ x φ, (6)
which is the basic model of Laplace Beltrami operator on 2-ARSs and whose self-adjointness will be studied in this paper. Notice that the operator "sum of squares",

∆ = X 2 1 + X 2 2 = ∂ 2 x + x 2 ∂ 2 y ( 7
)
has been deeply studied in the literature. This operator (which in this case is the principal symbol of ∆) can be obtained as the divergence of the gradient by taking as area, the standard Euclidean area on R 2 . However this operator does not give any information about ∆ because of the diverging first order term. Moreover in 2-ARS an intrinsic Laplacian with non-diverging terms cannot be defined naturally, since an intrinsic area which does not diverge on the singular set is not known.

Here by intrinsic we mean an area which depends only on the 2-ARS and not on the choice of the coordinates and of the orthonormal frame. Notice that even if all Riemannian quantities are not defined on M \ Z, classical geodesics do. See Section 3.1 for the explicit expressions of geodesics in the Grushin plane. Another interesting feature of the Grushin plane, is the fact that a bounded open set intersecting Z has finite diameter but infinite area.

The main purpose of this paper is to study the following question:

[Q] Let M be a 2-D manifold endowed with a 2-ARS, and let ∆ be the corresponding Laplace-Beltrami defined on C ∞ 0 (M \ Z). Is ∆ essentially self-adjoint?

Notice that a priory one expects a negative answer to this question, since, as explained below, a positive answer would imply that neither the heat flow, neither a quantum particle can pass through Z, while classical geodesics cross it with no singularities. Our main result is an unexpected positive answer to the question [Q] in the case in which there are no tangency points and in the case in which M is compact. More precisely we have the following.

Theorem 1. Let M be a 2-D compact orientable manifold endowed with a 2-ARS. Assume that

[HA] the singular set Z is an embedded one-dimensional submanifold of M ;

[HB] for every q ∈ M , (q) + [ , ](q) = T q M . Let dω be the corresponding Riemannian area and ∆ the corresponding Laplace-Beltrami operator, both defined on M \ Z. Then we have the following facts

1. ∆ with domain C ∞ 0 (M \ Z) is essentially self-adjoint on L 2 (M, dω).
2. The domain of ∆ (the self-adjoint extension denoted by the same letter) is given by

D(∆) = u ∈ L 2 (M, dω) ∆ D,g u ∈ L 2 (M, dω)
where ∆ D,g u is ∆u seen as a distribution in Ω = M \ Z.

3. The resolvent (-∆ + 1) -1 is compact and therefore its spectrum is discrete and consists of eigenvalues with finite multiplicity.

Hypothesis HA is generic (see [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF]). Hypothesis HB implies that every point is either a Riemannian point or a Grushin point (see Figure 1). The hypotheses that M is compact and that there no tangency points are technical and the same result should hold in much more general situations. The orientability of the manifold can be weakened. Indeed it is only necessary that each connected component of Z (which is diffeomorfic to S 1 ) admit an open tubular neighborhood diffeomorfic to (-1, 1) × S 1 . See Proposition 2 and 3.

Remark 1. Notice that 2-ARSs satisfying HA and HB do exist. See for instance [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF] where such a 2-ARS has been built on compact orientable manifolds of any genus. Notice moreover that a 2-ARS satisfying HA and HB are structurally stable in the sense that small perturbations of the local orthonormal frames in the C ∞ norm, do not destroy conditions HA and HB.

Theorem 1 has a certain number of implications. First it implies that a quantum particle, localized for t = 0 on a connected component of M \ Z, remains localized in this connected component for any time t ∈ R. The same phenomenon holds for the wave or the heat equation. Indeed, the essential self-adjointness means that our operator can be naturally and uniquely extended (by taking its closure) to a self-adjoint operator without adding any additional boundary condition. But, in our case, one possible extension is to take an extension for the operator defined on each connected component (for instance the Friedrichs extension) and to "concatenate" them. This possible extension is the one which separates the dynamics. For instance this is what we obtain when one defines the Friedrichs extension of the usual Laplace operator on R defined on function in C ∞ 0 (R \ {0}). We obtain two distinct dynamics with Dirichlet boundary condition at 0. Yet, this is of course not the only possible dynamic because the operator defined on function C ∞ 0 (R \ {0}) is not essentially self-adjoint. So, the essential self-adjointness means that the unique self-adjoint extension of our operator is the one that separates the dynamics on each connected components. That means that it is not necessary to add any boundary condition: the explosion of the area naturally acts as a barrier which prevents the crossing of the degeneracy zone by the particules.

More precisely we have the following.

Corollary 1. With the notations of Theorem 1, consider the unique solution u of the Schrödinger equation (according to the self-adjoint extension defined in the previous theorem),

i∂ t u + ∆u = 0 u(0) = u 0 ∈ L 2 (M, dω) (8)
with u 0 supported in a connected component Ω of M \ Z. Then, u(t) is supported in Ω for any t ≥ 0. The same holds for the solution of the heat or for the solution of the wave equation.

Second it is well known that in Riemannian geometry one can relate properties of the Riemannian distance to those of the corresponding heat kernels. For instance we have that Theorem 2 (Varadhan, Neel, Stroock). Let M be a compact, connected, smooth Riemannian manifold and d the corresponding Riemannian distance. Let p t (q 1 , q 2 ) be the heat kernel of the heat equation ∂ t φ = 1 2 ∆φ, where ∆ is the Laplace-Beltrami operator. Define

E t (q 1 , q 2 ) := -t log p t (q 1 , q 2 )
We have:

1) E t (q 1 , q 2 ) → 1 2 d(q 1 , q 2 ) 2 , uniformly on M × M as t → 0,

2) Let Cut(q 1 ) ⊂ M the cut locus from q 1 . Then q 2 ∈Cut(q 1 ) if and only if lim t→0 ∇ 2 E t (q 1 , q 2 ) = 1 2 ∇ 2 d(q 1 , q 2 ) 2 , while q 2 ∈Cut(q 1 ) if and only if lim sup t→0 ∇ 2 E t (q 1 , q 2 ) = ∞, where . is the operator norm.

The first result is due to Varadhan [START_REF] Varadhan | On the behavior of the fundamental solution of the heat equation with variable coefficients[END_REF], the second one to Neel and Stroock [START_REF] Neel | The small-time asymptotics of the heat kernel at the cut locus[END_REF][START_REF] Neel | Analysis of the cut locus via the heat kernel[END_REF]. Both these results do not depend upon the fact that one is using the Laplace-Beltrami operator or the principal symbol of the operator in a fixed frame to construct the heat kernel.

The situation is very different in 2-ARG. Both these results are false for the Laplace Beltrami operator. Indeed the distance between two points belonging to two different connected components of

M \ Z is finite while if q belongs to a connected component W of M \ Z then p t (q, •) is supported in W .
However a result in the spirit of the one of Varadhan has been obtained by Leandre in [START_REF] Léandre | Minoration en temps petit de la densité d'une diffusion dégénérée[END_REF] for the operator "sum of squares". Hence the Laplace-Beltrami operator defined above, has quite different properties with respect to the operator sum of squares. The last one is not intrinsic, but however keeps tracks of intrinsic quantities as the almost-Riemannian distance. In particular the corresponding heat flow crosses the set Z which is not the case for the Laplace-Beltrami operator. For other relations among the heat kernel and the distance in sub-Riemannian geometry see [START_REF] Barilari | Trace heat kernel asymptotics in 3d contact sub-riemannian geometry[END_REF].

To prove Theorem 1 we start by analyzing the Grushin case [START_REF] Barilari | Trace heat kernel asymptotics in 3d contact sub-riemannian geometry[END_REF]. We first compactify in the y variable by considering it on R x × T y .

By setting f = |x|g we are reduced to study the essential self adjointness of the following operator considered on L 2 with the usual euclidian metric

L = ∂ 2 x + x 2 ∂ 2 y - 3 4 1 x 2 . ( 9
)
If for a moment we forget the term x 2 ∂ 2 y which is not relevant for the selfadjointness of this operator (and becomes a non-positive potential -k 2 x 2 after performing Fourier transform in y), we are reduced to study the operator ∂ 2

x -3 4 1

x 2 , which is well known in the literature. Indeed we have the following result

Proposition 1. The operator -∂ 2 x + c |x| 2 defined on L 2 (]0, +∞[) with domain C ∞ 0 (]0, +∞[) is essentially self-adjoint if and only c ≥ 3 4 .
There are different proof of this result in the literature (see [START_REF] Reed | Methods of modern mathematical physics[END_REF] Chapter X for an introduction), often leading to some stronger statement (for instance for potential V ≥ 3 4x 2 )) and generalizations to higher dimensions.

The rest of the proof for an almost Riemannian structure consists in generalizing this result for a normal form around a connected component of the singular set. The main tools are Kato inequality and perturbation theory.

Remark 2. Notice that, as remarked in the introduction of [START_REF] Gérard | The Szegö cubic equation[END_REF], solutions of the Schrödinger equation, with the Laplacian defined as sum of square ∂ 2

x + x 2 ∂ 2 y , display a total lack of dispersion. It would be interesting to understand if the same holds for the Laplace-Beltrami operator defined by [START_REF] Barilari | Trace heat kernel asymptotics in 3d contact sub-riemannian geometry[END_REF]. It would be also interesting to study the behaviour of a wave packet moving towards the singular set, to understand if there is a reflection or a dispersion. Since in Theorem 1 we prove that the spectrum is discrete, it seems natural to expect a reflection.

The structure of the paper is the following. In section 2 we briefly recall the notion of almost-Riemannian structure. An original result is given in Proposition 2, in which we globalize the normal form of type (F 2 ) around a connected component of Z in the compact orientable case and under the assumption that there are no tangency points. In Section 3 we analyze the Grushin case, both for what concerns geodesics and the self-adjointness of the Laplace-Beltrami operator. The main result, is proven in Section 4.

As a byproduct of our studies, we obtain that the Laplace-Beltrami operator is not essentially self-adjoint in the case in which an orthonormal basis is given by the vectors (1, 0), (0, |x| α ) with α ∈ (0, 1) (see remark 4). This fact motivates some remarks on metrics of this kind (which do not enter in the standard framework of almost-Riemannian geometry, since the vector fields are not smooth). For instance we show that in this case, even if the area element explodes, the area of a bounded open set intersecting Z is finite (which is false for 2-ARSs). Moreover we show that there exist regular curves passing through Z with a velocity not belonging to the span of X 1 and X 2 , having finite length. See Section 3.3.

Finally, in the appendix, we discuss the Martinet case to show that this kind of phenomenon appears also in sub-Riemannian structure of constant rank but which are not equiregular i.e. in which the grow vector depends on the point (see for instance [START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF]). In this case the role of the Laplace-Beltrami operator is played by the intrinsic sub-Riemannian Laplacian defined via the Popp measure (see [START_REF] Agrachev | The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups[END_REF][START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF]). While aproaching the Martinet surface the Popp volume explodes and the first order coefficient of the sub-Riemannian Laplacian does as well.

Theorem 3. Consider the sub-Riemannian structure in M = T x × R y × T z for which an orthonormal basis is given by X 1 = (1, 0, y 2

2 ), X 2 = (0, 1, 0). Then the corresponding intrinsic sub-Riemannian Laplacian which is given by

∆ sr = (X 1 ) 2 + (X 2 ) 2 -1 y X 2 with domain C ∞ 0 (M \{y = 0}) is essentially self-adjoint on L 2 (M, dω), where dω = 1
|y| dx dy dz is the Popp volume.

Basic Definitions

In this section we recall some basic definitions in the framework of 2-ARSs following [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF][START_REF] Agrachev | Two-dimensional almost-Riemannian structures with tangency points[END_REF].

Let M be a smooth surface without boundary. Throughout the paper, unless specified, manifolds are smooth (i.e., C ∞ ) and without boundary; vector fields and differential forms are smooth. The set of smooth vector fields on M is denoted by Vec(M ). The circle is denoted by S 1 or T depending on the context. C ∞ 0 denotes the set of smooth functions with compact support.

2-Almost-Riemannian Structures

Definition 1. A 2-dimensional almost-Riemannian structure (2-ARS) is a triple S = (E, f, •, • )
where E is a vector bundle of rank 2 over M and •, • is a Euclidean structure on E, that is, •, • q is a scalar product on E q smoothly depending on q. Finally f : E → T M is a morphism of vector bundles, i.e., (i) the diagram

E f / / π E ! ! D D D D D D D D T M π M
commutes, where π : T M → M and π E : E → M denote the canonical projections and (ii) f is linear on fibers. Denoting by Γ(E) the C ∞ (M )-module of smooth sections on E, and by f * : Γ(E) → Vec(M ), the map σ → f * (σ) := f • σ. We require that the submodule of Vec(M ) given by = f * (Γ(E)) to be bracket generating, i.e., Lie q ( ) = T q M for every q ∈ M .

Here Lie( ) is the smallest Lie subalgebra of Vec(M) containing and Lie q ( ) is the linear subspace of T q M whose elements are evaluation at q of elements belonging to Lie( ). The condition that satisfies the Lie bracket generating assumption is known also as the Hörmander condition.

We say that a 2-ARS (E, f, •, • ) is orientable if E is orientable as a vector bundle. Notice that one can build non-orientable 2-ARSs on orientable manifolds and orientable 2-ARSs on nonorientable manifolds. See [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF] for some examples. We say that a 2-ARS (E, f,

•, • ) is trivializable if E is isomorphic to the trivial bundle M × R 2 .
A particular case of 2-ARSs is given by Riemannian surfaces. In this case E = T M and f is the identity.

Let S = (E, f, •, • ) be a 2-ARS on a surface M . We denote by (q) the linear subspace {V (q) | V ∈ } = f(E q ) ⊆ T q M . The set of points in M such that dim( (q)) < 2 is called singular set and denoted by Z. Since is bracket generating, the subspace (q) is nontrivial for every q and Z coincides with the set of points q where is one-dimensional.

The Euclidean structure on E allows to define a symmetric positive definite C ∞ (M )-bilinear form on the submodule by

G : × → C ∞ (M ) G(V, W ) = σ V , σ W where σ V , σ W are the unique 2 sections of E satisfying f • σ V = V, f • σ W = W .
At points q ∈ M where f| Eq is an isomorphism, i.e. on M \ Z, G is a tensor and the value G(V, W )| q depends only on V (q), W (q). In this case G defines a Riemannian metric g via

g q (v, w) = G(V, W )(q), v, w ∈ T q M,
where V and W are two vector fields such that v = V (q) and w = W (q). This is no longer true at points q where f| Eq is not injective.

If (σ 1 , σ 2 ) is an orthonormal frame for •, • on an open subset Ω of M , an orthonormal frame for G on Ω is given by (f • σ 1 , f • σ 2 ).
Orthonormal frames are systems of local generators of .

For every q ∈ M and every v ∈ (q) define

G q (v) = inf{ u, u q | u ∈ E q , f(u) = v}.
An absolutely continuous curve γ : [0, T ] → M is admissible for S if there exists a measurable essentially bounded function

[0, T ] ∋ t → u(t) ∈ E γ(t) , called control function such that γ(t) = f(u(t)) for almost every t ∈ [0, T ]. Given an admissible curve γ : [0, T ] → M , the length of γ is ℓ(γ) = ˆT 0 G γ(t) ( γ(t)) dt.
The Carnot-Caratheodory distance (or sub-Riemannian distance) on M associated with S is defined as

d(q 0 , q 1 ) = inf{ℓ(γ) | γ(0) = q 0 , γ(T ) = q 1 , γ admissible}.
It is a standard fact that ℓ(γ) is invariant under reparameterization of the curve γ. Moreover, if an admissible curve γ minimizes the so-called energy functional E(γ) = ´T 0 G γ(t) ( γ(t)) dt with T fixed (and fixed initial and final point) then v = G γ(t) ( γ(t)) is constant and γ is also a minimizer of ℓ(•). On the other hand a minimizer

γ of ℓ(•) such that v is constant is a minimizer of E(•) with T = ℓ(γ)/v.
The finiteness and the continuity of d(•, •) with respect to the topology of M are guaranteed by the Lie bracket generating assumption on the 2-ARS (see [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]). The Carnot-Caratheodory distance endows M with the structure of metric space compatible with the topology of M as differential manifold.

When the 2-ARS is trivializable, the problem of finding a curve minimizing the energy between two fixed points q 0 , q 1 ∈ M is naturally formulated as the distributional optimal control problem with quadratic cost and fixed final time

q = 2 i=1 u i X i (q) , u i ∈ R , ˆT 0 2 i=1 u 2 i (t) dt → min, q(0) = q 0 , q(T ) = q 1 .
where {X 1 , X 2 } is an orthonormal frame.

Geodesics

A geodesic for S is a curve γ : [0, T ] → M such that for every sufficiently small nontrivial interval

[t 1 , t 2 ] ⊂ [0, T ], γ| [t 1 ,t 2 ] is a minimizer of E(•). A geodesic for which G γ(t) ( γ(t)
) is (constantly) equal to one is said to be parameterized by arclength. The local existence of minimizing geodesics is a standard consequence of Filippov Theorem (see for instance [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]). When M is compact any two points of M are connected by a minimizing geodesic.

Locally, in an open set Ω, if {X 1 , X 2 } is an orthonormal frame, a curve parameterized by arclength is a geodesic if and only if it is the projection on Ω of a solution of the Hamiltonian system corresponding to the Hamiltonian

H(q, λ) = 1 2 ((λX 1 (q)) 2 + (λX 2 (q)) 2 ), q ∈ Ω, λ ∈ T * q Ω. ( 10
)
lying on the level set H = 1/2. This is the Pontryagin Maximum Principle [START_REF] Pontryagin | The Mathematical Theory of Optimal Processes[END_REF] in the case of 2-ARSs. Its simple form follows from the absence of abnormal extremals in 2-ARSs, as a consequence of the Hörmander condition see [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF]. Notice that when looking for a geodesic γ minimizing the energy from a submanifold T (possibly of dimension zero), one should add the transversality condition λ(0)T γ(0) T = 0.

Generic 2-ARSs and normal forms

A property (P ) defined for 2-ARSs is said to be generic if for every rank-2 vector bundle E over M , (P ) holds for every f in an open and dense subset of the set of morphisms of vector bundles from E to T M , endowed with the C ∞ -Whitney topology. Define 1 = and k+1 = k + [ , k ]. We say that S satisfies condition (H0) if the following properties hold: (i) Z is an embedded one-dimensional submanifold of M ; (ii) the points q ∈ M at which 2 (q) is one-dimensional are isolated; (iii) 3 (q) = T q M for every q ∈ M . It is not difficult to prove that property (H0) is generic among 2-ARSs (see [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF]). This hypothesis was essential to show Gauss-Bonnet type results for ARSs in [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF][START_REF] Agrachev | Two-dimensional almost-Riemannian structures with tangency points[END_REF][START_REF] Boscain | High-order angles in almost-Riemannian geometry[END_REF]. The following theorem recalls the local normal forms for ARSs satisfying hypothesis (H0) (see Figure 1).

Theorem 4 ([3]

). Consider a 2-ARS satisfies (H0). Then for every point q ∈ M there exist a neighborhood U of q and an orthonormal frame {X 1 , X 2 } of the ARS on U such that, up to a change of coordinates, q = (0, 0) and {X 1 , X 2 } has one of the forms

(F1) X 1 (x, y) = ∂ ∂x , X 2 (x, y) = e φ(x,y) ∂ ∂y , (F2) X 1 (x, y) = ∂ ∂x , X 2 (x, y) = xe φ(x,y) ∂ ∂y , (F3) X 1 (x, y) = ∂ ∂x , X 2 (x, y) = (y -x 2 ψ(x
))e ξ(x,y) ∂ ∂y , where φ, ψ and ξ are smooth functions such that φ(0, y) = 0 and ψ(0) = 0.

Let S be a 2-ARS satisfying (H0). A point q ∈ M is said to be an ordinary point if (q) = T q M , hence, if S is locally described by (F1). We call q a Grushin point if (q) is one-dimensional and 2 (q) = T q M , i.e., if the local description (F2) applies. Finally, if (q) = 2 (q) has dimension one and 3 (q) = T q M then we say that q is a tangency point and S can be described near q by the normal form (F3).

Notice that under hypotheses HA and HB of Theorem 1, H0 is fulfilled and there are no Tangency points.

In the compact case, the following proposition permits to extend the normal form (F2) to a neighborhood of a connected component of Z, when there are no tangency points. Proposition 2. Consider a 2-ARS on a compact orientable manifold satisfying (H0). Let W be a connected component of Z containing no tangency points. Then there exists a tubular neighborhood U of W and an orthonormal frame {X 1 , X 2 } of the 2-ARS on U such that, up to a change of coordinates, W = {(0, y), y ∈ S 1 } and {X 1 , X 2 } has the form

X 1 (x, y) = ∂ ∂x , X 2 (x, y) = xe φ(x,y) ∂ ∂y . (11)
Proof. The proof consists in using as first coordinates the distance from W and it is very similar to the proof of Theorem 4, which contains a local version of Proposition 2 (see [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF] pp. 813-814, proof of Lemma 1 and proof of Theorem 1 for Grushin points). Here we just explain which modifications are necessary. Since M is assumed to be compact then W is diffeomorphic to S 1 . Moreover, since M is assumed to be orientable, a sufficiently small open tubular neighborhood of W is diffeomorphic to (-1, 1) × S 1 . Then consider a smooth regular parametrization S 1 ∋ α → w(α) of W .

Let α → λ 0 (α) ∈ T * w(α) M be a smooth map satisfying H(λ 0 (α), w(α)) = 1/2, where H is the Hamiltonian of the PMP (10) and λ 0 (α) ⊥ T w(α) W . Notice that such a map exists since we are assuming that M is orientable.

Let E(t, α) be the solution at time t of the Hamiltonian system given by the Pontryagin Maximum Principle with initial condition (q(0), λ(0)) = (w(α), λ 0 (α)). With the same arguments given in the proof of Lemma 1 in [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF], one shows that E(t, α) is a local diffeomorphism around every point of the type (0, ᾱ), ᾱ ∈ S 1 . Using the fact that E is a global diffeomorphism from {0}× S 1 to W and by suitably reducing ε one gets that it is a diffeomorphism between (-ε, ε) × S 1 (for some ε > 0) and a tubular neighborhood of W . This permits to use as coordinates in U the pair (t, α). Indeed t is the distance from W which we proved to be smooth in U . As in [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF], one builds an orthonormal frame in U defining the vector field X 1 by

X 1 (t, α) = ∂ t E(t, α) = ∂ t .
As in [START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF] one obtains that the second vector field of the orthonormal frame has to be of the form te φ(t,α) ∂ ∂α . Remark 3. Notice that in Proposition 2, the hypothesis that M is orientable can be weakened by requiring that W admits an open tubular neighborhood diffeomorphic to (-1, 1) × S 1

The Grushin case

As already mentioned the Grushin plane is the trivializable almost Riemannian metric on the (x, y) plane for which an orthonormal basis is given by

X 1 = 1 0 , X 2 = 0 x . (12)
In the sense of Definition 1 it can be seen as a triple (E, f, •, • ) where E = R 2 ×R 2 , f((x, y), (a, b)) = ((x, y), (a, bx)) and •, • is the standard Euclidean metric.

Geodesics of the Grushin plane

In this section we recall how to compute the geodesics for the Grushin plane, with the purpose of stressing that they can cross the singular set with no singularities. Setting q = (x, y) and λ = (λ 1 , λ 2 ), the Hamiltonian ( 10) is given by

H(x, y, λ 1 , λ 2 ) = 1 2 (λ 2 1 + x 2 λ 2 2 ) ( 13
)
and the corresponding Hamiltonian equations are:

ẋ = λ 1 , λ1 = -xλ 2 2 ẏ = x 2 λ 2 , λ2 = 0 (14)
Geodesics parameterized by arclength are projections on the (x, y) plane of solutions of these equations, lying on the level set H = 1/2. We study the geodesics starting from i) a Grushin point, e.g. (0, 0) ii) an ordinary point, e.g. (-1, 0).

Case (x(0), y(0)) = (0, 0) In this case the condition H(x(0), y(0), λ 1 (0), λ 2 (0)) = 1/2 implies that we have two families of geodesics corresponding respectively to λ 1 (0) = ±1, λ 2 (0) =: a ∈ R. Their expression can be easily obtained and it is given by:

x(t) = ±t, y(t) = 0 if a = 0 x(t) = ± sin(at) a , y(t) = 2at-sin(2at) 4a 2 if a = 0 (15)
Some geodesics are plotted in Figure 2 together with the "front" i.e. the end point of all geodesics at time t = 1. Notice that geodesics start horizontally. The particular form of the front shows the presence of a conjugate locus accumulating to the origin.

Case (x(0), y(0)) = (-1, 0) In this case the condition H(x(0), y(0), λ 1 (0), λ 2 (0)) = 1/2 becomes λ 2 1 + λ 2 2 = 1 and it is convenient to set λ 1 = cos(θ), λ 2 = sin(θ), θ ∈ S 1 . The expression of the geodesics is given by: Some geodesics are plotted in Figure 3 together with the "front" at time t = 4.8. Notice that geodesics pass horizontally through Z, with no singularities. The particular form of the front shows the presence of a conjugate locus. Geodesics can have conjugate times only after intersecting Z. Before it is impossible since they are Riemannian and the curvature is negative.

         x(t) = t -1, y(t) = 0 if θ = 0 x(t) = -t -1, y(t) = 0 if θ = π x(t) = - sin(θ -t sin(θ)) sin(θ) y(t) = 2t -2 cos(θ) + sin(2θ-2t sin(θ)) sin(θ) 4 sin(θ) if θ / ∈ {0, π} (16)

The Laplace-Beltrami operator on the Grushin plane

In this section we illustrate in details the results of the paper in the Grushin case. For the study of the Laplace Beltrami operator it will be convenient to compactify the y direction considering x ∈ R and y ∈ T.

As mentioned in the introduction, the Laplace-Beltrami operator on the Grushin plane can be written as

∆ = ∂ 2 x + x 2 ∂ 2 y - 1 x ∂ x .
We expect that the Laplacian is the Friedrichs extension associated to the positive quadratic form

Q(u, u) = ˆR2 |∇u| 2 dω = ˆR2 1 |x| |∂ x u| 2 + |x| |∂ y u| 2 dxdy
Let us make the change of variable f = |x|g which is unitary from

L 2 (R × T, dω) to L 2 (R × T, dx dy), so that (f 1 , f 2 ) L 2 (R×T,dω) = (g 1 , g 2 ) L 2 (R×T,dx dy) .
We compute the operator in the new variable:

∆f = ∂ 2 x f + x 2 ∂ 2 y f - 1 |x| ∂ x f = |x| ∂ 2 x g + x 2 ∂ 2 y g - 3 4x 2 g =: |x|Lg.
Hence we are left to study the operator

Lg = ∂ 2 x g + x 2 ∂ 2 y g -3 4x 2 g on L 2 (R × T, dx dy).
Decomposing a function g(x, y) in the Fourier basis in the variable y, we get the decomposition

L 2 (R × T) ≈ ⊕ ⊥ k∈Z H k
where H k = L 2 (R) and the operator L acts on each H k by

(Lu) k (x) = ∂ 2 x u k -x 2 k 2 u k - 3 4x 2 u k = ∂ 2 x u k -V k u k . with V k (x) ≥ 3 4x 2 .
But, we know that in dimension 1, the operator [START_REF] Reed | Methods of modern mathematical physics[END_REF], Theorem X.10 for the proof of the limit point case at 0 and Theorem X.8 at +∞). Hence, each operator ∂ 2

-∂ 2 x + V with domain C ∞ 0 (]0, +∞[) is essentially self-adjoint on L 2 (]0, +∞[) if V ≥ 3 4x 2 (see
x u k -V k is essentially self-adjoint. As a consequence L is essentially self-adjoint as well.

Remark 4. V = 3 4x 2 is exactly the limit singular potential to have -∂ 2 x + V in the limit point case at zero and hence essentially self-adjoint (see [START_REF] Reed | Methods of modern mathematical physics[END_REF]Theorem X.10,page 159]). It is surprising to find the same constant 3 4 for the Grushin operator. Notice that if we consider as orthonormal basis of the metric the vector fields X = 1 0 and Y = 0 |x| α with α > 0, we obtain for the Laplace-Beltrami operator,

∆u = ∂ 2 x u + |x| 2α ∂ 2 y u - α x ∂ x u.
Making the change of variable u = |x| α/2 v (which is unitary from L 2 (R×T, dω) to L 2 (R×T, dx dy)) we get for the transformed operator

∆v = ∂ 2 x v - α 2 α 2 + 1 1 x 2 + |x| 2α ∂ 2 y v,
which is essentially self-adjoint if and only if α ∈ [1, +∞[.

1/α √ Grushin

In this section, motivated by Remark 4, we briefly discuss the generalized Riemannian structures on R 2 for which an orthonormal basis is given by

X 1 = 1 0 , X 2 = 0 |x| α , α ∈ (0, 1). ( 17
)
Notice that this generalized-Riemannian structure does not fit the definition of almost-Riemannian structures, since the vector fields are not smooth. As usual call Z the set in which the two vector fields are not linearly independent. It is interesting to notice the following facts, which in some cases are very different with respect to the standard Grushin case:

• The Lie bracket between X 1 and X 2 is not well defined on Z. Indeed [X 1 , X 2 ] = (0, α|x| (α-1) ) T . However it is not difficult to prove that the corresponding distance is continuous and endows R 2 with its standard topology (note that it is not the case if α > 0).

• The curvature is given by K = -α(1+α)

x 2

and the area element by dω = 1 |x| α dx dy. Hence the area element explodes, but the area of a bounded open set intersecting Z is finite (and its diameter is finite as well). This is certainly the reason why the Laplace-Beltrami operator is not essentially self-adjoint and, as a consequence, a quantum particle or the heat flow can pass through the set Z. Notice that for the Grushin plane a bounded open set intersecting Z has finite diameter but infinite area.

• There exists regular curves passing through Z with a velocity not belonging to the span of X 1 and X 2 and having finite length. For instance take α = 1/2 and the curve γ(t) := (t, t), t ∈ [-1, 1]. We have γ(0) / ∈ span{X 1 (γ(0)), X 2 (γ(0)} but its length is finite. Indeed it corresponds to the controls u 1 = 1 and

u 2 = 1/|t| 1/2 . Hence ℓ(γ) = ´1 -1 1 + |t| -1 dt = 2( √ 2 + arcsinh( 1
)) ∼ 4.59. Such phenomenon does not appear on the singular set of a 2-ARS, as a consequence of the smoothness of orthonormal frames.

4 Laplace-Beltrami on an Almost-Riemannian structure

Computation in local coordinates

The aim of this subsection is the proof of the following theorem.

Theorem 5. Let g a Riemannian metric on Ω = {x = (x, y) ∈ R × T |x = 0 } of the form diag 1, x -2 e -2Φ(x,y)
where Φ is a smooth function which is constant for large |x|. Let ∆ be the corresponding Laplace-Beltrami operator with domain C ∞ 0 (Ω).

Then, ∆ is essentially self-adjoint for L 2 (R×T, dω), where dω is the corresponding Riemannian area on Ω.

The same result holds by replacing Ω with Ω ± where

Ω -= {x = (x, y) ∈ R × T |x < 0 } and Ω + = {x = (x, y) ∈ R × T |x > 0 }.
A simple consequence of this theorem is that the only self-adjoint Laplace operator that can be constructed by extension of the natural one (i.e. the Laplace-Beltrami operator defined on

C ∞ 0 (Ω)) preserves the decomposition L 2 (Ω, dω) = L 2 (Ω -, dω) ⊕ ⊥ L 2 (Ω + , dω) .
Indeed, if we denote ∆ -(resp ∆ + ) the unique self-adjoint extension of the natural Laplace operator with domain C ∞ 0 (Ω -) (resp C ∞ 0 (Ω + )), we can define the operator ∆(u -+ u + ) = ∆ -u -+ ∆ + u + with the natural domain inherited from the one of ∆ -and ∆ + . The operator ∆ is a selfadjoint extension of the Laplace-Beltrami operator with domain C ∞ 0 (Ω) and it is the unique one by Theorem 5. Another way of seeing this fact is from the point of view of the evolution equation.

Corollary 2. With the notations of Theorem 5, consider the unique solution u of the Schrödinger equation (according to the self-adjoint extension defined in the previous theorem),

i∂ t u + ∆u = 0 u(0) = u 0 ∈ L 2 (R × T, dω) (18)
with u 0 supported in Ω + . Then, u(t) is supported in Ω + for any t ≥ 0. The same holds for the solution of the heat or for the solution of the wave equation.

This corollary is a simple consequence of the essential self-adjointness of the Laplace-Beltrami operator as discussed above. This is in strong contrast with the classical dynamics associated with this metric where the geodesics can "cross" the barrier {x = 0} (see Figure 3). The proof of Theorem 5 relies on a change of variable similar to the one we did for the Grushin operator which leads to an operator that can be written as "Grushin type + singular potential". Then, we follow the proof of Kalf and Walter [START_REF] Kalf | Note on a paper of Simon on essentially self-adjoint Schrödinger operators with singular potentials[END_REF] (which was himself inspired by B. Simon [START_REF] Simon | Essential self-adjointness of Schrödinger operators with singular potentials[END_REF]) using Kato inequality for the main part. The other part can be treated by perturbation theory. Note that some related results for singular potential were proven in [START_REF] Donnelly | Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues[END_REF] and [START_REF] Maeda | Essential selfadjointness of Schrödinger operators with potentials singular along affine subspaces[END_REF].

Proof of Theorem 5. In coordinates, the Laplace-Beltrami operator has the form

∆ = 1 ω i,j ∂ i ωg ij ∂ j
where we have denoted ω = √ detg the Riemannian volume.

For u ∈ L 2 (R × T, dω), let us make the change of variable u = ω -1/2 v which is unitary from L 2 (R × T, dω) to L 2 (R × T, dx dy) and let us compute its action in the new variable.

ω 1/2 ∆ω -1/2 v := ∆v = 1 ω 1/2 i,j ∂ i ωg ij ∂ j (ω -1/2 v) = 1 ω 1/2 i,j ∂ i ωg ij (∂ j ω -1/2 )v + g ij (ω 1/2 )∂ j v = 1 ω 1/2 i,j ∂ i g ij ω(∂ j ω -1/2 ) v + g ij ω(∂ j ω -1/2 )∂ i v + (∂ i g ij )ω 1/2 ∂ j v + g ij (∂ i ω 1/2 )∂ j v + g ij (ω 1/2 )∂ ij v = 1 ω 1/2 i,j ∂ i g ij ω(∂ j ω -1/2 ) v + g ij ω(∂ i ω -1/2 )∂ j v + (∂ i g ij )ω 1/2 ∂ j v + g ij (∂ i ω 1/2 )∂ j v + g ij (ω 1/2 )∂ ij v
where we have used the symmetry of g.

We have:

ω(∂ i ω -1/2 ) + (∂ i ω 1/2 ) = -1 2 ωω -3/2 ∂ i ω + 1 2 ω -1/2 ∂ i ω = 0. Moreover let µ := i,j 1 ω 1/2 ∂ i g ij ω(∂ j ω -1/2 ) = - i,j 1 2ω 1/2 ∂ i g ij (∂ j ω)ω -1/2 ) = i,j - 1 2 ∂ i (g ij )∂ j (ln ω) - 1 2ω g ij ∂ ij ω + 1 4ω 2 g ij (∂ i ω)(∂ j ω) .
Hence we get ∆v = i,j

g ij ∂ ij v + (∂ i g ij )∂ j v + µv
= div e (∇v) + µv

= div e (G -1 ∇ e v) + µv.

In the last formula we have used matrix notation where G -1 = (g ij ) and div e , ∇ e are the divergence and the gradient of the Euclidian space. Now let us specify the computation in our diagonal metric.

ω = 1 |x| e -Φ , ∂ 1 ω = - sgn(x) |x| 2 e -Φ - ∂ 1 Φ |x| e -Φ , ∂ 2 ω = - ∂ 2 Φ |x| e -Φ ; ∂ 2 2 ω = (∂ 2 Φ) 2 |x| e -Φ - ∂ 2 2 Φ |x| e -Φ , ∂ 2 1 ω = 2 |x| 3 e -Φ + 2sgn(x)∂ 1 Φ |x| 2 e -Φ + (∂ 1 Φ) 2 |x| e -Φ - ∂ 2 1 Φ |x| e -Φ , µ = - 1 |x| 2 - sgn(x)∂ 1 Φ |x| - (∂ 1 Φ) 2 2 + ∂ 2 1 Φ 2 + 1 4|x| 2 + sgn(x)∂ 1 Φ 2|x| + (∂ 1 Φ) 2 4 +x 2 e 2Φ (∂ 2 Φ) 2 + 1 2 ∂ 2 2 Φ -(∂ 2 Φ) 2 + 1 4 (∂ 2 Φ) 2 = - 3 4|x| 2 - sgn(x)∂ 1 Φ 2|x| - (∂ 1 Φ) 2 4 + ∂ 2 1 Φ 2 +x 2 e 2Φ 1 2 ∂ 2 2 Φ + 3 4 (∂ 2 Φ) 2 := - 3 4|x| 2 -µ 2 .
Now, we are able to conclude by the following two Lemmas. The first one proves that the main part without µ 2 is essentially self-adjoint while the second one treats µ 2 as a perturbation.

Lemma 1. Let G -1 given by diag 1, x 2 e 2Φ(x,y) with |Φ| ≤ C for C > 0.

The operator L : v → -div e (G -1 ∇ e v) + 3 4|x| 2 v defined on L 2 (R × T, dx dy) with domain C ∞ 0 (Ω) is positive and essentially self-adjoint.

Hence, since L defines a positive symmetric operator, its unique self-adjoint extension is the Friedrichs extension (in the following still denoted by L).

Lemma 2. The operator of multiplication by µ 2 (defined with domain C ∞ 0 (Ω))) is infinitesimally small with respect to L. So, by the theorem of Kato-Rellich (Theorem X.12 of [START_REF] Reed | Methods of modern mathematical physics[END_REF]), L + µ 2 is essentially self-adjoint.

To conclude the proof of Theorem 5 we are left to prove the two lemmas.

Proof of Lemma 1. We follow closely the proof of Kalf and Walter [START_REF] Kalf | Note on a paper of Simon on essentially self-adjoint Schrödinger operators with singular potentials[END_REF]. Actually, we mimick the proof of the 1-D case and we use the fact that our operator is the classical Laplacian with potential for function only depending on x: -∂ 2 1 + 3 4|x| 2 . It is enough to prove that Ran(L + I) is dense in L 2 , see Theorem X.26 of [START_REF] Reed | Methods of modern mathematical physics[END_REF]. So, let h ∈ L 2 (Ω) such that h ⊥ Ran(L + I) and let us prove h = 0 (we can assume h real valued without loss of generality).

Set β(x) = |x| 3/2 e -|x| 1+|x| 3/2 . We have β(x) ≥ 0 and β ∈ L 2 (Ω). We obtain, for x = 0,

-div e (G -1 ∇ e β) + 3 4|x| 2 β + β = -∂ 2 1 β + 3 4|x| 2 β + β = 3|x| 1/2 e -|x| 4(1 + |x| 3/2 ) 3 (|x| 2 + 4|x| 3/2 + 7|x| 1/2 + 4) := X(x) Let η, ζ ∈ C ∞ ([0, +∞[) so that η(s) = 0 if s ≤ 1/2 = 1 if s ≥ 1 ζ(s) = 1 if s ≤ 1 = 0 if s ≥ 2 Denote β n (x) = η(n|x|)ζ( |x| n )β(x) ∈ C ∞ 0 (Ω) so that Lβ n is well defined in L 2 (Ω)
. Moreover, we have

β(x) = O(|x| 3/2 ), |∇β(x)| = O(|x| 1/2 ), as x → 0 β(x) = O(e -|x| ), |∇β(x)| = O(e -|x| ), as x → ∞ so, Lβ n + β n = η(n|x|)ζ( |x| n )X -2∂ 1 η(n|x|)ζ( |x| n ) ∂ 1 β -∂ 2 1 η(n|x|)ζ( |x| n ) β converges weakly in L 2 to X ∈ L 2 (Ω) such that X(x) ≥ 0. So, we can write (X, |h|) L 2 = lim n→∞ (Lβ n + β n , |h|) L 2 ≥ 0
But, we will use the Kato inequality: Lemma 3 (Kato's inequality, see Lemma A of [START_REF] Kato | Schrödinger operators with singular potentials[END_REF]). For v ∈ L 1 loc (Ω) real valued such that div e (G -1 ∇ e v) ∈ L 1 loc (Ω), we have, in the sense of distributions on Ω

-div e (G -1 ∇ e |v|) ≤ -(sgnv)div e (G -1 ∇ e v).

Notice that the Kato inequality of [START_REF] Kato | Schrödinger operators with singular potentials[END_REF] can only be applied on Ω ε := {(x, y) ∈ Ω : |x| > ε} where the metric is positive definite. This does not create difficulties since we are applying this Lemma in the sense of distributions on Ω.

Applying the Kato Lemma to h and β n > 0 as test function, we get

(Lβ n + β n , |h|) L 2 ≤ (β n , (sgnh)(Lh + h)) L 2 ≤ 0. So (X, |h|) L 2 = 0 and h ≡ 0.
Proof of Lemma 2. The quadratic form associated to L + 1 is

Q(u, u) = ˆΩ(G -1 ∇ e u • ∇ e u) + 3 4|x| 2 |u| 2 + |u| 2 dx dy For u ∈ C ∞ 0 (Ω)
, and by Cauchy-Schwarz inequality, we have

Q(u, u) = (Lu, u) L 2 + u 2 L 2 ≤ Lu L 2 u L 2 + u 2 L 2 ≤ ε Lu 2 L 2 + 1 4ε + 1 u 2 L 2
Moreover, since Φ is smooth and constant for large x, we have

µ 2 u 2 L 2 ≤ C u |x| L 2 + u 2 L 2 ≤ CQ(u, u) ≤ Cε Lu 2 L 2 + (C ε + 1) u 2 L 2 .
This yields the result since ε is arbitrary small and the estimate can be extended to any u ∈ D(L) by taking closure.

The proof of Theorem 5 is concluded.

The next Lemma is the first step to establish the compactness of the resolvent of ∆ in the compact case. Of course, since here we are dealing with the case of R × T, we can not expect compactness without adding a cut-off function.

Lemma 4. Denote by ∆ the positive self-adjoint operator defined by Theorem 5. Then, the truncated resolvent ρ(x)(-∆ + 1) -1 , where ρ ∈ C ∞ 0 (R), is well-defined and compact from L 2 (R × T, dω) to itself.

Proof. The fact that (-∆ + 1) -1 is well defined comes from the positivity of -∆. For the compactness it is equivalent to prove that the operator ρ(x)(-∆ + 1) -1 is compact when defined on L 2 (R × T, dx dy) with the metric induced by the Lebesgue measure.

We begin by showing that it is the case for L defined above. Let u n be a real valued sequence in L 2 (R × T, dx dy) with L 2 norm bounded by 1. Denote

f n := ρ(x)(L + 1) -1 u n := ρ(x)f n . In particular, Q(f n , f n ) = ((L + 1)f n , f n ) = (u n , f n ) = ´Ω G -1 ∇ e f n • ∇ e f n + 3 4|x| 2 |f n | 2 + |f n | 2 dx dy is bounded in L 2 (R × T, dx dy). Set ε > 0. Take ϕ ∈ C ∞ 0 (R), supported in ] -ε, ε[ with 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 in a neighborhood of 0. Split f n = ϕ(x) f n + (1 -ϕ(x)) f n := f n,1 + f n,2 . Then, we have |xϕ(x)| 2 ≤ ε 2 and ϕf n 2 L 2 = ´|xϕ(x)| 2 fn |x| 2 ≤ ε 2 4 3 Q(f n , f n ) ≤ Cε 2 . This gives also f n,1 L 2 ≤ Cε. Now that ϕ is fixed, f n,2 = (1 -ϕ(x)) f n = (1 -ϕ(x))ρ(x)f n is supported in a fixed compact of Ω and bounded in H 1 . Indeed, ∇ e f n,2 2 L 2 ≤ 2( (1 -ϕ(x))ρ(x)∇ e f n 2 L 2 + ∂ x (ϕ(x)ρ(x))f n 2 L 2 ) ≤ CQ(f n , f n ) ≤ C
where the second inequality comes from the fact that there exist a constant C such that Id ≤ CG -1 (in the sense of quadratic forms) on the support of (1ϕ(x))ρ(x). We conclude by invoking the compact embedding of H 1 into L 2 on compact sets so that f n,2 is convergent up to extraction. Actually, we have proved that for any ε > 0, we can find an extraction γ such that f γ(n) can be written

f γ(n) = f γ(n),1 + f γ(n),2 with f γ(n),1 L 2 ≤ ε and f γ(n),2 convergent. By choosing ε = 1
p , p ∈ N, and by a diagonal extraction argument, we easily get that we can find a subsequence such that f γ(n) is a Cauchy sequence and converges. This gives that ρ(L + 1) -1 is compact. It remains to prove the same result for L + µ 2 . outside of the zone of degeneracy Z where u is C ∞ . So, we get -∆ i u i + u i ∈ C ∞ 0 (R * × T) and so -∆ i u i + u i ∈ L 2 (M, dω i ). This is true in the sense of distributions, but that means that for any ϕ ∈ C ∞ 0 (R * × T), we have

(u i , -∆ i ϕ + ϕ) L 2 (R * ×T,dω i ) ≤ C ϕ L 2 (R * ×T,dω i ) .
In particular, that means that u i belongs to the domain of the adjoint of the operator -∆ i + I with domain C ∞ 0 (R * × T). But Theorem 5 gives that this operator is essentially self-adjoint so, u i ∈ D(L i ) where L i is the self-adjoint (Friedrichs) extension of -∆ i + I on R * × T. In particular, we can write (the right hand side has to be understood as a limit for a sequence u i,n in C ∞ 0 (Ω) converging strongly to u i for the norm of the quadratic form)

(u i , -∆ i u i + u i ) L 2 (R * ×T,dω i ) = ∇ i u i 2 L 2 (R * ×T,dω i ) + u i 2 L 2 (R * ×T,dω i ) .
This quantity also makes sense when u i is considered on the manifold M and the extension of the metric was chosen so that -∆ i u i + u i = -∆u i + u i in the sense of ditributions and ∇ i u i = ∇u i .

The same result holds for i ∈ I corresponding to a Riemannian zone. Moreover, if i = j ∈ I, the common support of u i and u j do not intersect the degeneracy zone Z. So, these functions are C ∞ in this zone and we can write (u i , -∆u j + u j ) L 2 (M ) = (∇u i , ∇u j ) L 2 (M ) + (u i , u j ) L 2 (M )

By summing up, we get 0 = (u, -∆u + u) L 2 (M,dω) = (∇u, ∇u) L 2 (M,dω) + (u, u) L 2 (M,dω) and u ≡ 0.

From now on, we will use the same notation ∆ for the self-adjoint extension of the symmetric operator ∆.

To prove the second statement of Theorem 1 we have to prove the following.

Proposition 4. The domain of ∆ is given by

D(∆) = u ∈ L 2 (M, dω) ∆ D,g u ∈ L 2 (M, dω) (19)
where ∆ D,g u is ∆u seen as a distribution in Ω = M \ Z.

Proof. Let T be the Laplace operator defined with domain given by [START_REF] Franchi | Une métrique associée à une classe d'opérateurs elliptiques dégénérés[END_REF]. We easily see that

D(T ) = u ∈ L 2 (M, dω) |∃C : ∀ϕ ∈ C ∞ 0 (Ω), |(u, ∆ϕ) L 2 | ≤ C ϕ L 2 = D(∆ * ) = D(∆).
To prove the third statement of Theorem 1 we have to prove the following.

Proposition 5. Denote by ∆ the positive self-adjoint operator defined by Proposition 3, then the resolvent (-∆ + 1) -1 is well defined and compact from L 2 (M, dω) to itself with the measure defined by the metric. Therefore, the spectrum of ∆ is discrete and consists of eigenvalues of finite multiplicity.

Proof. The fact that (-∆ + 1) -1 is well defined comes from the positivity of -∆. Now let u n be a bounded sequence in L 2 (M, dω) and f n = (-∆ + 1) -1 u n . By density, we can assume u n in C ∞ 0 (M \Z) and f n ∈ C ∞ (M \Z) by elliptic regularity. Then, we have

´M |∇f n | 2 +|f n | 2 = (u n , f n ) L 2 bounded.
Consider the partition of unity (Ψ i ) i∈I introduced in the proof of the previous Theorem and denote f i,n := Ψ i f n .

Let i ∈ I be an index corresponding to a "Grushin zone". By the formula

(-∆ + I)f n,i = Ψ i u n -2∇Ψ i • ∇f n -(∆Ψ i )f n ,
we have that (-∆ + I)f n,i is bounded in L 2 (M ). By using Lemma 4, we get that ρ(x)(-∆ i + 1) -1 (-∆ + I)f n,i is compact for a function ρ ∈ C ∞ 0 (R) defined in some local coordinate charts. Here, we have denoted ∆ i the Laplacian for an extension of the local metric to R × T. To finish, we only have to notice that (-∆ i + 1) -1 (-∆ + I)f n,i = (-∆ i + 1) -1 (-∆ i + 1)f n,i = f n,i because of the support of f n,i (note that it is not the case for (-∆ i + 1) -1 because the resolvent depends on the extension).

So, choosing ρ such that ρ(x)f n,i = f n,i in local coordinates, we have proved that each sequence f n,i is compact for a Grushin zone. The same result holds for a Riemannian zone because the equivalent of Lemma 4 still holds for any Riemannian extension. By summing up, we get that the sequence f n is convergent, up to extraction, which yields the result.

A The Martinet case

In this Section, we prove Theorem 3. A sub-Riemannian structure is a triple (M, , g), where M is a smooth manifold, is a smooth vector distribution of constant rank satisfying the Hörmander condition, and g is a Riemannian metric on . Let 1 = and k+1 = k + [ , k ]. A sub-Riemannian structure is said to be equiregular if the dimension of k does not depend on the point.

Almost-Riemannian and sub-Riemannian structures can be treated in the unified setting of rank-varying sub-Riemannian structures, see [START_REF] Agrachev | Introduction to Riemannian and sub-Riemannian geometry[END_REF]Chapter 3] and [4, Definition 2]. Here for sake of readability we omit this point of view.

In this section, we briefly treat the Martinet sub-Riemannian manifold (see for instance [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF][START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF]), defined by M = R 3 , (q) = span{X 1 (q), X 2 (q)}, and g(X i , X j ) = δ ij , i, j = 1, 2, where

X 1 =   1 0 y 2 2   , X 2 =   0 1 0   . (20)
Notice that X

(Figure 1 :

 1 Figure1: The local representations established in[START_REF] Agrachev | A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF]. Tangency points are the most difficult to handle due to the fact that the asymptotic of the distance is different from the two sides of the singular set (see[START_REF] Bonnard | The Sphere and the Cut Locus at a Tangency Point in Two-Dimensional Almost-Riemannian Geometry[END_REF][START_REF] Boscain | A normal form for generic 2-dimensional almostriemannian structures at a tangency point[END_REF]).

Figure 2 :

 2 Figure 2: Geodesics and front for the Grushin plane, starting from the singular set.

Figure 3 :

 3 Figure 3: Geodesics and front for the Grushin plane, starting from a Riemannian point

3 = [X 1 , X 2 X 1 ,

 3121 X 2 and X 3 span the tangent space at any point outside the so called Martinet plane Z = {y = 0}, which is the region where the structure is not equiregular. However X4 = [[X 1 , X 2 ], X 2 ] = [X 3 , X 2 ] =the Hörmander condition is fulfilled on the whole space.

This research has been supported by the European Research Council, ERC StG

GeCoMethods, contract number 239748, by the ANR Project GCM, program Blanche, project number NT09-504490 and by the DIGITEO project CONGEO.

the uniqueness is consequence of the fact that we assume E of rank two and Lie bracket generating.

Again, let u n be a sequence in L 2 (R × T, dx dy) with L 2 norm bounded by 1. Denote f n := ρ(x)(L + µ 2 + 1) -1 u n := ρ(x)f n . Thanks to Lemma 2, we get

so that we get by absorption for ε small enough that (L + 1)f n is bounded. Since we have proved that ρ(L + 1) -1 is a compact operator, we get that f n = ρ(x)f n = ρ(L + 1) -1 (L + 1)f n is relativelly compact.

Case of a compact manifold

In this section, using the previous results, we prove the three statements of Theorem 1. For the first statement, we have to prove the following.

Proposition 3. Consider a 2-ARS on a compact manifold M satisfying hypotheses HA and HB.

Denote by ∆ the Laplace operator defined for functions of C ∞ 0 (Ω) where Ω = M \ Z. Then, ∆ is essentially self-adjoint on L 2 (M, dω).

Proof. Let M = i∈I Ω i be a finite covering of M such that for every connected component W of Z there exists i ∈ I such that W ⊂ Ω i and W ∩ Ω j = ∅ for j ∈ I, j = i and an orthonormal frame for the 2-ARS in Ω i is given by [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF]. Moreover assume that if Ω j (j ∈ I) does not contain any Grushin point, then an orthonormal frame for the 2-ARS in Ω i is given by the normal form (F1). This is possible thanks to Theorem 4 and Proposition 2.

Theorem 5 yields the result in local coordinates around a connected component of the singular set. We only have to extend it to the whole manifold. So, let (Ψ i ) i∈I be a partition of unity associated to

We can also assume that Ψ i ≡ 0 or Ψ i ≡ 1 in a neighborhood of Z, so that ∇Ψ i and ∆Ψ i are C ∞ compactly supported in Ω. Here we use the fact that we have global coordinates around every connected component of Z.

So again, let u ∈ L 2 (Ω) such that u ⊥ Ran(-∆ + I) and let us prove u = 0 (we can assume u real valued without loss of generality). We denote

Since the domain of definition of ∆ contains C ∞ 0 (Ω), we get that u is solution of -∆u + u = 0 in the sense of distributions on Ω. By elliptic regularity, we get that u ∈ C ∞ (Ω). So, the only problem is the possibility to compute integration by part around the degeneracy points.

Let Ω i with i ∈ I be an open set around which we can find a coordinate system so that the metric g takes the diagonal form 1, x -2 e -2Φ(x,y) that we extend arbitrarily on R × T with Φ smooth constant for large x. We will denote ∆ i , ∇ i and dω i the Laplacian, gradient and area corresponding to this extension on R × T. Since u i ∈ L 2 (Ω i ), we can consider the function u i in this local coordinates (in what follows we will not distinguish u i with its representent in local coordinates) and make the computation (in the sense of distributions of Ω and of R * × T in local coordinates):

Remark that in these local coordinates u has only a true meaning for small x but we can then extend this equality on R * × T since Ψ i is compactly supported. Moreover, ∇ i Ψ i and ∆ i Ψ i is supported With this structure, R 3 \ Z is a 3D sub-Riemannian contact manifold. On such a structure it is possible to define intrinsically a volume form which is given by dX 1 ∧ dX 2 ∧ dX 3 , where {dX 1 , dX 2 , dX 3 } is the dual basis to {X 1 , X 2 , X 3 }. Such a volume form is independent on the choice of the orthonormal basis which define the sub-Riemannian structure, and it is called the Popp measure (see [START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF] and Proposition 8 of [START_REF] Agrachev | The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups[END_REF]). One gets dX 1 = dx, dX 2 = dy and dX 3 = y 2 dx -1 y dz so that dX 1 ∧ dX 2 ∧ dX 3 = -1 y dx ∧ dy ∧ dz and the corresponding density is dω = 1 |y| dxdydz. This allows to define the sub-Riemannian Laplacian as the divergence of the sub-Riemannian gradient defined as in formula (1) (see Remark 14 of [START_REF] Agrachev | The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups[END_REF]),

Notice that the singularity in the first order term of ∆ sr appears similarly as it does in the Grushin case. We do the same reasoning.

For simplicity, we compactify in x and z and consider the same structure on M = T x × R y × T z . We will prove that ∆ sr is essentially self-adjoint on L 2 (M, dω) with domain C ∞ 0 (M \ Z). Again, we make the change of variable f = |y|g so that

So, we are left to prove that ∆ sr

We compute the Fourier transform in x and z and get the decomposition

where with V (k,l) (y) ≥ 3 4y 2 . We conclude as in the Grushin case. This result suggests the general conjecture that for a sub-Riemannian structure which is rankvarying or not equiregular on an hypersurface, the singular set acts as a barrier for the heat flow and for a quantum particle.


Serge Thibault

Laura Verdelli

PLANNING PROCESS, BETWEEN EXPERT PROJECT AND COLLECTIVE ACTION

Keywords: Project sciences, planning process, complexity

The content of this paper concerns some recent evolutions of the planning process in France.

Analyzing two study cases, we'll show that today, two types of process are coexisting and, in a way, confronting. The first one corresponds to the "classic" plan for which the project is the principal stage of a linear process. This linear plan is organized by two dominant actors, the contractor and the project manager. For the second, the project is not the main phase of a circular process which contains five main steps. This second emerging type questions the expert as the only authorized actor to establish the project. The project itself becomes second in regard to the definition of the term of the situation and the emergence of some "collectives of action" including all civil society. The described evolution

is not yet the object of a precise knowledge which participates of the construction of project sciences.

Our approach aims to contribute to project sciences.

Introduction

The content of this paper concerns some recent evolutions of the planning process in France. We will establish two major conceptions, one dating back to the moment of its formalization during the 1970s, the other one being contemporary. Our results come from cross checked data of numerous research projects developed by the CITERES Centre Interdisciplinaire CItés, TERritoires, Environnement et Sociétés laboratory. Planning process is enquired across different laboratory actions, even not being the central focus its constant presence in laboratory activities drove us a couple of years ago to design a transversal axis re-reading strategic planning under a different angle. Our approach aims to contribute to project sciences.

Generally speaking for the European countries the sixties are characterised by a strong economic growth. In France territorial development and planning depend from a central declared power. Since 1 Urban and Regional planning department, Polytech'Tours, UMR 6173 CNRS CITERES, University of Tours -Département aménagement, Polytech'Tours, 35, allée Ferdinand de Lesseps, 37200 Tours, France -serge.thibault@univ-tours.fr 2 Urban and Regional planning department, Polytech'Tours, UMR 6173 CNRS CITERES, University of Tours -Département aménagement, Polytech'Tours, 35, allée Ferdinand de Lesseps, 37200 Tours, France -laura.verdelli@univ-tours.fr the beginning of the eighties, in a deepness changed social, cultural, economic, historic context, the central power has promote the creation of local and regional powers, influencing, at the same time, the planning project process and its expectations. Corresponding to a transition period, the present times face the cohabitation and the confrontation of two major way of "doing" the project.

The first one is a "classical" scheme where the project is often presented as the dominant step of a linear planning process, framed by a previous step defining the conditions and the major expectations and a subsequent one: the operational realisation. Within this way the expert, as a figure, dominates the process, being the only legitimate to conceive the project.

The second one, whose existence is our hypothesis, obliges to relocate the project within the planning process. In fact we will face a sort of circular process made of the combination of, at least, five equivalent phases:

1. Definition of project expectations by a collective of action 2. Project conception 3. Project realisation 4. Project appropriation by users

Subsequent space transformations

The project will be less the result than a framework of/for collective action.

An example of treatment

We will analyze two study cases: two different periods, two projects, two different national organizations (between centralized system and decentralization) almost the same space: le Loire valley (France).

The aims is to show that the planning process, that for the first project could be defined as to be constituted of a "linear" sequence of actions, must be considered today as a circular process which mobilizes no longer only experts but different "collectives of action" emerging from civil society.

Project

Planning system

Core points

The Garden Metropolis project (1968)(1969)(1970)(1971)(1972)(1973)(1974)(1975)(1976)(1977) Scheme Since de development of national landplaning policies, during the second half of the twentieth century, the Loire basin and specifically the part around Orleans, Blois and Tours, has been object of some specific projects, all more or less characterised by the will to realize some operational structures able to control urban development and to double it with valorisation, management and conservation of natural, cultural and landscape heritages (one of the first richness of the region). The first, and probably the most ambitious and the more original, of these projects, identified by some innovations in terms of regional planning, but still unknown to the public, even planning professional, was the one called Garden Metropolis.

To understand some of the reasons that led to the conception of this project, as to its failure, we are briefly reminding some elements that contributed to the emergence of a volountarist planning politic of French territory, first brought by the national state, then, after the decentralisation, by the negotiation among some local and intermediate institutional scales.

The landplaning policy that will be established since 1950 and that will lead en 1964 to the creation of the Délégation à l'Aménagement du Territoire et à l'Action Régionale (DATAR) aims in balancing a centralisation that didn't help the (economic) development capacity of regional scales. At the same time it aims to a better urban growth distribution, through the "creation" of a system made of: a. eight Paris, but to consider that they can constitute a real urban "system" that can be metropolised as a whole, giving more importance to horizontal connections among peripheral components than to relations centre-periphery.

The project, last version approved in 1977, is based onto the evidence of the existence of a natural, geographical, cultural, historical, heritage axis highly structuring: the Loire river corridor. If the formalisation of such a metropolitan project had been organised by a series of planning documents, the progressive realisation of regional structures and of the premises of decentralisation had to affect its evolution and largely modify its original contents:

• July 1970, publication by OREALM of a project summary8 ;

• June 1971, presentation by OREALM of the diagnostic and basic reference for the conception of the project9 ;

• 5 July 1972, law establishing the regions as « établissement public à vocation spécialisée »;

• December 1973, the OREALM becomes OREAC (Organisation d'Etudes d'Aménagement de la Région Centre) enlarging the dimension of intervention to the whole Centre region;

• 1975 Approval of the master plan of the medium Loire, resized as to fit global regional interests;

• 1977 Publication of the book Aménagement de la Loire moyenne -schéma de la Métropole jardin, but the shift towards the regional urban network had already totally changed the specific goals of the project Starting from this major inflection the garden metropolis project loose its first orientation and its latest document develops more a series of sectors schemes being less the reinforcement of each part in connection with the whole than the application of the initial project to some geographical subgroups.

To resume: elaborated since 1968 by OREALM, within the framework of the metropolis of balance of Parisian basin, it proposed to realise a real axis, linking nature and urbanisation along the Loire river corridor. Of innovative conception, based on the conservation of the existing territorial structures, valorising natural, cultural and urban environments, it was approved by national directive in 1975, but due to the time changes it was never applied. The site is inscribed in the category of organically evolved landscape (resulting from an initial social, economic, administrative, and/or religious imperative and that developed its present form by association with and in response to its natural environment), in the sub-category of continuing landscape, which retains an active social role in contemporary society closely associated with the traditional way of life, and in which the evolutionary process is still in progress 11 . Waterways corridors in general seems to be a site of experimentation of spatial mechanisms, based on recovering and valorisation of historic accumulation of their occupation, after a period of abandoning, made possible by the extension of patrimony notions. Considering that inhabit cannot be resumed to the sole place of living, but has also to be enlarged to frequented space, either private or not, it become necessary to conceive new forms of spatial organization, taking into account sustainable development imperatives, increasing environment preoccupations, growing nature request expressed by citizens (disposing of rising spare time to dedicate to amusements and entertainments) as contributing factors. Along many waterways and rivers corridors still survive precious not built up spaces, preserved while often submitted to floods, which constitute actually a land reserve very close to towns. Those spaces, often combining high natural and cultural heritage quality, can awake specific interests within different planning actors, documents and instruments. Those spaces, preserved free of edification, appear today as possible sites for future projects, connected to sustainable development, patrimonialisation and economic growth (especially via the valorisation of the heritage targeting tourist market).

At the same time the expected impulse dynamics of the site overlap the ones of a "classical" UNESCO label operation: the increase of international visibility, of tourist frequentation, of civil society awareness… Moreover, the Loire valley site also constitutes the biggest site in France and is submitted to the recent UNESCO requirement of being provided of a management system.

To resume: concretized by the UNESCO inscription of 2000, within a certain global heritage "fever", it aims to valorise local heritages as a resource for economic and territorial development. Promoting a reorganization of the system of action and actors it allows to observe eventual territorial recompositions (on a non ordinary territorial cut-out base) and to observe ten years impacts.

Illustration n. 2 -The UNESCO perimeter (in dark green), and the river corridor (in light green)

Landplaning system and organization

The last forty years are marked by the end of the certainties and the emergence of social and environmental crises. The same spatial planning process involves some transformations which lead it to adapt to the new context while being one of the agents of the emergence of this context.

Project Pilot System Technical teams

The , at the same time deputy (1958-1973/1976-1997), and occasionally minister of (1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006), mayor of Saint-Germain-sur-Vienne (1971-1989), deputy (1997-2001), senator of the department (since 2001), member of Edgar Pisani cabinet (1963)(1964)(1965), chargé de mission at the presidency of the republic (1991)(1992)(1993)(1994)(1995), interministerial delegate Town and social and urban development (1988)(1989)(1990)(1991), president of the interministerial coordination mission for Big architectural and urban operations (1985)(1986)(1987)(1988), director of the sector Urbanism and landscapes at the equipment ministry (1982)(1983)(1984)(1985), president of the Natural regional park, special councillor for France at the UNESCO's world heritage centre,

UNESCO focal point at the French Senate…

To resume we can explore the hypothesis that the transformations that affected the planning process are related to the passage from the simple to the complex. This formula has probably more the merit to concentrate in a few terms an evolution, even if it means being grotesque, than to represent faithfully an evolution, probably complex in any time. Much more certainly, any project, whatever its period and its context, presents dimensions of certainty and uncertainty, but at levels that can be differentiated according to the reference period.

The study of the transformations has to take into account the various possible types of project, while they probably do not affect all types in a similar way. A first typology considers three main factors: 1.

the milieu and its scales; 2. the domain (environment, mobility, urban design, etc.); 3. the level (utopia, conceptual, normative, strategic or operational project) 12 .

Project Level of the project Declination

The strategic or operational? system that help the action(s)?

Voluntary adhesion to the project common frame:

the signature of the engagement charter, federating the actors under the major common objective 13 of "structuring a territorial project of sustainable valorisation, at the scale of the site, in an international perspective of economic, cultural and scientific exchange" 14

The operational component, aiming at federating wills and projects and to coordinate, animate and participate to the realisation and monitoring of the programmes of actions, has as core missions:

• "The appropriation of the values of the inscription by the local populations;

• The animation of actors and local authorities having a responsibility on the quality and the preservation of the inscribed Val de Loire 12 A conceptual project does not aim at establishing a particular transformation but aims at exploring the fields of the possible. The strategic project, generally established at a territorial scale, tends to define a common horizon for a set of operational projects, often concerning a very detailed scale, some restricted parts of a space and a particular domain. 13 Being the ensemble of the aims: "to CONTRIBUTE to the valorisation of the Val de Loire site; to REINFORCE the attractiveness of the site, pointing out on quality, heritages preservation, social and economic territorial development site;

• To support the organisations contributing to international radiance of the inscribed site, in particular the regions"15 .

And, as core operational projects to sustain:

• "the Loire castles

• The river to river cooperation

• The 'Loire à vélo' project

• The Loire navy and ports

• The landscape project

• The Rendez-vous of world heritage

• The world heritage vineyards" 16 We can also inquiry the relation between the whole and the parts. The first project is a global scheme, where the whole makes the parts; this global schemes guides the translation into operational through its mandatory inclusion in local and municipal planning documents. The second one cannot count on a formalised scheme, the project itself consist in the establishment of a frame, which means that the parts make the whole, which makes the parts. The whole, as an organised space, is an (intentionally?)

emergence.

The same for the actions network. The first project could count on a pre-organised functioning: it was structured on diagnostic, solutions, different scales frames, interlocking of planning documents. The second one is an auto-organised one: each situation lead to the establishment of a proper solution and process, all planning documents, even at different scale, has equivalent importance. Since 1968 the Garden metropolis project proposed to lay out the river corridor between Tours and Orleans, passing through Blois, through the articulation and co-presence of natural and urban dimensions of the site. It foresaw a real linear system, based upon a discontinuous urbanisation, regulated by green cuts. This articulation wanted on one hand to contain urban development and sprawl and, on the other, to identify a certain number of structural elements to take into consideration while planning, like heritages and landscapes. The project opened, since the seventies, to the taking into account of a series of nowadays unavoidable preoccupations.

Project

If we only take into consideration the two analysed examples, almost applied on the same regional space, but conceived at two different and separated timeframes17 , we could admit that they illustrate the call into question of planned action with the abandoning of the plan to the profit of more soft models. These models rely on negotiation, on actors' voluntary adhesion to project(s), on flexibility, etc. They are all based on local initiatives, communication, search for compromise, etc. That kind of re-questioning of plan correspond to the emergence of the "city by project" described by Boltanski and Thévenot [START_REF] Boltanski | Le nouvel esprit du capitalisme[END_REF], which is a way of acting marked by the flexibility, the capacity of adaptation, the transversality, etc. Even if its not popular anymore to defend the planned action, once the complexity encourage action models much more soft, the garden metropolis project still remain a very innovator project, that would have had a deepen and on a permanent basis influence in Centre region if it had been realised. We're not as sure that the action framework provided by the Loire valley UNESCO's world heritage owns the same valuable quality.

Finally, bigger concerns about planning problems, multiplication of actors, cooperation and coordination among actors, etc. hold today a very important place and become a result in itself at the same title that the very space transformation. The question of space organisation, at the same time what is organising and what is organised, push us to conclude that the river corridor of today is composed without a clear meta intentionality. Globally, reading different appreciations by each one, contradictory strategies of all, we can hardly find an integrating system. Any actor, even legitimate, is facing a complex and contradictory system, even if soft and multi answer one.

  Scheme of regional planning focusing essentially on the Loire valley between Tours and Orleans

  Parisian basin "balance" metropolis 3 , five equipped of a Organisme d'Etude d'Aménagement de l'aire Métropolitaine (OREAM) in charge of the elaboration of their new master plan; b. nine new towns 4 ; c. the first four urban areas 5 ; d. nine support towns of more than 100.000 inhabitants 6 . With the approval, in 1967, of the Loi d'Orientation Foncière, defining the new framework for urban planning, the national government impulses the further creation of the regions, and, at the beginning of the eighties, the advent of the decentralization. « Le 9 avril 1968, le Groupe Interministériel d'Aménagement du Bassin Parisien (GIABP) prescrit l'élaboration d'un schéma d'aménagement à long terme de la vallée de la Loire Moyenne qui exprime deux vocations : zone d'appui du Bassin Parisien et section 'clé' du système constitué par le fleuve et ses affluents » 7 . The privileged situation of the Loire axis had not escape to the stakeholders carrying the file of landplaning, they detected in these space close to Ile-de-France, a possible location to reduce urban pressure on Paris city region. The conception of the landplaning project is assign to the Organisation d'Etudes d'Aménagement de la Loire Moyenne (OREALM), created by state government in 1968. The multidisciplinary team of this organisation deliberately choose not to separate the different towns, linking each one directly to

  Illustration n.1 -Principles of the urbanisation following a linear discontinuous scheme 10The second study case concerns the UNESCO's inscription of The Loire Valley between Chalonnes and Sully-sur-Loire on the World Heritage List, as Cultural landscape, on November of the year 2000. The inscription is the final act of a long process, made by several partial (in space and/or aims) projects developed on, at least, two decades, which led to the patrimonialisation of the Loire river corridor along 260 km. At the same time it is also perceived as the threshold concretising a structural change is the system of action, where all the territorial institutions (at all the different levels) become equally legitimate actors.

  DATAR: Délégation à l'Aménagement du OREALM: Organisme d'Etude et the prefect of the region (state), with the participation of the presidents of the regions, the presidents of the departments, the mayors of the majors towns, the presidents of intermunicipalities, the presidents of urban areas, the presidents of some specific dispositive such as Etablissement public Loire and Natural regional park The development committee: proposition component in charge of actions and agreement in concert. It is open to all concerned actors, especially in the fields of economy, tourism, environment, heritage, culture and education The Mission Val de Loire: operational component. It's a syndicat mixte interrégional (interregional mixed union), chaired alternatively by one of the two concerned regions, composed by a union committee, made by regional elected representative and by an on-field structure made by professionals The functioning transition was accompanied by a similar translation in French planning documents and instruments. From a very central and normative system composed by, at the specific local level: SDAU Schéma Directeur d'Aménagement et Urbanisme and POS Plan d'Occupation des Sols, we arrive to a more participative democracy and collaborating system made by: SCoT Schéma de Cohérence Territoriale and PLU Plan Local d'Urbanisme.Which evidences the knowledge of these two different systems can bring to our hypothesis of a global relocation of the place of the project within the planning process? Especially if we consider that we had just taken into account the official system and not the individual, sometimes very powerful, contributions to it? opposed by a down able to intervene on top -the example of the former mayor of Tours

  down using the top and able to federate all local actors around a common projectthe example of the former mayor of Chinon

  , tourist reception; to USE the UNESCO label following the established rules; to COOPERATE with all the others institutional, public and private actors as to create a permanent exchange of information about valorisation activities; to TAKE PART in cooperation actions together with local institutions, especially along the world heritage list's rivers; to DEVELOP programmes of actions, concrete and coordinated, of valorisation". Charte d'engagement -Val de Loire -patrimoine mondial de l'UNESCO, signed in Orleans, the 25th of November 2002, p. 5 (summary and translation by the authors) 14 Charte d'engagement -Val de Loire -patrimoine mondial de l'UNESCO, signed in Orleans, the 25th of November 2002, p. 6 (translation by the authors)

  reports on the first steps of a research which is trying to translate the evolution ongoing in the contemporary conception of urban and regional planning into an understandable narrative. Study cases are two examples of project or action which diachronic illustrates this evolution. The first one corresponds to the elaboration of a regional plan for the Loire valley (France) called Métropole-Jardin, established during the seventies, which, being certainly innovative, was nevertheless characterized by the domination of the expert activity, in a frame of very centralized state regime. The second corresponds to some projects actually developed after the inscription of a large portion of the Loire valley on the World Heritage list of UNESCO, in a period marked by the decentralization and the more or less actual application of the participation of all types of actors to territorial planning.

Lille-Roubaix-Tourcoing, Nancy-Metz-Thionville, Strasbourg, Lyon-Saint-Étienne-Grenoble, Marseille-Aix-en-Provence-Delta du Rhône, Toulouse, Bordeaux et Nantes-Saint-Nazaire

4 Marne la vallée, Cergy Pontoise, Saint Quentin en Yvelines, L'Ile d'Abeau, Villeneuve d'Ascq, Evry, Sénart, Val de Reuil, Etang de

Berre 5 Bordeaux, Lille, Lyon,

Strasbourg6 They are divided in four groups : Caen, Orléans et Tours, Amiens et Rouen, Le Havre,

Reims et Troyes[START_REF] Cohen | Métropoles d'équilibre[END_REF] Aménagement de la Loire moyenne -schéma de la Métropole jardin, Schéma général d'aménagement de la France, Travaux et recherches de prospective -Aménagement du territoire, Paris, La documentationfrançaise, 1977, p. 3

"Eléments pour un livre blanc"

Livre Blanc "Vers la métropole jardin

In: Vers la métropole jardin[START_REF]Vers la métropole jardin[END_REF]), Livre Blanc, OREALM, juin 1971, carte 21, p. 122

Excerpt from the official web site : http://www.valdeloire.org (4 June 2010) -(translation by the authors)

One marked by a centralised territorial planning organisation, the other one decentralised, but partially based on the contractual negotiation between local institutional partners and the national state government

the territorial actors as equivalent

To expose our results we will employ the example of the analysis of the text of a selected number of municipal planning tools. As test sample we choose 7 municipalities along the river; within the Tours urban area; in different geographical, pedologic, geomorphologic and centre-periphery situation; along a gradient between urban and rural.

We wish to understand how the institutional actors in charge of planning mobilise the "world heritage project" as to achieve some specific wills composing with territorial sustainable development, protection and valorisation of heritages and landscapes, protection of population towards the floods risks, economic growth and agriculture decline.

The municipalities having their municipal territory totally inscribed in flood prone areas are: La Riche and La ville aux Dames: urban, low presence of natural and built heritage, high urban pressure;

Berthenay, St-Genouph: rural, low presence of built heritage, some agriculture left, low urban pressure.

The municipalities between the low valley and the plateau are:

Luynes, St-Etienne-de-Chigny: at the limit of the urban area, high patrimonial identity and a low value image of the agriculture on the plateau; Montlouis-sur-Loire: at the limit of the urban area, high patrimonial identity and important presence of cultural landscape, high value agriculture on the plateau (labelled vine production).

What we can observe is that all of them, according to their specific situation, make an instrumental use of the heritage discourse as to pursuit some common (and competitive) wills. Within the world heritage engagement, which constitute the common frame and the common project, we can pinpoint four different and contradictory positions (on only seven municipalities) using at the best the super local characteristics of the territory: once ceasing to urban pressure (through the sacrifice of an "ordinary" agriculture or through the prosecution of the officially forbidden buildings in flooding zones); once promoting a proximity "sustainable" agriculture; once protecting high quality agriculture also promoted as "natural" area; once valorising historic and landscape heritage for touristic purposes.