Camille Laurent
email: laurent@ann.jussieu.fr

Matthieu Léautaud
email: matthieu.leautaud@math.u-psud.fr.

Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller

Keywords: Observability, eigenfunctions, spectral inequality, heat equation, tunneling estimates, control cost. 2010 Mathematics Subject Classification: 35K05, 35L05, 93B07, 93B05, 35P20

This article is concerned in the first place with the short-time observability constant of the heat equation from a subdomain ω of a bounded domain M. The constant is of the form e K T , where K depends only on the geometry of M and ω. Luc Miller [Mil04a] conjectured that K is (universally) proportional to the square of the maximal distance from ω to a point of M. We show in particular geometries that K may blow up like | log(r)| 2 when ω is a ball of radius r, hence disproving the conjecture. We then prove in the general case the associated upper bound on this blowup. We also show that the conjecture is true for positive solutions of the heat equation.

The proofs rely on the study of the maximal vanishing rate of (sums of) eigenfunctions. They also yield lower and upper bounds for other geometric constants appearing as tunneling constants or approximate control costs.

As an intermediate step in the proofs, we provide a uniform Carleman estimate for Lipschitz metrics. The latter also implies uniform spectral inequalities and observability estimates for the heat equation in a bounded class of Lipschitz metrics, which are of independent interest.

Introduction and main results

We are interested in several constants appearing in the study of eigenfunctions concentration and control theory, and the links between them. In the whole paper, we are given a connected compact Riemannian manifold (M, g) with or without boundary ∂M, we denote by ∆ g the (negative) Laplace-Beltrami operator on M. In case ∂M = ∅, we denote by Int(M) the interior of M, so that M = ∂M Int(M) (see e.g. [Lee13, Chapter 1]). For readability, we first focus in the next section on results concerning the observability constant for the heat equation.

The control cost for the heat equation

Here, we study the so-called cost of controllability of the heat equation. It is well known since the seminal papers of Lebeau-Robbiano [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] and Fursikov-Imanuvilov [START_REF] Fursikov | Controllability of evolution equations[END_REF] that for any time T > 0, the heat equation is controlable to zero. More precisely, by duality, the controlability problem is equivalent to the observability problem for solutions of the free heat equation (see e.

Here, (e t∆g ) t>0 denotes the semigroup generated by the Dirichlet Laplace operator on M (otherwise explicitely stated). The observability constant C T,ω is then directly related to the cost of the control to zero and has been the object of several studies. It has been proved by Seidman [START_REF] Thomas | Two results on exact boundary control of parabolic equations[END_REF] in dimension one (in the closely related case of a boundary observation) and by Fursikov-Imanuvilov [START_REF] Fursikov | Controllability of evolution equations[END_REF] in general (see also [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF] for obtaining this result via the Lebeau-Robbiano method), that the cost in small time blows up at most exponentially:

ω = ∅ =⇒ there is C, K > 0 such that C T,ω ≤ Ce K T
for all T > 0.

(2)

Guïchal [START_REF] Güichal | A lower bound of the norm of the control operator for the heat equation[END_REF] in one dimension and Miller [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] in the general case proved that exponential blowup indeed occurs: ω = M =⇒ there is c > 0 such that C T,ω ≥ ce which, according to the abovementionned results satisfies K heat (ω) < ∞ as soon as ω = ∅ and K heat (ω) > 0, as soon as ω = M. This constant depends only on the geometry of the manifold (M, g) and the subset ω. It is expected to contain geometric features of short time heat propagation, and has thus received a lot of attention in the past fifteen years [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF][START_REF] Miller | How violent are fast controls for Schrödinger and plate vibrations?[END_REF][START_REF] Miller | On exponential observability estimates for the heat semigroup with explicit rates[END_REF], TT07, Mil10, EZ11b, TT11, BP17, DE19, EV18, NTTV18, Phu18]. In this direction, the result of Miller [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] is actually more precise and provides a geometric lower bound: for all (M, g), ω, we have

K heat (ω) ≥ L(M, ω) 2 4 ,
where, for E ⊂ M, we write

L(M, E) = sup x∈M dist g (x, E). (4)
The proof relies on heat kernel estimates. In [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF][START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF], Luc Miller also proved that in case ω satisfies the Geometric Control Condition in (M, g) (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]) we have

K heat (ω) ≤ α * L 2 ω ,
where L ω is the maximal length of a "ray of geometric optics" (i.e. geodesic curve in case ∂M = ∅) not intersecting ω, and α * ≤ 2 is an absolute constant (independent of the geometry). Based on these results and the idea that the heat kernel provides the most concentrated solutions of the heat equation, he formulated the following conjecture [Mil04a, Section 2.1]-[Mil06b, Section 3.1].

Conjecture 1.1 (Luc Miller). For all (M, g) and ω ⊂ M such that ω = M, we have K heat (ω) = L(M,ω) 2

4

.

Note that it has been proved in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] that, in the related context of the 1D heat equation with a boundary observation, the factor 1 4 might not be correct (and should be replaced by 1 2 , see Section 1.4 below). Our first result disproves Conjecture 1.1 in a stronger sense.

Theorem 1.2 (Counterexamples). Assume (M, g) is one of the following 1. M = S n ⊂ R n+1 and g is the canonical metric (see Section 3.1); 2. M = S ⊂ R 3 is a surface of revolution diffeomorphic to the sphere S 2 , and g is the metric induced by the Euclidean metric on R 3 (with additional non degeneracy conditions, see Section 3.2);

3. M = D = (x 1 , x 2 ) ∈ R 2 x 2 1 + x 2 2 ≤ 1 ⊂ R 2 is the unit disk, g the Euclidean metric and Dirichlet conditions are taken on ∂M (see Section 3.3).

Then, for any C > 0, there exists ω ⊂ M so that K heat (ω) ≥ CL(M, ω) 2 and K heat (ω) ≥ C.

More precisely, assume that x 0 is either 1. any point in S n , 2. one of the two points that intersect the axis of revolution of S ⊂ R 3 , 3. the center of D.

Then, there exists C > 0 and r 0 > 0 so that we have

K heat (B g (x 0 , r)) ≥ C| log(r)| 2 (5)
for any 0 < r ≤ r 0 .

Here, B g (x 0 , r) denotes the geodesic ball of M centered at x 0 of radius r. The results we obtain are slightly more precise. In particular, the constant C is an explicit geometric constant. The lower bounds are related to an appropriate Agmon distance associated to the problem. We refer to Corollary 1.10 below for more precise estimates.

Note also that this blowup of K heat (B(x 0 , r)) for small r does not always happen and is due here to a particular (de)concentration phenomenum. For instance on M = T 1 , the set ω = B(x 0 , r) always satisfies the Geometric Control Condition for any time T > 1 -2r. Abstract results (see (15) below for more details) give K heat (B(x 0 , r)) ≤ α * ≤ 2 for any r > 0 and blowup does not occur.

Our next result shows that the blowup given by ( 5) is actually optimal as far as the asymptotics of K heat for small balls is concerned. We prove the following observability result from small balls (closely related to previous results of Jerison-Lebeau [START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF], see Section 1.3.2 below).

Theorem 1.3. For all x 0 ∈ M, there exist C > 0 such that for all r > 0 we have K heat (B(x 0 , r)) ≤ C| log(r)| 2 + C.

Note that Bardos and Phung [START_REF] Bardos | Observation estimate for kinetic transport equations by diffusion approximation[END_REF][START_REF] Dang | Carleman commutator approach in logarithmic convexity for parabolic equations[END_REF] recently proved independently that K heat (B(x 0 , r)) ≤ C r + C for all > 0 in case M ⊂ R n is star-shaped w.r.t. x 0 .

These results seem to suggest that L(M, ω) is not the only appropriate parameter needed for estimating K heat (ω). There are indeed some solutions of the heat equation concentrating more than the heat kernel for small times. Our last result concerning the heat equation goes actually in the opposite direction. It provides with a large class of solutions of the heat equation, namely positive solutions, that do not concentrate more than the heat kernel, thus proving Conjecture 1.1 when restricted to this class of solutions. Recall that L(M, E) is defined in (4).

Theorem 1.4. Assume that (M, g) has geodesically convex boundary ∂M. Then, for any nonempty open set ω ⊂ M and z 0 ∈ M, for any ε > 0, there exist C, D > 0 so that for any 0 < T ≤ D, we have

u(T ) 2 L 2 (M) ≤ C T e (1+ε)(L(M,ω)+ε) 2 2T T 0 u(t, •) 2 L 2 (ω) dt, (6)
u(T ) 2 L 2 (M) ≤ C T e (1+ε)(L(M,z 0 )+ε) 2 2T T 0 u(t, z 0 ) 2 dt, (7)
for all u 0 ∈ L 2 (M) such that u 0 ≥ 0 a.e. on M and associated solution u to (∂ t -∆ g )u = 0 on R + * × Int(M), u| t=0 = u 0 in Int(M), ∂ ν u = 0 on R + × ∂M. Theorem 1.4 follows from classical Li-Yau estimates [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF]. Notice that here, Neumann boundary conditions are taken (ν denotes a unit vector field normal to ∂M), and an additional geometric assumption is made (convexity of ∂M). The result still holds without the convexity assumption up to replacing (1 + ε) in the exponent by a geometric constant, see Remark 5.2. We also recall that for nonnegative initial data u 0 ≥ 0, the solution of the heat equation remains nonnegative for all times. Of course, the counterexamples of Theorem 1.2 prevent these estimates to hold in general. Estimate (7) is particularly surprising (even without considering the value of the constants) and of course only true for positive solutions (otherwise just taking z 0 in a nodal set of an eigenfunction of ∆ g invalidates (7)). Finally, let us mention that the constants C and D are explicitely estimated by geometric quantities (see Remark 5.4).

Let us now put these results in a broader context, and introduce several related geometric constants appearing in tunneling estimates and control theory.

Tunneling constants in control theory, and their links

The lower bounds of Theorem 1.2 are proved using very particular solutions to the heat equation arising from by eigenfunctions (exhibiting a very strong concentration far from x 0 as well as a strong deconcentration near x 0 ). It is therefore natural to study related constants measuring such (de)concentration properties. In this section, we introduce all geometric constants studied in the paper and collect known links between them.

We first introduce spectral subspaces of the Laplace operator ∆ g (with Dirichlet boundary conditions if ∂M = ∅), which are at the core of most results presented here. Namely, for λ ∈ Sp(-∆ g ), the space E λ := span{ψ ∈ L 2 (M), -∆ g ψ = λψ} denotes the eigenspace associated to the eigenvalue λ and, for all λ > 0, E ≤λ := span{E λj , λ j ∈ Sp(-∆ g ), λ j ≤ λ}, the space of linear combinations of eigenfunctions associated to eigenvalues ≤ λ.

Let us now introduce the constants studied in the article, else than that involved in (1)-(2). For any nonempty open subset ω ⊂ M, we recall the following results:

• Vanishing of eigenfunctions [START_REF] Donnelly | Nodal sets of eigenfunctions on Riemannian manifolds[END_REF][START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]: there exist C, K such that we have

ψ L 2 (M) ≤ Ce K √ λ ψ L 2 (ω)
, for all λ ∈ Sp(-∆ g ) and ψ ∈ E λ .

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• Vanishing of sums of eigenfunctions (so-called Lebeau-Robbiano spectral inequality) [LR95, JL99, LZ98]: there exist C, K such that we have

u L 2 (M) ≤ Ce K √ λ u L 2 (ω)
, for all λ > 0 and all u ∈ E ≤λ .

• Infinite time observability of the heat equation [START_REF] Fursikov | Controllability of evolution equations[END_REF]: there exist C, K such that we have

R + e -2K t e t∆g u 2 L 2 (M) dt ≤ C R + e t∆g u 2 L 2 (ω) dt, for all u ∈ L 2 (M). (10)
• Approximate observability for the wave equation [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF],

(∂ 2 t -∆ g )u = 0, u| (0,T )×∂M = 0, (u, ∂ t u)| t=0 = (u 0 , u 1 ) :

For all T > 2L(M, ω), there exist C, K, µ 0 > 0 such that we have

(u 0 , u 1 ) L 2 (M)×H -1 (M) ≤ Ce Kµ u L 2 ((0,T )×ω) + 1 µ (u 0 , u 1 ) H 1 0 (M)×L 2 (M)
, for all µ ≥ µ 0 and all (u 0 , u 1 ) ∈ H 1 0 (M) × L 2 (M), and u solution to (11).

Recall the definition of L(M, ω) in (4). Remark that this last estimate is equivalent to (see [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF] or Corollary 2.2 below)

(u 0 , u 1 ) H 1 0 (M)×L 2 (M) ≤ C e K Λ u L 2 ((0,T )×ω) , Λ = (u 0 , u 1 ) H 1 0 (M)×L 2 (M) (u 0 , u 1 ) L 2 (M)×H -1 (M) ,
for all (u 0 , u 1 ) ∈ H 1 0 (M) × L 2 (M), and u solution to (11).

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Note that in the reference [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF], the observation term in the right hand-side of these inequalities is u L 2 (0,T ;H 1 (ω)) instead of u L 2 ((0,T )×ω) . That the stronger inequalities above holds is proved in [LL17, Section 5.3] (see also [START_REF] Laurent | The cost function for the approximate control of waves[END_REF]).

In all these inequalities, we are interested in the "best constant K" such that the estimate holds for some C. More precisely, we are interested in the way it depends on the geometry of (M, g) and ω (and, in the case of (12), the time T ). Let us first formulate the precise definitions of these constants. These are the analogues to that of K heat (ω) given in (3). Definition 1.5. Given ω ⊂ M an open set, we define K eig (ω), K Σ (ω), K ∞ (ω), K wave (ω, T ) to be the best exponents in the above estimates (8)-(12), namely:

K eig (ω) = inf {K > 0, ∃C > 0 s.t. (8) holds} , K Σ (ω) = inf {K > 0, ∃C > 0 s.t. (9) holds} , K ∞ (ω) = inf {K > 0, ∃C > 0 s.t. (10) holds} , K wave (ω, T ) = inf {K > 0, ∃C > 0, µ 0 > 0 s.t. (12) holds} = inf {K > 0, ∃C > 0, s.t. (13) holds} . ( 14
)
A proof of the equality in ( 14) is given in Corollary 2.2 below. Note that we may write K wave (ω, T ) = +∞ if T < 2L(M, ω) since ( 12)-(13) are known not to hold (see the discussion in [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF]). However, K wave (ω, T ) < +∞ as soon as T > 2L(M, ω), by virtue of (12)-(13).

Let us now collect some known facts concerning these constants, in addition to the already discussed bound K heat (ω) ≥ L(M,ω) 2 4 [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF]. A first trivial (but useful) fact is that K eig (ω) ≤ K Σ (ω). The following properties can also be found in the literature:

1. For all (M, g), ω such that ω = M, we have

K Σ (ω) ≥ L(M,ω) 2 , see [Mil10, Theorem 5.3] (that K Σ (ω) > 0 had already been proved in [JL99]). 2. K ∞ (ω) ≤ K heat (ω), [Mil06b, Theorem 1]. 3. For all (M, g), ω, we have K ∞ (ω) ≥ d1(ω) 2 4 , with d 1 (ω) = sup {r > 0, ∃x ∈ M, B(x, r) ⊂ M \ ω}, see [FCZ00] and [Zua01, Section 4.1].
4. Assume ω satisfies the Geometric Control Condition in (M, g) and denote by L ω the maximal length of a ray of geodesic optics not intersecting ω. Then, we have

K heat (ω) ≤ α * L 2 ω ( 15
)
with α * = 2 36 37 2 , see [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF][START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF] (improved to α * = 3/4 in [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF] and to 0.6966 in [DE19]).

5. Assume ω satisfies the Geometric Control Condition in (M, g) and denote by L ω the maximal length of a ray of geometric optics not intersecting ω. Then, we have [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF] Corollary 1 and Section 2.4] (see also [START_REF] Thomas | How violent are fast controls[END_REF] for a proof of

K ∞ (ω) ≤ 1 16 L 2 ω , see [EZ11b, Theorem 1.1]. 6. K heat (ω) ≤ 4K Σ (ω) 2 , see
K heat (ω) ≤ 8K Σ (ω) 2 ).
7. If (ω, T ) satisfy the geometric control condition [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], then K wave (ω, T ) = 0 (more precisely, (12)-(13) hold with K = K = 0). Conversally, if (M, g) is real-analytic and (ω, T ) does not satisfy the geometric control condition (for a ray that only intersects ∂M transversally), then K wave (ω, T ) > 0, see [START_REF] Lebeau | Contrôle analytique. I. Estimations a priori[END_REF].

Notice that in all these statements, the constants K heat and K ∞ (heat equation) are homogeneous to a square of a distance (as for the heat kernel), whereas the other ones are homogeneous to a distance (as for the wave kernel). Remark also that every comparison statement above follows, in the associated reference, from a proper inequality (the above statements being only a weak form of those).

Also notice that the converse inequality K Σ (ω) 2 ≤ CK heat (ω) for a universal constant C does not seem to hold in general. For instance, in the related situation of boundary control on an interval (0, 1) (see Section 1.4), K heat ({0}) is finite while a dimensional analysis shows that no spectral inequality holds true, i.e. K Σ ({0}) is infinite.

We first complete the above list of comparison results by the following proposition.

Proposition 1.6 (Other links between the constants). We have

K eig (ω) 2 4 ≤ K heat (ω), K eig (ω) 2 4 ≤ K ∞ (ω).
Also for all T > 0, we have K eig (ω) ≤ K wave (ω, T ).

Note that the last statement is empty if T < 2L(M, ω) since ( 12)-(13) are known not to hold (see the discussion in [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF]), but is nonempty if we have K wave (ω, T ) < ∞, that is if T > 2L(M, ω), by virtue of (12)-(13).

Hence, in order to produce lower bounds for K Σ (ω), K heat (ω), K ∞ (ω), K wave (ω, T ), we shall product lower bounds for K eig (ω), i.e. construct sequences of eigenfunctions having a maximal vanishing rate on ω. Note also that, summarizing the inequalities so far, we have:

K eig (ω) 2 4 ≤ K ∞ (ω) ≤ K heat (ω) ≤ 4K Σ (ω) 2 , (16)
so that the understanding of concentration properties for eigenfunctions and sums of eigenfunctions essentially contains those of the heat equation. Therefore, our main focus in the following is to produce:

• maximally vanishing eigenfunctions in particular geometries to yield a lower bound for K eig ;

• a uniform Lebeau-Robbiano spectral inequality on small balls to yield an upper bound for K Σ .

Note that reducing our attention to K eig in the seek of lower bounds is already very restrictive! Indeed, as soon as the Schrödinger equation on (M, g) is observable from ω in finite time (in particular if ω satisfies the geometric control condition, see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF][START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF]), then K eig (ω) = 0 (more precisely, (8) holds with K = 0).

Before starting to state these lower/upper bounds, let us give a link between K heat (ω) and K wave (ω, T ), consequence of a result of Ervedoza-Zuazua [START_REF] Ervedoza | Observability of heat processes by transmutation without geometric restrictions[END_REF] (weak observability with exponential cost for the wave equation implies observability of the heat equation).

Proposition 1.7. There exist universal constants α 1 , α 2 > 0 so that for any S > 0, we have

K heat (ω) ≤ α 1 S 2 + α 2 K wave (ω, S) 2 .
The proof of this result in Section 2.3 is a little more precise about this estimate. In particular, several values of (α 1 , α 2 ) can be deduced from it. The value of α 1 is thought to be related to the cost of the boundary control of the 1D heat equation. Note that, as in (16), this yields

K eig (ω) 2 4 ≤ K ∞ (ω) ≤ K heat (ω) ≤ α 1 S 2 + α 2 K wave (ω, S) 2 , for all S > 0.
However, this upper bound seems for the moment less useful than that of (16), since the proof of (12)-(13) in [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF] is more technically involved than that of (9) in [LR95, JL99, LZ98]. The computation of K wave (ω, S) seems thus more intricate than that of K Σ (ω).

Main results

Constructing maximally vanishing eigenfunctions: lower bound for K eig

In this Section, we provide lower bounds for K eig in three different geometries. This then proves Theorem 1.2 as a direct corollary of Proposition 1.6.

The sphere We first state the results we obtain on two dimensional sphere S 2 , since they are particularly simple. The higher dimensional case S n is completely similar. The sphere S 2 is parametrized by (s, θ) ∈ (0, π) × S 1 . We denote by N (resp. S) the north pole described by s = 0 (resp. the south pole described by s = π), and remark that s is the geodesic distance to the point N .

Theorem 1.8. For k ∈ N, the function

ψ k (s, θ) = c k sin(s) k e ikθ , c k = k 1/4 2 1/2 π 3/4 1 + O( 1 k ) as k → +∞ satisfies -∆ g ψ k = k(k + 1)ψ k on S 2 , ψ k ∈ C ∞ (S 2 ), ψ k L 2 (S 2 ) = 1, |ψ k (s, θ)| = c k sin(s) k ≤ c k s k for s ∈ [0, π], k ∈ N, ψ k 2 L 2 (B(N,r)) = c 2 k π k + 1 sin(r) 2k+2 cos(r) (1 + R), |R| ≤ tan(r) 2 2k + 2 for r ∈ [0, π 2 ), k ∈ N.
This result is a much more explicit, more precise (and simpler to prove) version of the general results we obtain on surfaces of revolution. We turn to the general case and shall explain at the end of the section the links with Theorem 1.8.

Surfaces of revolutions

The precise description of the geometry of the surfaces we consider is given in Section 3.2 and we only give here the features required to state the result. We consider M = S ⊂ R 3 a smooth compact surface diffeomorphic to the sphere S 2 . We assume moreover that it has revolution invariance around an axis, that intersects S in two points, the north and the south poles, respectively N, S ∈ S. These points are the only invariant points of the revolution symmetry. The surface is then endowed with the metric g inherited from the Euclidean metric on R 3 , which itself enjoys the rotation invariance. Then, we describe (almost all) the surface by two coordinates, namely s = dist g (•, N ), the geodesic distance to the north pole and θ, the angle of rotation. The variable s is in (0, L) where L = dist g (N, S). The surface is characterized by the function R(s) associating to s the Euclidean distance in R 3 to the symmetry axis, which, by definition, is rotationally invariant, and satisfies R(0) = 0 = R(L). This function R is the "profile" of the revolution surface S.

We shall now assume that R reaches at s 0 a global maximum, and introduce the relevant Agmon distance to the "equator" s = s 0 , defined by the eikonal equation

d A (s) 2 - 1 R(s) 2 - 1 R(s 0 ) 2 = 0, d A (s 0 ) = 0, sgn(d A (s 0 )) = sgn(s -s 0 ), (17)
or, more explicitely, for s ∈ (0, L), by

d A (s) = s s0 1 R(y) 2 - 1 R(s 0 ) 2 dy . (18)
A more intrinsic definition of d A is given in Remark 3.3 below (and requires additional notation).

Theorem 1.9. Assume that s → R(s) admits a non-degenerate strict global maximum at s 0 ∈ (0, L).

Then, for all k ∈ N, there exists ψ k ∈ C ∞ (S), and λ k ≥ 0 such that

λ k = k 2 R(s 0 ) 2 + k |R (s 0 )| R 3 (s 0 ) + O(k 1/2 ), ψ k L 2 (S) = 1, -∆ g ψ k = λ k ψ k .
Moreover, there exist C, C * , C 0 , k 0 > 0 such that, for all k ∈ N, k ≥ k 0 and all 0 ≤ r ≤ s 0 , we have the estimate

ψ k L 2 (B(N,r)) ≤ Cλ C0 k e -d A (r)(R(s0) √ λ k -C * ) .
Note that one can choose any C * > 1 2 |R (s 0 )|R(s 0 ) in this result. This statement has to be completed by the asymptotic behavior of d A (proved in Lemma 3.8) when s → 0, namely

d A (s) = -log(s) + O(1), as s → 0 + . ( 19
)
That is to say that the equator and the poles are infinitely distant to each other for the Agmon distance d A (as opposed to the geodesic distance dist g ). Note that at first order, d A does not depend on the geometry of the surface S close to the north pole N (s = 0). A similar statement holds close to the south pole S (s = L). This, together with Definition 1.5 and Proposition 1.6, yields the following direct corollary.

Corollary 1.10. Under the assumptions of Theorem 1.9, for all 0 ≤ r ≤ s 0 , we have the estimate

K eig (B g (N, r)) ≥ d A (r)R(s 0 ).
This yields also

K Σ (B g (N, r)) ≥ d A (r)R(s 0 ), K wave (B g (N, r), T ) ≥ d A (r)R(s 0 ), for any T > 0, K ∞ (B g (N, r)) ≥ d A (r)R(s 0 ) 2 4 , K heat (B g (N, r)) ≥ d A (r)R(s 0 ) 2 4 .
Note also that Theorem 1.9, combined with the explicit asymptotic expansion (19) of the Agmon distance d A implies the following result.

Corollary 1.11 (Rate of vanishing). With (λ k , ψ k ) as in Theorem 1.9, there exist C, C * , C 0 , k 0 > 0 such that, for all k ∈ N, k ≥ k 0 and all r ≥ 0, we have

ψ k L 2 (B(N,r)) ≤ Ce C √ λ k r R(s0) √ λ k -C * ,
and, in any local chart centered at N , we have

∂ α ψ k (N ) = 0 for all |α| < R(s 0 ) √ λ k -C * -n/2.
As on the sphere, these eigenfunctions saturate the maximal vanishing rate predicted by the Donnelly-Fefferman Theorem [START_REF] Donnelly | Nodal sets of eigenfunctions on Riemannian manifolds[END_REF].

Note that in these estimates, R(s 0 ) √ λ k ∼ k does not depend on the geometry.

The proofs rely on classical semiclassical decay estimates for eigenfunctions [START_REF] Simon | Instantons, double wells and large deviations[END_REF][START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF]. We refer to the monographs [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF][START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] for the historical background and more references. An additional difficulty here is linked to the degeneracy of the function R close to the north and south poles.

Note also that, to our knowledge, the idea of constructing such examples on surfaces of revolution is due to Lebeau [Leb96] and Allibert [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF].

The disk Recall that D = {(x, y) ∈ R 2 , x 2 + y 2 ≤ 1}. Our results on the disk are quite similar to the previous results on revolution surfaces. They are proved in Section 3.3. Note the construction is more explicit there since it involves Bessel functions. As in the above example, the concentration is related to an Agmon distance to the maximum of the radius r, which corresponds to the boundary ∂D here.

Theorem 1.12 (Whispering galleries on the disk). Denote, for r ∈ (0, 1],

d A (r) = -(tanh(α(r)) -α(r)) , with α(r) = cosh -1 (1/r). (20)
Then, for all k ∈ N, there exists

ψ k ∈ C ∞ (D) ∩ H 1 0 (D), and λ k ≥ 0 such that λ k = k 2 + O(k 4/3 ), ψ k L 2 (S) = 1, -∆ g ψ k = λ k ψ k .
Moreover, there exist C, β, k 0 > 0 such that for all k ≥ k 0 and 0 < r ≤ 1 -βλ

-1/3 k , we have ψ k L ∞ (B(0,r)) ≤ exp - λ k -Cλ 1/6 k d A (r) + Cλ 1/6 k .
That d A indeed represents an Agmon distance in the present context is justified in the next paragraph. Note that d A still satisfies d A (r) ∼ r→0 + log( 1 r ) here, so that the analogues of Corollaries 1.10 and 1.11 still hold in this setting.

Remarks on the Agmon distance In this paragraph, we compare the three geometries discussed above. In particular, we stress the fact that the results obtained on the sphere are refinements of those on general surfaces of revolution, and explain the similarities in the case of the disk.

Remark 1.13 (Agmon distance on the sphere). Note that the coordinates (s, θ) introduced on the unit sphere are the same as those defining general surfaces of revolution, with L = π, s ∈ (0, π), R(s) = sin(s) and the maximum of R is reached at s 0 = π 2 . In particular, recalling the definition of the Agmon distance in (18), we obtain, for s ∈ (0, π),

d A (s) = s s0 1 R(y) 2 - 1 R(s 0 ) 2 dy = s π/2 1 sin(y) 2 -1dy = s π/2 cos(y) sin(y) dy = | log(sin(s))|.
This can be rewritten intrinsically as

d A (m) = -log sin(dist g (m, N )) , m ∈ S 2 , (recall dist g (m, N ) + dist g (m, S) = π).
In view of this identity for the sphere, the estimates on the eigenfunctions ψ k of Theorem 1.8 can be reformulated as (λ k = k(k + 1))

|ψ k (s, θ)| = c k e -kd A (s) for s ∈ [0, π], k ∈ N, ψ k 2 L 2 (B(N,r)) = c 2 k π k + 1 e -(2k+2)d A (r) cos(r) (1 + R), |R| ≤ tan(r) 2 2k + 2 for r ∈ [0, π 2 ), k ∈ N.
These two statements (pointwise estimate and fine asymptotics of the L 2 norm) are much more precise than those of Theorem 1.9 on general surfaces of revolution. Note that one can put the disk in a general setting of surfaces of revolution with boundary. In this context, one can give a proof of (a slightly weaker version of) Theorem 1.12 following that Theorem 1.9 (and only relying on Agmon estimates), see [START_REF] Laurent | On uniform observability of gradient flows in the vanishing viscosity limit[END_REF]. As opposed to the proof of Theorem 1.12, the latter proof does not make use of the explicit knowledge of eigenfunctions on the disk and properties of Bessel functions.

Remark 1.14 (Agmon distance in the disk). Recalling the definition of d A in (20), we have α

(r) = -1 r 2 1 1 r 2 -1
, so that

(d A (r)) 2 = α (r) 2 1 cosh 2 (α(r)) -1 2 = 1 r 2 1 1 -r 2 (r 2 -1) 2 = 1 r 2 -1, and d A (1) = 0.
As a consequence, d A is exactly the Agmon distance to the boundary r = 1, and we have

d A (r) = - 1 r 2 -1, r ∈ (0, 1].
Note again that d A (r) ∼ r→0 + log( 1 r ) and, in particular, the center of the disk is at infinite Agmon distance to the boundary: d A (0) = +∞.

Uniform Lebeau-Robbiano spectral inequalities: upper bound for K Σ

The counterpart of Corollary 1.11 is due to Donnelly-Fefferman [START_REF] Donnelly | Nodal sets of eigenfunctions on Riemannian manifolds[END_REF], and roughly states that eigenfunctions vanish at most like r C √ λ+C on balls of radius r (λ is the eigenvalue). It has been generalized in some sense to sums of eigenfunctions by Jerison and Lebeau [START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF]. We prove here a variant of this result under the form of a uniform Lebeau-Robbiano spectral inequality with observation on small balls.

Theorem 1.15 (Uniform Lebeau-Robbiano spectral inequality with observation on small balls). Let (M, g) be a compact Riemannian manifold with (or without) boundary ∂M. For all x 0 ∈ M, there exist constants C 1 , C 2 > 0 such that for all r > 0, λ ≥ 0 and ψ ∈ E ≤λ , we have

ψ L 2 (M) ≤ e (C1 √ λ+C2)(1+log( 1 r )) ψ L 2 (B(x0,r)) .
Note that a careful inspection of the proofs (of all Carleman estimates used, that are stable by small perturbations) shows that the constant C 1 , C 2 can actually be taken independent of the point x 0 . Note that we prove the result in the case of Neumann boundary conditions as well. This uniform Lebeau-Robbiano spectral inequality directly implies Theorem 1.3 using [Mil10, Corollary 1] (recalled in Lemma 2.6 below).

One of the tools we develop for the proof of Theorem 1.3 also yields a uniform Lebeau-Robbiano inequality in a class of Lipschitz metrics. Even though not completely related to the main results of the paper, we choose to state is here since we believe it is of independent interest.

On the manifold M, we denote here by g a metric and (λ g j ) j∈N the spectrum of the associated Laplace-Beltrami operator -∆ g (with Dirichlet boundary condition if ∂M = ∅) and by (ψ g λj ) j∈N an associated Hilbert basis of eigenfunctions, in order to stress the dependence with respect to the metric. We also write E g ≤λ = span{ψ g λj , λ g j ≤ λ}, which of course, depends on the metric g. Now, given a reference Lipschitz metric g 0 , we define

Γ ε,D (M, g 0 ) = g Lipschitz continuous metric on M, g W 1,∞ (M) ≤ D, εg 0 ≤ g ≤ Dg 0 .
Theorem 1.16 (Uniform Lebeau-Robbiano spectral inequality in a class of metrics). Let M be a compact Riemannian manifold with (or without) boundary ∂M, g 0 be a Lipschitz continuous Riemannian metric on M, and ω ⊂ M a nonempty open set. Then, for all D ≥ ε > 0, there exist constants C, c > 0 such that for all g ∈ Γ ε,D (M, g 0 ), λ ≥ 0 and w ∈ E g ≤λ , we have

w L 2 (M) ≤ Ce c √ λ w L 2 (ω) . (21)
Note that the above estimate is valid whatever the choice of L 2 -norm (i.e. w.r.t. g or g 0 ) since all these norms are uniformly equivalent for metrics g the class Γ ε,D (M, g 0 ). This result could be reformulated by saying that (21) holds for all w ∈ g∈Γ ε,D (M,g0) E g ≤λ . This uniform Lebeau-Robbiano spectral inequality directly implies the following uniform estimate on the cost of the heat equation, using [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF] Corollary 1], recalled in Lemma 2.6 below (in which the constants are explicitely computed in terms of the constants in the spectral inequality).

Corollary 1.17. Let M be a compact Riemannian manifold with (or without) boundary ∂M, g 0 be a Lipschitz continuous Riemannian metric on M, and ω ⊂ M a nonempty open set. Then, for all D ≥ ε > 0, there exist constants C, K > 0 such that for all g ∈ Γ ε,D (M, g 0 ), we have

e T ∆g u 2 L 2 (M) ≤ Ce 2K T T 0 e t∆g u 2 L 2 (ω) dt,
for all T > 0 and all u ∈ L 2 (M).

Note that the proofs of Theorem 1.16 and Corollary 1.17 are completely constructive, and, as such, provide with explicitely computable constants.

The case of a barrel: upper bound for K wave and K heat

To conclude with the upper bounds on the constant, we present in this section some applications of results obtained by Allibert in [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF]. In case of a "barrel-type surface" with boundary (a geometric setting close to that of the surfaces of revolution described above), Allibert estimates the attainable space for the controlled wave equation. As corollaries, we deduce from this result estimates of K wave and, in view of Proposition 1.7, of K heat .

We first present the geometric context which (very close to that of surfaces of revolution described above). In this section, M = S is a surface of revolution of R 3 with boundary, parametrized by the equation

S = {(x, y, z) ∈ R 3 , z ∈ [0, L], x 2 + y 2 = R(z)},
where R is a strictly positive smooth function on [0, L], that admits at the point z 0 ∈ (0, L) a unique local (and therefore global) non degenerate maximum (i.e. R (z 0 ) < 0). Observation takes place at the boundary, only on the bottom side, that is Γ = {(x, y, 0) ∈ R 3 ; x 2 + y 2 = R(0)}. We may also describe S by (z, θ), with (x, y) = (R(z) cos θ, R(z) sin θ). We refer to Remark 3.4 to explain the link between the two parametrizations of revolution surfaces by s and z (and in particular, that we may write z = z(s) and R(s) = R(z(s))).

As above, we define the Agmon distance to the point z 0 , which in this z-parametrization writes (note that it is almost the same as (18) but in different coordinates):

d A (z) = z z0 1 + R 2 (y) 1 R(y) 2 - 1 R(c) 2 dy .
We also need the following definition of a critical time T 1 (see Allibert [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF] for more details), which, roughly speaking, represents the smallest period of the geodesic flow, modulo rotation. More precisely, the principal symbol of the wave operator on R × S is given by

p(t, z, θ, τ, ζ, η) = ζ 2 1 + R 2 (z) + η 2 R 2 (z) -τ 2 ,
where (τ, ζ, η) denote the dual variable to (t, z, θ). For any (generalized) bicaracteristic curve γ of p, bouncing on the boundary according to the reflection law ζ → -ζ, we denote T (γ) the smallest period of the function Π z (γ) where Π z is the projection on the component z. Then, T 1 is defined by

T 1 = sup{T (γ), γ bicharacteristic curve of p},
and we have T 1 ≥ 2L(M, Γ) (this critical time is larger than the time of unique continuation from Γ).

In this context, we define similarly K heat (Γ) and K wave (Γ, T ) with exactly the same definition as in (2) and Definition 1.5 with u L 2 ([0,T ]×ω) replaced by ∂ ν u L 2 ([0,T ]×Γ) in (1) and (12). Note that ∂ ν u is in L 2 ([0, T ] × Γ) for initial data in L 2 (resp. H 1 0 × L 2 ) for the heat (resp. wave) equation thanks to hidden regularity. We deduce from [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF] the following result.

Theorem 1.18. Under the above geometric assumptions, we have the estimates

K wave (Γ, T ) ≤ R(z 0 )d A (Γ), for all T > T 1 , (22)
K heat (Γ) ≤ α(T 1 (Γ) 2 + R(z 0 ) 2 d A (Γ) 2 ), (23)
for some universal constant α > 0.

The first estimate (22) follows simply from [All98, Théorème 2] (see Proposition 2.7 below), which is stated in terms of analytic spaces with respect to the rotation variable θ. Then, (22) implies (23) thanks to Proposition 1.7. Note that (22) also proves an analogue of Theorem 1.9 in this geometry, so that in fact:

K eig (Γ) = R(z 0 )d A (Γ), and K wave (Γ, T ) = R(z 0 )d A (Γ) for all T > T 1 . ( 24
)
He also proves upper and lower estimates for T ∈ (2L(M, Γ), T 1 ) (which do not coincide). The proof of Theorem 1.18 in Proposition 2.7 yields the according estimates of K wave (Γ, T ). Note finally that in the paper [START_REF] Laurent | On uniform observability of gradient flows in the vanishing viscosity limit[END_REF], using the methods of Allibert [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF], we also prove that K eig (B g (N, r)) = d A (r)R(s 0 ) in the context of Theorem 1.9 and Corollary 1.10.

Previous results

Except for the bounds (24) (and that of K eig in [START_REF] Laurent | On uniform observability of gradient flows in the vanishing viscosity limit[END_REF]) following from Allibert's result and the compu-

tation of K ∞ ({0}) on M = [0, L] in [FR71],
we are not aware of other situations in which the constants described in the previous paragraph are known exactly. We collect in this section previous results on the constants K heat and K wave , which received a lot of attention in the past fifteen years.

Parabolic equations in dimension one

The most studied case concerns the constant K heat , with observation/control at the boundary in the one dimensional case, say M = [-1, 1]. Yet, it seems that the constant K heat ({-1, 1}) is still unknown. Note that the latter has a particular importance since it has applications to higher dimensions (with geometric conditions) via the transmutation method of Luc Miller [START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF].

Here, we list previous results on M = [-1, 1] with Neumann trace observation (Dirichlet control) on both sides of the interval. Note also that each improvement of the constant was also the occasion of finding new techniques of proofs.

• K heat ({-1, 1}) ≤ 2 36 37 2 Miller [START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF], using the transmutation method;

• K heat ({-1, 1}) ≤ 3 4 Tenenbaum-Tucsnak [TT07]
, using results of analytic number theory;

• K heat ({-1, 1}) ≥ 1 2 , Lissy [Lis15], using complex analysis arguments; • K heat ({-1, 1}) ≤ 0, 7, Dardé-Ervedoza [DE19]
, combining Carleman estimates and complex analysis.

Note that in this setting, the analogue of Conjecture 1.1 would be K heat ({-1, 1}) = 1 4 , which [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] disproved in this context (by a factor 2). However, this result does not in general prevent the existence of a universal constant C > 0 so that

K heat (ω) = CL(M, ω) 2 .
As noticed in [START_REF] Ervedoza | Sharp observability estimates for heat equations[END_REF], the result in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] implies that on the interval (0, L), we have

K ∞ ({0}) = L 2 4
(and [START_REF] Ervedoza | Sharp observability estimates for heat equations[END_REF] even prove (10) for the critical K = L 2 4 ).

Parabolic equations in higher dimensions

There are many papers concerning the controllability properties of the heat equation. We only mention those providing with estimates on the constants studied in this paper.

The first computable estimates were obtained using the transmutation method to give estimates similar to (15). We can find several references improving the universal constant involved, see [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF][START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF][START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF][START_REF] Dardé | On the cost of observability in small times for the one-dimensional heat equation[END_REF].

In [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF], the authors prove

K Σ (ω * ) ≤ 3 log( (4πe) N |ω * | )
where M = (0, π) N is a cubic domain and |ω * | is the volume of the biggest rectangle included in ω. The proof of this result uses a number theoretic argument of Turán concerning families of the complex exponential (e ikx ) k∈Z (which can be interpretated as an estimate of K Σ (I) for I a subinterval of T). Remark that in this particular flat-torus geometry, we have no idea of what the right constant should be.

In [START_REF] Bardos | Observation estimate for kinetic transport equations by diffusion approximation[END_REF], the authors prove K Σ (B(0, r)) ≤ Cε r ε for all ε > 0 in convex geometries. This has just been extended by Phung [START_REF] Dang | Carleman commutator approach in logarithmic convexity for parabolic equations[END_REF]. Our Theorem 1.3 improves this result. Note also that [START_REF] Nakić | Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators[END_REF] gave results related to this in a periodic setting, tracking uniformity with respect to several parameters.

In the Euclidian space R n where ∆ is the usual flat Laplacian, spectral estimates as (9) can be interpretated as a manifestation of the uncertainty principle. Several results relying on this fact have been recently stated. We refer for instance to the recent articles [START_REF] Egidi | Sharp geometric condition for null-controllability of the heat equation on R d and consistent estimates on the control cost[END_REF] and [START_REF] Wang | Observable set, observability, interpolation inequality and spectral inequality for the heat equation in R n[END_REF] and the references therein.

The wave equation Lebeau [START_REF] Lebeau | Contrôle analytique. I. Estimations a priori[END_REF] proved in the analytic setting a result close to the fact that K wave (ω, T ) is finite for any open set ω and in optimal time T > 2L(M, ω). It was only very recently shown to be finite by the authors [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF] in a general C ∞ context. We refer the reader to the introduction of [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF] for a detailed discussion of the literature on unique continuation for waves, and estimates like (12)-( 13).

Estimates on analytic spaces of controllable data were computed by Allibert in the above described examples. We refer to Section 2.4 for more details about why they have implications on the constant K wave (and therefore K heat by Proposition 1.7). In [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF], he studies the example of the barrel as we describe it in Section 1.3.3. In [All99], he studies the example of a cylinder (0, π) × S 1 . The results he obtain in that paper should imply K wave (Γ, T ) ≤ C δ T 1-δ where Γ = {0} × S 1 and T > 2π. Notice finally that the blowup of the observability constant for the wave equation, when the time tends to the minimal geometric control time, has recently been investigated in [START_REF] Laurent | Uniform observability estimates for linear waves[END_REF].

Plan of the paper

The paper is divided in four main parts. In Section 2, we give the links between the different constants, proving in particular Propositions 1.6 and 1.7. We also interpret the description of the reachable set as an upper bound on the constant K wave (ω, T ).

In Section 3, we construct the various counterexamples on rotationally invariant geometries, presented in Section 1.3.1. This proves in particular Theorem 1.2.

Section 4 is devoted to the proof of the uniform Lebeau-Robbiano inequality on small balls, stated in Theorem 1.15.

Finally, we prove in Section 5 the observability inequality of Theorem 1.4 concerning positive solutions of the heat equation.

The paper ends with two appendices, in the first of which, Appendix A, we prove a uniform Carleman estimate for bounded families of Lipschitz metrics. Such an estimate is used as an intermediate in the proof of Theorem 1.15. The result also yields Theorem 1.16.

Note finally that in the companion paper [START_REF] Laurent | On uniform observability of gradient flows in the vanishing viscosity limit[END_REF], we apply similar techniques for the problem of uniform observability/controllability of transport equations in the vanishing viscosity limit.

Lemma 2.1. Let µ 0 ≥ 0, K ≥ 0 and assume that Λ > 0 and X ≥ 0 satisfy

1 Λ ≤ e Kµ X + 1 µ , for all µ > µ 0 . (25)
Then, for all α > 0, we have

1 ≤ 1 Λ+α≤µ0 µ 0 -α α e Kµ0 + 1 Λ+α>µ0 e Kα α Λ(Λ + α)e KΛ X. (26)
Let F : R + → R + be a nondecreasing function and assume that Λ > 0 and X ≥ 0 satisfy

Λ ≥ 1 and 1 ≤ F (Λ)X. (27)
Then, we have

1 Λ ≤ F (µ)X + 1 µ , for all µ > 0. ( 28
)
As a direct consequence of this lemma, we obtain the following corollary, clarifying the definition of K wave (ω, T ).

Corollary 2.2. Assume (12) with constants K, C, µ 0 > 0. Then, there is C > 0 such that

(u 0 , u 1 ) H 1 0 (M)×L 2 (M) ≤ C Λ 2 e KΛ u L 2 ((0,T )×ω) , Λ = (u 0 , u 1 ) H 1 0 (M)×L 2 (M) (u 0 , u 1 ) L 2 (M)×H -1 (M) , for all (u 0 , u 1 ) ∈ H 1 0 (M) × L 2 (M)
, and u solution to (11),

(u 0 , u 1 ) L 2 (M)×H -1 (M) ≤ C µ 2 e Kµ u L 2 ((0,T )×ω) + 1 µ (u 0 , u 1 ) H 1 0 (M)×L 2 (M) , for all µ > 0 and all (u 0 , u 1 ) ∈ H 1 0 (M) × L 2 (M)
, and u solution to (11). Reciprocally, if (13) holds with constants K , C > 0, then (12) holds with K = K , C = C , and µ 0 = 0 (and for all µ > 0).

In particular, we have

K wave (ω, T ) = inf {K > 0, ∃C > 0, µ 0 > 0 s.t. (12) holds} = inf {K > 0, ∃C > 0, s.t. (13) holds} = inf {K > 0, ∃C > 0, s.t. ( 12
) holds with µ 0 = 0 (and all µ > 0)} .

Proof of Lemma 2.1. Let α > 0. In case Λ + α > µ 0 , the assumption (25

) with µ = Λ + α > µ 0 yields 1 Λ 1 - Λ Λ + α ≤ e K(Λ+α) X,
and hence

1 ≤ 1 α e Kα Λ(Λ + α)e KΛ X. ( 29
) If now Λ + α ≤ µ 0 (and, in particular, α < µ 0 ), that is 1 Λ ≥ 1 µ0-α > 0, the assumption (25) implies 1 µ 0 -α ≤ 1 Λ ≤ e Kµ X + 1 µ , for all µ ≥ µ 0 .
This yields in particular

X ≥ 1 µ 0 -α - 1 µ e -Kµ , for all µ ≥ µ 0 ,
and hence X ≥ max µ≥µ0

1 µ0-α -1 µ e -Kµ ≥ α µ0-α e -Kµ0 > 0. With (29), this proves (26). Let us now prove (28). If Λ ≥ µ, then 1 Λ ≤ 1 µ and (28) holds. If Λ ≤ µ, then (27) gives 1 Λ ≤ 1 ≤ F (Λ)X ≤ F (µ)
X and (28) also holds in this case, concluding the proof.

2.2

The constant K eig (ω) as a lower bound for K heat (ω), K ∞ (ω), K wave (ω, T ): Proof of Proposition 1.6

We prove a slightly more precise version of Proposition 1.6.

Lemma 2.3. Assume that (1) holds with constants K, C > 0. Then, we have

ψ L 2 (M) ≤ C 2λ e 2 √ Kλ ψ L 2 (ω) , for all λ ∈ Sp(-∆ g ) \ {0} and ψ ∈ E λ . (30)
In particular,

K eig (ω) 2 4 ≤ K heat (ω). ( 31
)
Assume that (10) holds with constants K, C > 0. Then, there exists C > 0 such that

ψ L 2 (M) ≤ C λ 1/8 e 2 √ Kλ ψ L 2 (ω) , for all λ ∈ Sp(-∆ g ) \ {0} and ψ ∈ E λ . (32)
In particular

K eig (ω) 2 4 ≤ K ∞ (ω). ( 33
)
Assume that (13) holds in time T with constants C , K . Then, we have

ψ L 2 (M) ≤ T λ C e K √ λ ψ L 2 (ω) , for all λ ∈ Sp(-∆ g ) \ {0} and ψ ∈ E λ . (34)
In particular, for all T > 0, we have K eig (ω) ≤ K wave (ω, T ).

Proof of Proposition 1.6. From (1), applied to u(t, x) = e -tλ ψ(x) with λ ∈ Sp(-∆ g ) \ {0} and ψ ∈ E λ , we have

e -2T λ ψ 2 L 2 (M) ≤ Ce 2K T T 0 e -2tλ ψ 2 L 2 (ω) dt = Ce 2K T 1 -e -2T λ 2λ ψ 2 L 2 (ω)
, for all T > 0.

Taking T = D √ λ , with D > 0 to be chosen, this implies

ψ 2 L 2 (M) ≤ Ce 2T λ e 2K T 1 2λ ψ 2 L 2 (ω) = C 2λ e 2 √ λ(D+ K D ) ψ 2 L 2 (ω) .
Minimizing the exponent with respect to D leads to choosing D = √ K, which implies (30) when taking the square root. From (30), (31) follows directly when taking the infimum over all K.

Let us now prove the second statement of the proposition. From (10), again applied to u(t, x) = e -tλ ψ(x) with λ ∈ Sp(-∆ g ) \ {0} and ψ ∈ E λ , we have

R + e -2K t e -2tλ ψ 2 L 2 (M) dt ≤ C R + e -2tλ ψ 2 L 2 (ω) dt = C 2λ ψ 2 L 2 (ω) . (35)
The left hand-side may also be computed asymptotically for λ → +∞ using Laplace method, setting µ = √ λ, as

R + e -2K t e -2µ 2 t dt = R + e -2 √ Kµ( 1 s +s) √ K µ ds = (1 + o(1)) √ K µ R e -2 √ Kµ(2+(s-1) 2 ) ds = (1 + o(1)) √ K µ e -4 √ Kµ π 2 √ Kµ = (1 + o(1)) π √ K 2µ 3 1 2 e -4 √ Kµ .
From (35), we then obtain that, for all eigenfunction ψ associated to the eigenvalue µ 2 , for µ → ∞, we have

(1 + o(1)) π √ K 2µ 3 1 2 e -4 √ Kµ ψ 2 L 2 (M) ≤ C 2µ 2 ψ 2 L 2 (ω) .
Coming back to λ = µ 2 , this implies that the existence of C, λ 0 > 0 such that for all λ ≥ λ 0

ψ 2 L 2 (M) ≤ C λ 1/4 e 4 √ Kλ ψ 2 L 2 (ω) ,
and hence the sought result of (32). That of (33) follows as above.

Let us now prove the last statement of the proposition. We want to apply (13) to the function

u(t, x) = cos(t √ λ)ψ with λ ∈ Sp(-∆ g ) \ {0}
and ψ ∈ E λ , which is a particular solution to (11). We have

Λ = (u|t=0,∂tu|t=0) H 1 0 (M)×L 2 (M) (u|t=0,∂tu|t=0) L 2 (M)×H -1 (M) = ψ H 1 0 (M) ψ L 2 (M)
= √ λ and (13) then yields

√ λ ψ L 2 (M) = ψ H 1 0 (M) = u| t=0 , ∂ t u| t=0 ) H 1 0 (M)×L 2 (M) ≤ C e K Λ u L 2 ((0,T )×ω)
, where

u 2 L 2 ((0,T )×ω) = T 0 cos 2 (t √ λ) ψ 2 L 2 (ω) dt ≤ T ψ 2 L 2 (ω) .
This finally implies (34). The last result follows from Corollary 2.2. This concludes the proof of the proposition.

2.3 Link between K heat (ω) and K wave (ω, T ): Proof of Proposition 1.7

The proof will follow very closely the method of Ervedoza-Zuazua [START_REF] Ervedoza | Observability of heat processes by transmutation without geometric restrictions[END_REF], but with a different assumption. Note that this strategy was applied to approximate controllability problems for parabolic equations in [START_REF] Laurent | Tunneling estimates and approximate controllability for hypoelliptic equations[END_REF]. We first summarize the results of [START_REF] Ervedoza | Observability of heat processes by transmutation without geometric restrictions[END_REF][START_REF] Ervedoza | Sharp observability estimates for heat equations[END_REF] we need in the next proposition for readibility.

Proposition 2.4 ([EZ11a, EZ11b]). Let T, S > 0 and α > 2S 2 . Let L be a negative self adjoint operator.

Then, there exists a kernel function k T (t, s) such that

• if y is solution of the heat equation ∂ s w -Lw = 0, then w(s) = T 0 k T (t, s)y(t)dt is solution of ∂ 2 s w -Lw = 0, for s ∈] -S, S[, (w, ∂ s w)| s=0 = 0, T 0 ∂ s k T (t, 0)y(t)dt = 0, T 0 e -α( 1 t + 1 T -t ) y(t)dt ; (36)
• for all δ ∈]0, 1[ and all (t, s) ∈]0, T [×] -S, S[, we have

|k T (t, s)| ≤ |s| exp 1 min {t, T -t} s 2 δ - α (1 + δ) . ( 37
)
Note that this last estimate is most useful for δ sufficiently close to one so that α ≥ S 2 (1 + 1 δ ). We first prove from this proposition an observability inequality for data in E ≤λ (i.e. at low frequency) as a consequence of the approximate observability result for waves (13) (coming from [START_REF] Laurent | Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves[END_REF]), with a precise dependence on the cutoff frequency λ and the control time T . Combined with an argument of Miller [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF], this allows to prove observability for all data in L 2 (M) (still keeping track of the constants), and we finally conclude the proof of Proposition 1.7 at the end of the section.

Lemma 2.5. Assume that (13) holds on the time interval (0, 2S) and with constant K . Then, there are C, α 0 > 0 such that we have

e T ∆g y 0 2 L 2 (M) ≤ C(1 + λ)e 2K (1+λ) 1 2 e 18S 2 T T 0 e t∆g y 0 2 L 2 (ω) dt,
for all 0 < T ≤ α 0 , λ > 0 and y 0 ∈ E ≤λ .

Proof. For α > 2S 2 (to be fixed later on), we use the kernel k T described in Proposition 2.4. Let w(s) be associated to y by w(s) = T 0 k T (t, s)y(t)dt, where y(t) = e t∆g y 0 with y 0 ∈ E ≤λ . Then, in (36), W 0 is of the particular form W 0 = 0,

T 0 e -α( 1 t + 1 T -t ) y(t)dt , so that a calculation (see [EZ11a, Equation (3.3)]) yields W 0 2 L 2 ×H -1 L ≥ (1 + λ) -1 W 0 2 H 1 L ×L 2 = (1 + λ) -1 T 0 e -α( 1 t + 1 T -t ) y(t)dt 2 L 2 ≥ (1 + λ) -1 i |y i | 2 e -2λiT T 0 e -α( 1 t + 1 T -t ) dt 2 .
The integral can be estimated by Laplace method

T 0 e -α( 1 t + 1 T -t ) dt = T 1 0 e -α T ( 1 s + 1 1-s ) ds ≥ CT T α 1/2 e -4 α T , for α T ≥ 1,
since the non degenerate minimum of 1 s + 1 1-s is 4 reached at s = 1/2 and the function is positive. We have thus obtained

W 0 2 L 2 ×H -1 L ≥ C(1 + λ) -1 T 3 α -1 e -8α T y(T ) 2 L 2 . ( 38
)
Moreover, we have

W 0 ∈ E ≤λ × E ≤λ so that W 0 H 1 0 ×L 2 W 0 L 2 ×H -1 ≤ (1 + λ) 1 2 .
As a consequence, (13) on the time interval (-S, S) (which, by time translation invariance, is the same as on (0, 2S)) with constant K implies

W 0 L 2 ×H -1 L ≤ Ce K (1+λ) 1 2 w L 2 (]-S,S[×ω) . (39)
Using Cauchy-Schwarz inequality, we have

w 2 L 2 (]-S,S[×ω) ≤ ]0,T [×]-S,S[ k T (t, s) 2 dt ds T 0 ω |y(t, x)| 2 dx dt. (40)
Now, we use (37) with δ ∈ (0, 1) fixed sufficiently close to one so that α ≥ S 2 (1+δ) δ (which is possible since we have assumed α > 2S 2 ). This yields

]0,T [×]-S,S[ k T (t, s) 2 dt ds ≤ CS 2 ]0,T [×]-S,S[ exp 1 min {t, T -t} S 2 δ - α (1 + δ) dt ds ≤ CS 3 T. (41)
Combining ( 38), (39), ( 40) and (41) then gives the sought result, since the estimates true for any α > 2S 2 .

The following result, taken from [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF], deduces observability from low frequency observability. The values of the constants are tracked precisely.

Lemma 2.6 (Miller [Mil10]). Let T 0 , a, b, C > 0 and assume

e T ∆g y 0 2 L 2 (M) ≤ Ce 2aλ 1 2 + 2b T T 0 e t∆g y 0 2 L 2 (ω) dt,
for all 0 < T < T 0 and all y 0 ∈ E ≤λ .

Then, we have Proof. The result is not stated exactly that way, but the author proves this as an intermediate result of [Mil10, Theorem 2.2]. More precisely, the assumptions of our Lemma are exactly estimate (10) in [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF],

with α = 1/2 and β = 1. It gives the result with

c * = 4b 2 a + 2 √ b - √ a -4 = 1 4 a + 2 √ b + √ a 4 = a + √ b + a 2 + 2a √ b 2 .
With these two lemmata in hand, we now conclude the proof of Proposition 1.7.

Proof of Proposition 1.7. To simplify notations, we prove the existence of universal constants so that K heat (ω) ≤ α 3 S 2 + α 4 K wave (ω, 2S) 2 for all S > 0.

Let K > K wave (ω, 2S) so that there exists C > 0 so that we have the estimate (see Corollary 2.2 for the equivalence)

(u 0 , u 1 ) H 1 0 (M)×L 2 (M) ≤ Ce K Λ u L 2 ((-S,S)×ω) , Λ = (u 0 , u 1 ) H 1 0 (M)×L 2 (M) (u 0 , u 1 ) L 2 (M)×H -1 (M) ,
for all (u 0 , u 1 ) ∈ H 1 0 (M) × L 2 (M), and u solution to (11).

(42)

Note that when compared to (12)-( 13), we have changed the interval (0, 2S) to (-S, S) which gives the same result by conservation of energy. The proof is a direct consequence of above Lemmata 2.5 and 2.6.

Link between K wave (ω, T ) and analytic spaces

As already mentioned, Theorem 1.18 is a corollary of observability estimates in spaces of ultradistributions (implying by duality that some spaces of analytic functions are attainable/controllable for the control problem) obtained by Allibert [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF]. The following proposition explains (in the general setting of the paper) the link between such estimates and ( 12)-(13) (see also [START_REF] Lebeau | Contrôle analytique. I. Estimations a priori[END_REF]).

Proposition 2.7. Assume there are C 0 , C > 0 such that for all (u 0 , u 1 ) ∈ H 1 0 (M)×L 2 (M) and associated u solution of (11), we have

e -C0

√

-∆g (u 0 , u 1 )

L 2 (M)×H -1 (M) ≤ C u L 2 ((0,T )×ω) (resp. ≤ C ∂ ν u L 2 ((0,T )×Γ) ). (43)
Then (12) is satisfied with constant K = C 0 and all µ > 0. In particular, we have

K wave (ω, T ) ≤ C 0 , (resp. K wave (Γ, T ) ≤ C 0 ).
Again, in this statement, ∆ g denotes the Laplace operator with Dirichlet boundary conditions.

Proof. Given µ > 0, we decompose the data (u 0 , u 1 ) as u 0 = 1 √ -∆g≤µ u 0 + 1 √ -∆g>µ u 0 (and similarly for u 1 ). Here 1 √ -∆g≤µ denotes the orthogonal projector on the spectral space of -∆ g associated to eigenfunctions λ j with λ j ≤ µ. Remarking that

1 √ -∆g>µ (u 0 , u 1 ) L 2 (M)×H -1 (M) ≤ 1 µ 1 √ -∆g>µ (u 0 , u 1 ) H 1 0 (M)×L 2 (M) ≤ 1 µ (u 0 , u 1 ) H 1 0 (M)×L 2 (M) ,
we obtain

(u 0 , u 1 ) L 2 (M)×H -1 (M) ≤ 1 √ -∆g≤µ (u 0 , u 1 ) L 2 (M)×H -1 (M) + 1 µ (u 0 , u 1 ) H 1 0 (M)×L 2 (M) ≤ e C0µ e -C0 √ -∆g (u 0 , u 1 ) L 2 (M)×H -1 (M) + 1 µ (u 0 , u 1 ) H 1 0 (M)×L 2 (M) ≤ Ce C0µ u L 2 ((0,T )×ω) + 1 µ (u 0 , u 1 ) H 1 0 (M)×L 2 (M) ,
where we used the assumption (43) in the last inequality. This concludes the proof of (12), and that of the proposition.

We now extract an estimate like (43) on some surfaces of revolution from [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF]. Indeed, a combination of several estimates in [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF] gives the following result on barrel-type surfaces.

Theorem 2.8 (Allibert [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF]). Under the geometric assumptions of Section 1.3.3, for any T > T 1 and C 0 > R(z 0 )d A (Γ), there exists C > 0 so that e -C0

√

-∆g (u 0 , u 1 )

H 1 0 ×L 2 ≤ C ∂ ν u L 2 ((0,T )×Γ) (44)
for any (u 0 , u 1 ) ∈ H 1 0 (M) × L 2 (M) and associated solution u of (11). The result is not stated exactly this way in the article. It is also more precise since it involves analytic spaces only in the θ variable. More precisely, denoting E k 0 the spaces of functions in H 1 0 × L 2 of the form f (s)e ikθ , the following estimate is proved in [All98, Theorème 2, Définition 3 and Proposition 1]:

(u 0 , u 1 ) H 1 0 ×L 2 ≤ C(k) ∂ ν u L 2 ((0,T )×Γ) (45)
for any (u 0 , u 1 ) ∈ E k 0 , where C(k) satisfies

lim sup n→+∞ ln C(k) k = d A (Γ).
In particular, for any δ > 0, there is

k 0 ∈ N such that C(k) ≤ e k(d A (Γ)+δ) .
Recalling that 1/R has a unique minimum at z = z 0 , together with the action of ∆ z,θ on functions of the form f (s)e ikθ (see (53) in Remark 3.4, or the formula of

P n in [All98]), we see that -∆ z,θ (f (s)e ikθ ), f (s)e ikθ L 2 (M) ≥ k 2 f R 2 L 2 ≥ k 2 R(z0) 2 f (s)e ikθ 2
L 2 (M) (and a similar formula in H 1 0 ). Denoting λ k,n the n-th eigenvalue of the operator restricted to the space

E k 0 , this yields λ k,n ≥ k 2 R(z0) 2 and thus C(k) ≤ e (d A (Γ)+δ)R(z0) √ λ k,n for all k ≥ k 0
and n ∈ N. As a consequence of (45), we obtain for k ≥ k 0 e -(d A (Γ)+δ)R(z0) √ -∆g (u 0 , u 1 )

H 1 0 ×L 2 ≤ ∂ ν u L 2 ((0,T )×Γ) for all (u 0 , u 1 ) ∈ E k 0 .
This finally gives (44) for any C 0 > R(z 0 )d A , when taking into account the orthogonality of the subspaces E k 0 for the norm of H 1 0 × L 2 and the norm of the observation.

With Theorem 2.8 in hand, Theorem 1.18 is now a straightforward consequence of Propositions 2.7 and 1.7.

Reformulation of the definition of the constants in terms of localization functions

This section is aimed at giving an alternative definition for the geometric constants K eig (ω), K Σ (ω), K heat (ω) in terms of localization functions.

Definition 2.9. Let ω ⊂ M be an open set. We set:

Loc eig (ω, λ) = inf ψ L 2 (ω) ψ L 2 (M) , ψ ∈ E λ \ {0} ∈ [0, 1], λ ∈ Sp(-∆ g ), Loc Σ (ω, λ) = inf u L 2 (ω) u L 2 (M) , u ∈ E ≤λ \ {0} ∈ [0, 1], Loc heat (ω, T ) = inf e t∆ u 0 L 2 ((0,T )×ω) e T ∆ u 0 L 2 (M) , u 0 ∈ L 2 (M) \ {0} .
Note that if the Schrödinger equation is observable from ω in finite time (in particular if ω satisfies the geometric control condition, see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF][START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF]), then, there exists C > 0 so that Loc eig (ω, λ) ≥ C for all λ ∈ Sp(-∆ g ). Under the sole assumption that ω = ∅, we have

Loc eig (ω, λ) ≥ C -1 e -C √ λ [DF88, LR95], Loc Σ (ω, λ) ≥ C -1 e -C √ λ [LR95, JL99, LZ98] and Loc heat (ω, T ) ≥ C -1 e -C T [FI96, Mil10].
Lemma 2.10. We have

K eig (ω) = lim sup λ→+∞,λ∈Sp(-∆g) -log Loc eig (ω, λ) √ λ , K Σ (ω) = lim sup λ→+∞ -log Loc Σ (ω, λ) √ λ , K heat (ω) = lim sup T →0 + -T log Loc heat (ω, T ),
Note that we do not have a similar formulation for the constants K ∞ (ω) and K wave (ω, T ) since they do not correspond to an asymptotic régime (like T → 0 or λ → +∞).

Proof. We only prove the second statement, the other proofs being similar. Setting

C Σ (ω) = lim sup λ→+∞ -log Loc Σ (ω, λ) √ λ ,
we want to prove that C Σ (ω) = K Σ (ω). Assume K, C satisfy (9), then we have

Loc Σ (ω, λ) ≥ 1 C e -K √ λ ,
and hence

-log Loc Σ (ω, λ) √ λ ≤ K √ λ + log(C) √ λ .
Taking the lim sup λ→+∞ , this implies C Σ (ω) ≤ K. Taking the infimum over all such K and recalling Definition 1.5, we obtain C Σ (ω) ≤ K Σ (ω).

We now prove the converse inequality. The definition of C Σ (ω) implies that for all ε, there exists λ 0 (ε) such that for all λ ≥ λ 0 (ε),

-log Loc Σ (ω, λ) √ λ ≤ C Σ (ω) + ε, that is Loc Σ (ω, λ) ≥ e -(CΣ(ω)+ε) √ λ .
This, together with the fact that Loc Σ (ω, λ) > 0 does not vanish on

[0, λ 0 (ε)], implies the existence of a constant C(ε) > 1 such that Loc Σ (ω, λ) ≥ 1 C(ε) e -(CΣ(ω)+ε)
√ λ for all λ ≥ 0. This is precisely estimate (9) with K = C Σ (ω) + ε and C = C(ε). Taking the infimum over all such K and recalling Definition 1.5, we obtain K Σ (ω) ≤ C Σ (ω) + ε for all ε > 0, and hence K Σ (ω) ≤ C Σ (ω), which concludes the proof.

3 Construction of maximally vanishing eigenfunctions

The sphere

In this section, we consider the simplest case of our results that is, the unit sphere in R 3 :

S 2 = {(x 1 , x 2 , x 3 ) ∈ R 3 , x 2 1 + x 2 2 + x 2 3 = 1} = {x ∈ R 3 , |x| = 1}.
Eigenfunctions and eigenvalues of the Laplace-Beltrami operator on S 2 are well-understood : eigenfunctions are restrictions to S 2 of harmonic homogeneous polynomials of R 3 , associated to the eigenvalue k(k + 1),

where k is the degree of the polynomial. We are particularly interested in so called equatorial spherical harmonics, given by

u k = P k | S 2 ∈ C ∞ (S 2 ), P k (x 1 , x 2 , x 3 ) = (x 1 + ix 2 ) k ,
known to concentrate exponentially on the equator given by x 3 = 0.

Since it can be written P k = z k where z = x 1 + ix 2 ∈ C, it is easy to check that P k is holomorphic as a function of z, and hence harmonic as a function of (x 1 , x 2 , x 3 ) ∈ R 3 . Moreover, P k is homogeneous of degree k. Therefore, see e.g. [Shu01, Proposition 22.2 p169], the function u k is an eigenfunction of the Laplace-Beltrami operator on S 2 :

-∆ S 2 u k = λ k u k with λ k = k(k + 1)
(this fact can also be checked directly with the expression in ( 46)). Note that we have

|u k (ω)| 2 = (x 2 1 + x 2 2 ) k = (1 -x 2 3 ) k , ω = x |x| .
We denote by N = (0, 0, 1) and S = (0, 0, -1), the north and south poles, and have coordinates :

(0, π) × S 1 → S 2 \ {N, S} (s, θ) → (sin s cos θ, sin s sin θ, cos s)
Remark that s(x) = dist g (x, N ), for x ∈ S 2 . In these coordinates, the metric is given by ds 2 + (sin s) 2 dθ 2 , the Riemannian volume element is dω = sin sdsdθ, and the sequence u k is defined by

u k (s, θ) = sin(s) k e ikθ . ( 46
)
Remark 3.1. The construction works equally well in the unit sphere S n ⊂ R n+1 , n ≥ 2. The coordinates has to be changed by

(0, π) × S 1 × S n-2 → S n \ {N, S} (s, θ, t) → (sin s cos θ, sin s sin θ, t cos s)
and we can still consider the eigenfunction

u k = (x 1 +ix 2 ) k | S n with -∆ S n u k = λ k u k and λ k = k(k +n-1).
With the above choice of the eigenfunction u k , we have

|u k (x)| 2 = (1 -x 2 3 ) k = (sin s) 2k = | sin dist g (x, N )| 2k = e -2kd A (x) , d A (x) = -log sin dist g (x, N ).
Note that d A is actually the Agmon distance to the equator (s = π 2 ) where S 2 is seen as a surface of revolution, see Remark 1.13. Also, given f ∈ L 1 (S 2 ), we have

S 2 f (ω)|u k (ω)| 2 dω = (0,π)×S 1 f (s, θ)(sin s) 2k+1 dsdθ = 2π (0,π) F (s)(sin s) 2k+1 ds, F (s) = 1 2π S 1 f (s, θ)dθ.
In case f = 1, this yields the asymptotics of the norm of u k , given by the Laplace method (see e.g. [START_REF] Erdélyi | Asymptotic expansions[END_REF][START_REF] Copson | Asymptotic expansions[END_REF]):

1 2π u k 2 L 2 (S 2 ) = 1 2π S 2 |u k (ω)| 2 dω = 1 -1 (1 -x 2 3 ) k dx 3 = 1 -1 e k log(1-x 2 3 ) dx 3 = (1 + O( 1 k )) R e -kx 2 3 dx 3 = π k (1 + O( 1 k )),
and hence u k L 2 (S 2 ) ∼ 2 1/2 π 3/4 k -1/4 as k → +∞.

We have the elementary estimate

u k 2 L 2 (B(N,r)) = 2π r 0 (sin s) 2k+1 ds ≤ π k + 1 r 2k+2 .
This can be slightly refined, e.g. by writing

u k 2 L 2 (B(N,r)) - π k + 1 (sin r) 2k+2 = u k 2 L 2 (B(N,r)) -2π r 0 cos s(sin s) 2k+1 ds = 2π r 0 (1 -cos s)(sin s) 2k+1 ds ≤ r 2 2 2π r 0 (sin s) 2k+1 ds = r 2 2 u k 2 L 2 (B(N,r)) .
To be a little more precise, let us now prove an equivalent for u k 2 L 2 (B(N,r)) as k → ∞, which is uniform in r.

Lemma 3.2. For all k ∈ N * and all r ∈ [0, π

2 ), we have

u k 2 L 2 (B(N,r)) = π k + 1 sin(r) 2k+2 cos(r) (1 + R) , with |R| ≤ tan(r) 2 2k + 2 .
This furnishes an optimal lower/upper bound for this quantity which is uniform with respect to r in any compact set [0, α] with α < π 2 . Proof. We write a = -log sin r > 0, change variable y = -log sin s, and want to have an asymptotic expansion of

1 2π u k 2 L 2 (B(N,r)) = r 0 (sin s) 2k+1 ds = +∞ a e -(2k+2)y 1 √ 1 -e -2y
dy.

This integral is of the form

I(a, k) := +∞ a e -(2k+2)y f (y)dy,
where

f (y) = 1 √ 1-e -2y is smooth on [a, +∞). Writing |f (y) -f (a)| ≤ (y -a) sup [a,∞) |f | ≤ (y -a)
e -2a (1 -e -2a ) 3/2 , since f (y) = -e -2y (1 -e -2y ) -3/2 and integrating on (a, +∞), we obtain

I(a, k) -f (a) e -(2k+2)a 2k + 2 ≤ e -(2k+2)a (2k + 2) 2 e -2a (1 -e -2a ) 3/2 .
Coming back to the original notation, this is precisely

1 2π u k 2 L 2 (Oε) - sin(r) 2k+2 (2k + 2) cos(r) ≤ sin(r) 2k+4 (2k + 2) 2 cos(r) 3 = sin(r) 2k+2 (2k + 2) 2 cos(r) tan(r) 2 ,
which concludes the proof of the lemma.

Note that the eigenfunctions we have constructed are complex valued. Yet, since u k = (sin(s)) k e ikθ , its real part Re(u k ) = (sin(s)) k cos(kθ) is a real-valued eigenfunction to which the same estimates hold, except that S 1 |e ikθ | 2 dθ = 2π should be replaced by S 1 cos(kθ) 2 dθ = π.

General surfaces of revolution

In this section we consider a revolution surface S ⊂ R 3 being diffeomorphic to a sphere S 2 , generalizing the results of Section 3.1. We follow [Bes78, Chapter4 B p95] for the precise geometric description of such manifolds.

Assume that (S, g) is an embedded submanifold of R 3 (endowed with the induced Euclidean structure), having S 1 = (R/2πZ) ∼ SO(2) as an effective isometry group. The action of S 1 on S, denoted by θ → R θ (such that R θ S = S) has exactly two fixed points denoted by N, S ∈ S (the so-called North and South poles).

We now describe a nice parametrization of (S, g). Let L = dist g (N, S) and γ 0 be a geodesic from N to S (thus with length L). For any θ ∈ S 1 , the isometry R θ transforms the geodesic γ 0 into R θ (γ 0 ), which is another geodesic joining N to S. Set U = S \ {N, S}. For every m ∈ U , there exists a unique θ ∈ S 1 such that m belongs to R θ (γ 0 ). The geodesic R θ (γ 0 ) can be parametrized by arclength

ρ : [0, L] → R θ (γ 0 ), ρ(0) = N, ρ(L) = S, s = dist g (ρ(s), N ) = L -dist g (ρ(s), S),
and there exists a unique s ∈ (0, L) such that ρ(s) = m. We use (s, θ) as a parametrization of U ⊂ S:

ζ : U = S \ {N, S} → (0, L) × S 1 m → ζ(m) = (s, θ).
We define two other exponential charts (U N , ζ N ) and (U S , ζ S ) centered at the fixed points N and S by

U N = {N } ∪ ζ -1 0, L 2 × S 1 = B g N, L 2 ⊂ S, U S = {S} ∪ ζ -1 L 2 , L × S 1 = B g S, L 2 ⊂ S, ζ N : U N → B R 2 0, L 2 , ζ N (N ) = 0, ζ S : U S → B R 2 0, L 2 , ζ S (S) = 0,
with the transition maps

ζ N • ζ -1 : ζ U ∩ U N = 0, L 2 × S 1 → ζ N U ∩ U N = B R 2 0, L 2 \ {0} (s, θ) → s cos(θ), s sin(θ) , and
ζ S • ζ -1 : ζ U ∩ U S = L 2 , L × S 1 → ζ S U ∩ U S = B R 2 0, L 2 \ {0} (s, θ)
→ (L -s) cos(θ), (L -s) sin(θ) .

On the cylinder (0, L) × S 1 , the metric g is given by

(ζ -1 ) * g = ds 2 + R(s) 2 dθ 2
for some smooth function R : (0, L) → R + * (see below Remark 3.4 for the geometric interpretation of R). Since g is a smooth metric on S, [Bes78, Proposition 4.6] gives that R extends to a

C ∞ function [0, L] → R + satisfying R(0) = R(L) = 0, R (0) = 1, R (L) = -1, R (2p) (0) = R (2p) (L) = 0 for any p ∈ N. (47)
In these coordinates, the Riemannian volume form is hence R(s)dsdθ, the Riemannian gradient of a function is

∇ g f = ∂ s f ∂ ∂s + 1 R(s) 2 ∂ θ f ∂ ∂θ , with g(∇ g f, ∇ g f ) = |∂ s f | 2 + 1 R(s) 2 |∂ θ f | 2 (48)
and the Laplace-Beltrami operator is given by

∆ s,θ = 1 R(s) ∂ s R(s)∂ s + 1 R(s) 2 ∂ 2 θ . (49)
Another important operator is the infinitesimal generator X θ of the group (R θ ) θ∈S 1 , defined, for f ∈ C ∞ (S), by

X θ f = lim θ→0 θ -1 (f • R θ -f ). (50)
In the chart (U, ζ), the action of R θ is given by (ζ -1 ) * R θ (u, θ ) = (u, θ + θ), so that (ζ -1 ) * X θ = ∂ θ . Let us now check that X θ is a smooth vector field. Indeed, we have

(ζ -1 N ) * X θ = (ζ -1 N ) * ζ * ∂ θ = d ζ N • ζ -1 • ∂ θ , and hence (ζ -1 N ) * X θ s cos(θ), s sin(θ) = (-s sin(θ)∂ x1 + s cos(θ)∂ x2 ) s cos(θ), s sin(θ) , that is (ζ -1 N ) * X θ (x 1 , x 2 ) = -x 2 ∂ x1 + x 1 ∂ x2 . Since (ζ -1
N ) * X θ (0) = 0 (and since the computation is similar in U S ), we have obtained that X θ is smooth. Note also that X θ (N ) = X θ (S) = 0 and that its norm is given by g

(X θ , X θ )(s, θ) = R(s) (in the coordinates of U ).
We define by L 2 (S) := L 2 (S, d Vol g ) the space of square integrable functions, which is also invariant by the action of (R θ ) θ∈S 1 . Now, remark that (R θ ) θ∈S 1 acts as a (periodic) one-parameter unitary group on L 2 (S) by f → f • R θ . The Stone Theorem (see e.g. [RS80, Theorem VIII-8 p266]) hence implies that its infinitesimal generator is iA, where A is a selfadjoint operator on L 2 (S) with domain D(A) ⊂ L 2 (S). Since iAf = X θ f for f ∈ C ∞ (S) (which is dense in D(A)) according to (50), we have that A is the selfadjoint extension of X θ i . From now on, we slightly abuse the notation and still denote X θ i for its selfadjoint extension A. Since g is invariant by the action of R θ , we have

[X θ , ∆ g ] = 0.
Moreover, ∆ g has compact resolvent, so that the operators ∆ g and X θ share a common basis of eigenfunctions: indeed, X θ /i is selfadjoint and preserves each (finite dimensional) eigenspace of ∆ g , and it can be diagonalized on these spaces. In U a common eigenfunction can be written as

e ikθ f (s) with k ∈ Z, f ∈ C ∞ (0, L) ∩ L 2 ((0, L), R(s)ds) solution of - 1 R(s) ∂ s (R(s)∂ s f ) + k 2 R(s) 2 f = λf, (51)
for some λ ≥ 0, eigenvalue for -∆ g . To prove this assertion, take u a necessarily smooth common eigenfunction of ∆ g and X θ . In U (with the coordinates (s, θ)), we have u = u(s, θ) with (see (49) for the definition of ∆ s,θ )

-∆ s,θ u(s, θ) = λu(s, θ), and

∂ θ i u(s, θ) = µu(s, θ), (s, θ) ∈ (0, L) × S 1 , (52)
for some λ, µ ∈ R. Setting f (s) := u(s, 0), the second identity in (52) implies u(s, θ) = e iµθ f (s). The function u being smooth on (0, L) × S 1 , it is 2π-periodic in θ so that µ = k ∈ Z. Hence, u(s, θ) = e ikθ f (s) and the first identity in (52) directly yields (51). We will call these normalized eigenfunctions ϕ k,n = e ikθ f k,n (s) with eigenvalues λ k,n for -∆ g , where n ∈ N. In particular, we can write

L 2 (S) = ⊕ ⊥ (k,n)∈Z×N span(ϕ k,n ). We will denote L 2 k = ker(X θ -ik)) = ϕ ∈ L 2 (S); ϕ |U = e ikθ f (s), f ∈ L 2 ((0, L), R(s)ds) and H 2 k = H 2 (S) ∩ L 2 k . The commutation property implies that ∆ g H 2 k ⊂ L 2 k , so we can define the operator ∆ k = ∆ g|L 2 k which is self-adjoint with domain H 2
k . This can be seen for instance directly on the simultaneous diagonalization which implies an isometry L 2 (S) ≈ 2 (Z × N) where L 2 k ≈ {(k, n) |n ∈ N } as a closed subspace of 2 (Z × N). The fact that ∆ g has compact resolvent implies the same for ∆ k .

Remark 3.3. Note that the introduction of X θ allows to give a more intrinsic definition of d A introduced in (17): given any point m 0 on the "strict global non-degenerate equator" of S, the Agmon distance d A is the unique continuous function such that

X θ d A = 0, d A (m 0 ) = 0, |∇ g d A | 2 g (m) - 1 g(X θ , X θ )(m) - 1 g(X θ , X θ )(m 0 ) = 0.
All properties of Lemma 3.8 can be formulated intrinsically since s measures the geodesic distance to the north pole, and hence s(m) = dist g (m, N ), L -s(m) = dist g (m, S), and s(m) -s 0 = dist g (m, equator).

Remark 3.4. (Another possible parametrization) Some of the surfaces of revolution described above admit the following "cylindrical" parametrization on the set U : with z -< z + and the two poles N = (0, 0, z + ) and S = (0, 0, z -), we have

(z -, z + ) × S 1 → U = S \ {N, S} ⊂ R 3 (z, θ) → (R(z) cos θ, R(z) sin θ, z),
where R : [z -, z + ] → (0, ∞) is the profile of the surface, that is, a smooth positive function on (z -, z + ) satisfying R(z ± ) = 0 and lim z→z± R (z) = ∓∞. Note that R(z) represents the distance of S to the revolution axis {x = y = 0} at height z. Note that (except for the shape/topology of the surface) this parametrization is the same as that of Allibert [START_REF] Allibert | Contrôle analytique de l'équation des ondes et de l'équation de Schrödinger sur des surfaces de révolution[END_REF], see Section 1.3.3. We have

   dx 1 = R (z) cos θdz -R(z) sin θdθ dx 2 = R (z) sin θdz + R(z) cos θdθ dx 3 = dz,
so that the metric on S induced by the Euclidean structure is given by

g = dx 2 1 + dx 2 2 + dx 2 3 = (1 + R (z) 2 )dz 2 + R(z) 2 dθ 2 .
As a consequence, the Riemannian volume element is V(z)dzdθ with V(z) = R(z) 1 + R (z) 2 and the Laplace-Beltrami operator is given in this coordinates, by

∆ z,θ = 1 V(z) ∂ z V(z) 1 + R (z) 2 ∂ z + 1 R(z) 2 ∂ 2 θ , (53)
with a suitable selfadjoint extension on L 2 (z -, z + ) × S 1 , V(z)dzdθ . The link between s and z is the following diffeomorphism

s(z) = z z- 1 + R (t) 2 dt,
and we have L =

z+ z- 1 + R (t) 2 dt, together with R(s(z)) = R(z)(= g(X θ , X θ )).
In particular, we see that R(s) indeed measures the distance to the axis of revolution.

Remark 3.5 (The sphere). Note that, in the z-parametrization, the sphere is given by z ± = ±1 and r(z) = √ 1 -z 2 and hence r (z) = -z √ 1-z 2 and V(z) = 1 is smooth (which is not the general case if the surface is flat near the poles).

Let us first prove existence of the particular eigenfunctions under interest in Theorem 1.9. We then study their concentration/deconcentration properties.

Lemma 3.6. Assume that s → R(s) admits a non-degenerate local maximum at s 0 ∈ (0, L). Then, for all k ∈ N, there exists

ψ k ∈ C ∞ (S) ∩ L 2 k , and µ k ∈ R such that µ k = 1 R(s0) 2 + 1 k |R (s0)| R 3 (s0) + O( 1 k 3 2
),

ψ k L 2 (S) = 1, and we have -∆ g ψ k = k 2 µ k ψ k .
Note that the assumption of the lemma is R (s 0 ) = 0 and R (s 0 ) < 0. In the proofs below, we shall often consider h = k -1 as a semiclassical parameter.

Proof. We first construct a family of sufficiently accurate quasimodes (i.e. approximate eigenfunctions) compactly supported in U and of the form (in the coordinates (s, θ) of U ) e ikθ u k (s). The function u k (s) should thus solve (51) approximately. Setting h = k -1 and µ = λh 2 in that equation, we are left with the following semiclassical eigenvalue (or approximate eigenvalue) problem in the limit h → 0

+ (P h -µ)f = - h 2 R(s) ∂ s (R(s)∂ s f ) + 1 R(s) 2 -µ f = 0.
Now, according to the assumption, the potential 1 R(s) 2 is positive, tends to plus infinity near 0 and L, and admits 1 R(s0) 2 as a nondegenerate local minimum. Namely, this is R (s 0 ) = 0 and R (s 0 ) < 0. The construction is classical (harmonic approximation) and follows e.g. that of [DS99, Theorem 4.23] in a simpler setting. The idea is to approximate the operator P h by its harmonic approximation at s 0 , namely

Ph := - h 2 R(s 0 ) ∂ s R(s 0 )∂ s + 1 R(s 0 ) 2 + 1 R 2 (s 0 ) (s -s 0 ) 2 2 = -h 2 ∂ 2 s + 1 R(s 0 ) 2 - 2R (s 0 ) R 3 (s 0 ) (s -s 0 ) 2 2 (54)
Recall that the spectrum of the operator -h 2 ∂ 2 y + c 0 y 2 on R (c 0 > 0) is given by E n (h) = hE n (1) = h(2n + 1) √ c 0 , associated with the eigenfunctions u h n (y) = h -1 4 u 1 n (y/ √ h) where u 1 n (y) = p n (y)e -√ c0 y 2 2 (p n being a Hermite polynomial). Here, this applies with c 0 = |R (s0)| R 3 (s0) . We consider a cutoff function χ ∈ C ∞ c (0, L) such that χ = 1 in a neighborhood of s 0 . We set

u h (s) = χ(s)u h 0 (s), with u h 0 (s) = Ch -1 4 e -√ c0 (s-s 0 ) 2 2h , ( 55
)
where C is a normalizing constant, and prove this is an approximate eigenfunction (quasimode). First notice that we have, with Ph defined in (54), that

Ph u h = χ Ph u h 0 + [ Ph , χ]u h 0 = 1 R(s 0 ) 2 + h √ c 0 χu h 0 + [-h 2 ∂ 2 s , χ]u h 0 .
In this expression, [-h 2 ∂ 2 s , χ] is a first order differential operator supported away from zero, where u h 0 and its derivatives are exponentially small. This yields

Ph u h - 1 R(s 0 ) 2 + h √ c 0 u h L 2 (0,L),R (s)ds
= O(e -c/h ).

Now we consider, with norms L 2 (0, L), R(s)ds

P h - 1 R(s 0 ) 2 + h √ c 0 u h L 2 ≤ P h -Ph u h L 2 + Ph u h - 1 R(s 0 ) 2 + h √ c 0 u h L 2 ≤ h 2 R(s) ∂ s R(s)∂ s -h 2 ∂ 2 s u h L 2 + 1 R(s) 2 - 1 R(s 0 ) 2 -c 0 (s -s 0 ) 2 u h L 2 + Ce -c/h .
According to the Taylor formula and the definition of c 0 , we have

1 R(s) 2 -1 R(s0) 2 -c 0 (s -s 0 ) 2 = O((s -s 0 ) 3 ) on the support of χ, so that 1 R(s) 2 - 1 R(s 0 ) 2 -c 0 (s -s 0 ) 2 u h 2 L 2 ≤ C R |(s -s 0 ) 3 h -1 4 e -√ c0 (s-s 0 ) 2 2h | 2 dz ≤ Ch 3 .
We now estimate the term

h 2 R(s) ∂ s R(s)∂ s -h 2 ∂ 2 s u h L 2 = hR (s) R(s) h∂ s u h L 2 Notice that h∂ s u h = hχ u h 0 + hχ∂ s u h 0 = O L 2 (e -c/h ) - √ c 0 (s -s 0 )χu h 0 ,
where we have used the expression of u h 0 in (55). Moreover, since R (s 0 ) = 0, the Taylor formula yields

hR (s) R(s) h∂ s u h L 2 ≤ Ce -c/h + C h(s -s 0 ) 2 χu h 0 L 2 ≤ Ch 2 .
Now, combining the above estimates finally yields the existence of constants D, h 0 > 0 such that for all h < h 0 , we have, with

ν h = 1 R(s0) 2 + h √ c 0 , (P h -ν h )u h L 2 (0,L),R(s)ds ≤ Dh 3/2 ≈ Dh 3/2 u h L 2 (0,L),R (s)ds
. Now, we define in coordinates in U ⊂ S, f k (s, θ) = e ikθ u h (s), h = k -1 . This function is smooth and compactly supported in U thanks to the cutoff χ, and can therefore be extended as a function in C ∞ (S)∩L 2 k , still denoted f k , which satisfies

(h 2 ∆ k -ν h )f k L 2 k ≤ Dh 3/2 ≈ Dh 3/2 f k L 2 k . Hence, if ν h / ∈ Sp(-h 2 ∆ k ), this implies (-h 2 ∆ k -ν h ) -1 L 2 k →L 2 k ≥ 1 Dh 3/2 . Finally, the operator h 2 ∆ k being selfadjoint on L 2 k , we have, for z ∈ C \ Sp(-h 2 ∆ k ), (-h 2 ∆ k -z) -1 = 1 d(z,Sp(-h 2 ∆ k )) , so that, if ν h / ∈ Sp(-h 2 ∆ k ), 1 d(ν h , Sp(-h 2 ∆ k )) ≥ 1 Dh 3/2 .
In any case, this implies d(ν h , Sp(-h 2 ∆ k )) ≤ Dh 3/2 , and using that the spectrum of h 2 ∆ k is purely pointwise, this proves the sought result.

The next step is to study the behavior of the eigenfunction ψ k constructed in the previous lemma (and under a stronger assumption on the point s 0 ). This is the goal of the so-called Agmon estimates. We first need the following integration-by-parts lemma.

Lemma 3.7. For all Ψ ∈ W 1,∞ (S) real valued and all w ∈ H 2 (S), we have

S |∇ g (Ψw)| 2 g d Vol g - S |∇ g Ψ| 2 g |w| 2 d Vol g = Re S |Ψ| 2 (-∆ g w)w d Vol g .
Proof. For Ψ ∈ C 2 (S), this is a direct consequence of the integration by parts formula (also valid when S has a boundary ∂S and w| ∂S = 0)

S |∇ g (Ψw)| 2 g d Vol g = - S ∆ g (Ψw)Ψwd Vol g = Re S -Ψ(∆ g w) -(∆ g Ψ)w -2∇ g Ψ • ∇ g w Ψwd Vol g = Re S |Ψ| 2 (-∆ g w)w d Vol g + A with A = Re S -(∆ g Ψ)Ψ|w| 2 -2∇ g Ψ • ∇ g wΨw d Vol g = Re S |∇ g Ψ| 2 |w| 2 + ∇ g Ψ • ∇ g (|w| 2 )Ψ -2∇ g Ψ • ∇ g wΨw d Vol g = S |∇ g Ψ| 2 |w| 2 d Vol g ,
where we integrated by parts in the second line. This is the sought estimate in case Ψ ∈ C 2 (S). The result of the lemma follows by a classical approximation argument, see e.g. [DS99, Proof of Proposition 6.1].

We shall now assume that R reaches at s 0 a strict global non-degenerate maximum, and introduce the relevant Agmon distance to the "equator" s = s 0 . The latter is defined in the coordinates of U by the eikonal equation (17), or, more explicitely, for s ∈ (0, L), by (18). Lemma 3.8 (Properties of d A ). Assume that R reaches at s 0 a strict global non-degenerate maximum. Then, d A ∈ C 2 (0, L), and we have

d A (s) = -log(s) + O(1), as s → 0 + , d A (s) = -log(L -s) + O(1), as s → L -, (56)
d A (s) = 1 2 -R (s 0 ) R 3 (s 0 ) (s -s 0 ) 2 + O((s -s 0 ) 3 ), as s → s 0 . (57)
Proof. Remark that according to (47), we have 1 R(y) → +∞ as y → 0 + or y → L -, with R(s) = s + O(s 3 ), when s → 0 + , and R(s) = L -s + O((L -s) 3 ), when s → L -.

As a consequence, with (18), we obtain d A (s) = s s0 1 y (1 + O(y 2 ))dy = -log(s) + O(1), as s → 0 + (and similarly when s → L -), that is (56).

Let us also study the behavior of d A near s 0 . Denoting

V (s) = 1 R(s) 2 -1 R(s0) 2 , we have V (s 0 ) = V (s 0 ) = 0 and V (s 0 ) = -2R (s0)
R 3 (s0) > 0. This implies (57) and that d A is of class C 2 near s 0 , by Taylor expansion of d A and its derivatives.

We can now state the following relatively precise result. All results concerning surfaces of revolution are corollaries of this one.

Theorem 3.9 (Agmon estimate). Assume that R reaches at s 0 a strict global non-degenerate maximum, and consider the associated numbers µ k and functions ψ k given by Lemma 3.6. There exist

C, C 0 , k 0 > 0 such that, for all k ∈ N, k ≥ k 0 , ψ k ∈ L 2 (S, e kd A d Vol g ) with the estimate S e 2kd A (m) |ψ k | 2 (m)d Vol g (m) ≤ Ck 2C0 .
Here, we have denoted d A (m) for d A (s(m)) with a slight abuse of notation. Note that d A (N ) = d A (S) = +∞. We first draw corollaries of this result, concluding the proof of Theorem 1.9, and then prove Theorem 3.9 at the end of the section. Using that d A is decreasing on (0, s 0 ], we obtain the following direct Corollary.

Corollary 3.10. Under the assumptions of Theorem 3.9, there exist C, C 0 , k 0 > 0 such that, for all k ∈ N, k ≥ k 0 and all s 1 ≤ s 0 , we have

B(N,s1) |ψ k | 2 d Vol g ≤ Ck 2C0 e -2d A (s1)k .
From this result, we may now derive a proof of Theorem 1.9 and Corollary 1.11.

Proof of Theorem 1.9 and Corollary 1.11. The eigenfunctions constructed in Lemma 3.6 satisfy

λ k = k 2 1 R(s0) 2 + 1 k |R (s0)| R 3 (s0) + O( 1 k 3 2
) . In particular, for any

C * > 1 2 |R (s 0 )|R(s 0 ), there is k 0 ∈ N such that k ≥ √ λ k R(s 0 ) -C * for k ≥ k 0 .
This gives e -2kd A (s1) ≤ e 2C * d A (s1) e -2d A (s1)R(s0) √ λ k . Then, Theorem 1.9 follows directly from Corollary 3.10 up to changing the constants involved. The second part of Theorem 1.9 follows directly from Proposition 1.6.

Corollary 1.11 follows from the asymptotics (56) of d A and the fact than Theorem 1.9 is uniform for r small. Indeed, for an appropriate constant C, we have d A (s) ≥ -log(s) -C for all 0 < s 1 ≤ s 0 , .

Finally, for fixed λ k and using the uniformity for r small, we obtain the order of vanishing using the general Lemma B.1 of the Appendix.

We will need a very simple Lemma Lemma 3.11. Let ϕ ∈ W 1,∞ (S) ∩ L 2 k , then, we have the pointwise estimate on U

|∇ g (ϕ)| 2 g ≥ k 2 g(X θ , X θ ) |ϕ| 2 .
Proof. We have, in the coordinates of U , that ϕ writes ϕ(s, θ) = e ikθ f (s), with, according to (48),

|∇ g (ϕ)| 2 g = |∂ s f | 2 + 1 R(s) 2 |∂ θ (e ikθ f (s))| 2 = |∂ s f | 2 + k 2 R(s) 2 |e ikθ f (s)| 2 ≥ k 2 R(s) 2 |e ikθ f (s)| 2 = k 2 g(X θ , X θ ) |ϕ| 2 ,
which is the sought result.

Let us now give a proof of Theorem 3.9, following that of [Hel88, Proposition 3.3.5].

Proof of Theorem 3.9. As in the above proof, we use the notation h = k -1 , considered as a semiclassical parameter. We define, for some constant C 0 > 1, h 0 > 0 and h ∈ (0, h 0 ) the sets

Ω -= {s ∈ (0, L), d A (s) ≤ C 0 h}, Ω + = {s ∈ (0, L), d A (s) > C 0 h}, We set φ(s) = d A (s) -C 0 h log(C 0 ), for s ∈ Ω -, = d A (s) -C 0 h log(d A (s)/h), for s ∈ Ω + . For M > 1, set φ M = min(φ, M ) and Ω M = φ -1 M ({M }). Moreover, on Ω -, we have φ = d A -C 0 h log(C 0 ) ≤ d A ≤ C 0 h < C 0 h 0 , so for M ≥ C 0 h 0 , we have Ω -∩ Ω M = ∅. Hence, we have a partition Ω -(Ω + \ Ω M ) (Ω + ∩ Ω M ).
Note that it will be very important in what follows that all the estimates are independent on M while C 0 will be defined later on. The function φ M is Lipschitz on (0, L), and can be pulled back to a (R θ ) invariant Lipschitz function defined on U , and extended to S by φ M (N ) = φ M (S) = M . We call S + , S -, S M ⊂ S, the (R θ ) invariant regions on S associated to Ω -, Ω + , Ω M , respectively, so that

S = S -(S + \ S M ) (S + ∩ S M ).
We now apply the formula of Lemma 3.7 with Ψ = e φ M h with φ M given above and M large, and w = ψ h (note that ψ h ∈ C ∞ (S) since it is an eigenfunction of ∆ g , so the Lemma applies).

S |∇ g (Ψψ h )| 2 g d Vol g - S |∇ g Ψ| 2 g |ψ h | 2 d Vol g = k 2 µ h S |Ψ| 2 |ψ h | 2 d Vol g . Applying now Lemma 3.11 since Ψψ h ∈ W 1,∞ (S) ∩ L 2 k and using |∇ g Ψ| 2 g = k 2 |φ M (s)| 2 e 2φ M /
h in U and so almost everywhere in S, we get

S 1 R(s) 2 -|φ M (s)| 2 -µ h e 2φ M /h |ψ h | 2 d Vol g ≤ 0.
Using the expression of φ M on Ω -and of µ h = 1 R(s0) 2 + O(h), this yields, for some C > 0 (independent of h and M ),

S+ 1 R(s) 2 -|φ M (s)| 2 -µ h e 2φ/h |ψ h | 2 d Vol g ≤ Ch S- e 2d A (s)/h |ψ h | 2 d Vol g ≤ Che 2C0 S- |ψ h | 2 d Vol g ≤ Che 2C0 , since ψ h is normalized.
Note also that on

Ω M ∩ Ω + , we have d A ≥ C 0 h and so d A ≥ d A -C 0 h log(C 0 ) ≥ φ ≥ M ≥ 1. Hence, since d A is continuous, there is a constant ε > 0 so that s ∈ Ω M ∩ Ω + implies |s -s 0 | ≥ ε.
In particular, since s 0 is a nondegenerate maximum for R, there is η > 0 so that it also implies

1 R(s) 2 -1 R(s0) 2 ≥ η. On S M ∩ S + , we thus have 1 R(s) 2 -|φ M (s)| 2 -µ h = 1 R(s) 2 - 1 R(s 0 ) 2 + O(h) ≥ 0,
for h < h 0 for h 0 only depending on the geometry, and not on M . Therefore, we have obtained

S+\S M 1 R(s) 2 -|φ (s)| 2 -µ h e 2φ/h |ψ h | 2 d Vol g ≤ Che 2C0 . ( 58
) Next, on Ω + \ Ω M , we have φ = d A -C 0 h d A d A and hence 1 R(s) 2 -|φ | 2 -µ h = -h |R (s 0 )| R 3 (s 0 ) + O(h 3 2 ) + 2C 0 h (d A ) 2 d A -C 2 0 h 2 (d A ) 2 d 2 A ≥ -h |R (s 0 )| R 3 (s 0 ) + O(h 3 2 ) + C 0 h (d A ) 2 d A ,
where we used that d A ≥ C 0 h. According to (57),

(d A ) 2 d A → 2 -R (s0) R(s0) 3 > 0 and (d A ) 2
d A can thus be extended by continuity at s 0 . Since d A (s) = 0 iff s = s 0 (R reaches at s 0 its unique global maximum), the extended function is uniformly bounded from below on any compact of (0, L). Moreover, according to (56), we have

(d A ) 2 d A (s) ∼ s→0 + 1 s 2 log(s -1 )
, and

(d A ) 2 d A (s) ∼ s→L - 1 (L -s) 2 log((L -s) -1 )
.

Hence, there is a constant C 1 > 0 such that

(d A ) 2 d A ≥ C 1 on (0, L), and we have 1 R(s) 2 -|φ | 2 -µ h ≥ h C 0 (d A ) 2 d A - |R (s 0 )| R 3 (s 0 ) + O(h 1 2 ) ≥ C 0 2 h (d A ) 2 2d A ,
when taking C 0 large w.r.t. C -1 1 and h ≤ h 0 with h 0 depending on C 0 , C 1 . We can now fix C 0 , h 0 . From (58), we have thus obtained

Ch S+\S M (d A ) 2 d A e 2φ/h |ψ h | 2 d Vol g ≤ Che 2C0 .
Our next task is to replace φ by d A in this expression. Note that e 2φ(s)/h = e 2d A (s)/h h d A (s)

2C0

. In particular, this yields

Ch S+\S M (d A ) 2 d A e 2d A (z)/h h d A (s) 2C0 |ψ h | 2 d Vol g ≤ Ch.

Now, the function

(d A ) 2 d 1+2C 0 A
is positive on (0, s 0 ) ∪ (s 0 , L), tends to +∞ at s 0 , and satisfies, as above ((L-s) -1 )) 1+2C 0 → +∞, as s → L -. Hence, it is bounded from below on (0, L) by a constant, and we obtain

(d A ) 2 d 1+2C0 A ∼ 1 s 2 (log(s -1 )) 1+2C0 → +∞, as s → 0 + ,

and similarly

(d A ) 2 d 1+2C 0 A ∼ 1 (L-s) 2 (log
S+\S M e 2d A (z)/h |ψ h | 2 d Vol g ≤ Ch -2C0 ,
which, combined with the already remarked fact that S-e 2d A (z)/h |ψ h | 2 Vol g ≤ Cte, gives

S\S M e 2d A (z)/h |ψ h | 2 d Vol g ≤ Ch -2C0 .
Since all the constants are independent on M , it gives the sought result by dominated convergence (for fixed h) making M tends to infinity.

The disk

Denote D = (x, y) ∈ R 2 x 2 + y 2 ≤ 1 ⊂ R 2 the unit disk. We denote by ∆ the (negative) flat Laplace operator in R 2 . In polar coordinates, x = r cos θ, y = r sin θ, we have

∆ = ∂ 2 x + ∂ 2 y = ∂ 2 ∂r 2 + 1 r ∂ ∂r + 1 r 2 ∂ 2 ∂θ 2 .
Then, it can be seen that

ψ n,k (r, θ) = J n (z n,k r)e inθ (59)
is an orthogonal basis of L 2 (D), where

• J n is the Bessel function of order n, namely:

J n (z) = 1 2π π -π e iz sin θ e -inθ dθ, n ∈ Z, z ∈ C \ R -, (60)
• 0 < z n,1 < z n,2 < z n,3 < • • • is the sequence of the positive zeros of J n .

We refer for instance to [START_REF] Vasy | Partial differential equations[END_REF]Chapters 14.4 and 15] for an elementary introduction. In particular, the functions defined in (59) satisfy

-∆ψ n,k = λ n,k ψ n,k in Int(D), with λ n,k = z 2 n,k and ψ n,k | ∂D = 0.
Roughly speaking, the index n encodes the oscillation in the θ variable while the index k will contain an oscillation in the radial variable. We refer to [START_REF] Nalini Anantharaman | Delocalization of quasimodes on the disk[END_REF] for a description of concentration/delocalization properties of general eigenfunctions (or, more generally, quasimodes) on the disk. Here, we want to analyse some eigenfunctions corresponding to the so-called whispering gallery modes that are concentrated close to the boundary of D. They "rotate" very fast and concentrate towards one of the two trajectories of the billiard contained in S * ∂D. This phenomenon corresponds to n → +∞ and k small, typically k = 1. In the following, we thus focus on:

ψ n,1 (r, θ) = J n (z n,1 r)e inθ ,
and hence on the function J n (z n,1 r). This requires information on z n,1 .

A huge amount of information is known on the Bessel functions and its zeros. But we will need very few of them. First, we need to normalize them. For instance, [BGT03, Lemma 5.1] taken for k = 1 (which is that of interest for us) yields

ψ n,1 L 2 (D) ≈ n -2 3 .
We also need a rough estimate on the asympotic of the z n,1 , see [BGT03] Lemma 4.3 for instance, namely,

z n,1 = n + O(n 1/3 ), z n,1 > n.
To estimate the norm of ψ n,1 on B(0, ε), ε < 1, we first prove the following lemma.

Lemma 3.12. For all α ≥ 0 and n ∈ N, we have

J n n cosh(α) ≤ e n(tanh(α)-α) .
Note that in [Cop65, Section 32 p79], for fixed α, a full asymptotics in terms of n is proved, with principal term:

J n n cosh(α) ≈ e n(tanh(α)-α) 2πn tanh(α) . ( 61
)
Here, we need only the principal term but also a uniform bound in terms of α. Note that the short proof below is not very informative, and the reader is referred to [Cop65, Section 32] for a complete steepest descent approach to this asymptotic expansion.

Proof of Lemma 3.12. We start from formula (60), in which we write ν = n cosh(α) , and use the holomorphy of the integrand, together with the fact that e iν(sin z-z cosh α) is a periodic function of Re(z) to change the contour. This yields:

J n (ν) = 1 2π π -π e i( n cosh(α) ) sin θ e -inθ dθ = 1 2π π -π e iν(sin θ-θ cosh α) dθ = 1 2π π-iα -π-iα e iν(sin z-z cosh α) dz = 1 2π π -π
e iν(sin x cosh α-i cos x sinh α-x cosh α+iα cosh α) dx.

This implies

|J n (ν) | ≤ 1 2π π -π
e ν(cos x sinh α-α cosh α) dx ≤ e ν(sinh α-α cosh α) = e n(tanh α-α) , and concludes the lemma.

Lemma 3.13. There exist C, β, n 0 > 0 such that for all n ≥ n 0 and 0 < r ≤ 1 -βn -2/3 , we have

ψ n,1 L ∞ (B(0,r)) ≤ exp -nd A (r) + Cn 1/3 .
Recall the definition of d A in (20). See also Remark 1.14. Note that for r ∈ (0, 1) fixed, the asymptotic formula (61) implies that such eigenfunctions have indeed the decay given by Lemma 3.13.

Proof. We have zn,1 n = 1 + O(n -2/3 ) and zn,1 n > 1. Hence recalling that |d A | is decreasing on (0, 1], we have, as long as

rzn,1 n ≤ 1, d A ( rz n,1 n ) -d A (r) ≤ Cn -2/3 r|d A (r)| = Cn -2/3 r 1 r 2 -1 = Cn -2/3 1 -r 2 .
Thus we obtain from Lemma 3.12

|J n (z n,1 r)| = |J n (n z n,1 n r)| ≤ exp -nd A ( z n,1 n r) ≤ exp -nd A (r) + Cn 1/3
for all n ∈ N and 0 < r ≤ n zn,1 .

The combination of the previous estimates give Theorem 1.12.

4 Maximal vanishing rate of sums of eigenfunctions, and observability from small balls

In this section, we prove Theorem 1.15, i.e. the Lebeau-Robbiano spectral inequality with observation in balls of (small) radius r and constants uniform in r.

We follow the proof proposed by Jerison and Lebeau in [JL99, middle of p231]. There are three main steps, that we summarize in three lemmata. We then prove Theorem 1.15 from these lemmata, and prove the lemmata afterwards.

In the following, for β > 0, we set X β = (-β, β) × M, and denote P = -∂ 2 s -∆ g . In the set X 2S = (-2S, 2S) × M, we denote by (s, x) the running point and by B r a geodesic ball (for the metric Id ⊗g) of radius r (its center being implicit in the notation). We also use the rescaled H 1 norm on an open set U , denoted H 1 r (U ) and defined by

F 2 H 1 r (U ) = F 2 L 2 (U ) + r 2 ∇ g F 2 L 2 (U ) . ( 62
)
This will only be used on small geodesic balls or annuli, namely U = B αr or U = B αr \ B βr .

The three key lemmata

In this section, we state the three key lemmata needed for the proof of Theorem 1.15. The first lemma is a classical global Lebeau-Robbiano interpolation inequality, [LR95, Section 3, Estimate (1)].

Lemma 4.1 (Global interpolation inequality from unit balls to the whole space). Let S > 0 and let U ⊂ X 2S be any nonempty open set, then there is C > 0 and α 0 ∈ (0, 1) such that we have

F H 1 (X S ) ≤ C P F L 2 (X 2S ) + F H 1 (U ) α0 F 1-α0 H 1 (X 2S ) . for all F ∈ H 2 (X 2S ) such that F | (-2S,2S)×∂M = 0.
The next lemma states a local interpolation inequality. Its specificity is that the observation term is on a small ball B r and the constants are uniform in r small. For this, the exponent has to depend on r as | log(r)| -1 . Lemma 4.2 (Local interpolation inequality from small balls to unit balls). Let P = -∂ 2 s -∆ g and let B r denote balls centered at (s 0 , x 0 ) ∈ X T , away from the boundary. Then, there exists r 1 > 0 such that for all 0 < r 0 ≤ r 1 , there is C > 0 such that for all r ∈ (0, r0 10 ), and F ∈ H 2 (B r0 ), we have

F H 1 (B r 0 4 ) ≤ C P F L 2 (Br 0 ) + F H 1 r (Br) αr F 1-αr H 1 (Br 0 ) , α r = log 2 log 2r0 r + log 2 .
A proof of this Lemma is given in Section 4.3, starting from a Carleman estimate (with singular weight) due to Aronszajn [START_REF] Aronszajn | A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order[END_REF] (see also [START_REF] Aronszajn | A unique continuation theorem for exterior differential forms on Riemannian manifolds[END_REF][START_REF] Donnelly | Nodal sets of eigenfunctions on Riemannian manifolds[END_REF][START_REF] Donnelly | Nodal sets of eigenfunctions: Riemannian manifolds with boundary[END_REF]).

The last lemma is an interpolation inequality with boundary observation term. All terms are taken on sets of size r, and the important feature of this estimate is that the constants are uniform in r.

Lemma 4.3 (Uniform local interpolation at the boundary on small balls). Let (0, x 0 ) ∈ {0} × M, dist g (x 0 , ∂M) > 0 and consider balls centered at (0, x 0 ). Then, there exists C > 0, r 0 > 0 and α 0 ∈ (0, 1) such that we have for all 0 < r < r 0

F H 1 r (Br) ≤ C r 2 P F L 2 (B2r) + r 3/2 ∂ s F | s=0 L 2 (B2r∩{0}×M) α0 F 1-α0 H 1 r (B2r) for all F ∈ H 2 (X 2S ) such that F | (-2S,2S)×∂M = 0.
This lemma is proved in Section 4.4, consequence of a uniform Carleman estimate proved in Appendix A.

Concluding the proof of Theorem 1.15 from the three lemmata

From these three lemmata, we may now give a proof of Theorem 1.15. We first formulate a straightforward corollary of the three lemmata to prepare the proof.

Corollary 4.4. Let P = -∂ 2 s -∆ g and (0, x 0 ) ∈ {0} × Int(M) and consider balls centered at (0, x 0 ). Then, there exist r 0 > 0, C > 0 and α 0 ∈ (0, 1) such that, for all r ∈ (0, r0 10 ) and F ∈ H 2 (X 2S ) with P F = 0 and F | (-2S,2S)×∂M = 0, we have

F H 1 (X S ) ≤ C F α0 H 1 (B r 0 4 ) F 1-α0 H 1 (X 2S ) , F H 1 (B r 0 4 ) ≤ C F αr H 1 (Br) F 1-αr H 1 (X 2S ) , α r = log 2 log 2r0 r + log 2 , F H 1 (Br) ≤ C ∂ s F | s=0 α0 L 2 (B2r∩{0}×M) F 1-α0 H 1 (X 2S ) .
Proof of Theorem 1.15. Let us first treat the case where ∂M = ∅, or ∂M = ∅ but the center of the balls, x 0 is in Int(M). The case x 0 near ∂M will be treated afterwards.

We reformulate (again) these three results as (in a form close to that of [START_REF] Donnelly | Nodal sets of eigenfunctions on Riemannian manifolds[END_REF])

F H 1 (X 2S ) F H 1 (B r 0 4 ) ≤ C F H 1 (X 2S ) F H 1 (X S ) 1 α 0 , F H 1 (X 2S ) F H 1 (Br) ≤   C F H 1 (X 2S ) F H 1 (B r 0 4 )   1 αr , F H 1 (X 2S ) ∂ s F | s=0 L 2 (B2r∩{0}×M) ≤ C F H 1 (X 2S ) F H 1 (Br) 1 α 0
, and combine them to obtain

F H 1 (X 2S ) ∂ s F | s=0 L 2 (B2r∩{0}×M) ≤ C 1 α 0 C 1 α 0 αr C 1 α 2 0 αr F H 1 (X 2S ) F H 1 (X S ) 1 α 2 0 αr . ( 63
)
We then follow [LR95, JL99, LZ98, LRL12], and, given ψ ∈ E ≤λ take the function

F (s) = sinh(s -∆ g ) -∆ g Π + ψ + sΠ 0 ψ,
where ∆ g is the Dirichlet Laplacian, Π 0 the orthogonal projector on ker(∆ g ) and Π + = Id -Π 0 , that is F is the unique solution to

(-∂ 2 s -∆ g )F = 0, F | (-2S,2S)×∂M = 0, (F, ∂ s F )| s=0 = (0, ψ).
Classical computations (see e.g. [LRL12, Proof of Theorem 5.4]) show that there is C > 1 such that for all λ ≥ 0 and ψ ∈ E ≤λ , we have

1 C ψ L 2 (M) ≤ F H 1 (X S ) ≤ F H 1 (X 2S ) ≤ Ce 3S √ λ ψ L 2 (M) .
As a consequence, (63) yields for some C, κ > 0, for all λ ≥ 0, ψ ∈ E ≤λ , and r ∈ (0, r0 4 ) Proposition 4.5. Let P = -∂ 2 s -∆ g and let (ρ, t) ∈ (0, r 1 ) × S n be geodesic polar coordinates around a point (s 0 , x 0 ) ∈ X S away from the boundary. Then, there exists a function ρ(ρ) with

ψ L 2 (M) ψ L 2 (B M (x0,2r)) ≤ C κ+ 1 αr e (κ+ 1 αr ) √ λ . ( 64
ρ = ρ + O(ρ 2 ), as ρ → 0 + , ( 65
)
and constants τ 0 , C, r 0 > 0, such that we have

C |ρ -τ P u| 2 ρ -1 dρdt ≥ |ρ -τ ∇u| 2 + |ρ -τ u| 2 ρ -1 dρdt, for all τ ≥ τ 0 , u ∈ C ∞ 0 (B r0 \ {0}).
With this Carleman-Aronszajn estimate in hand, we now give a proof of Lemma 4.2.

Proof of Lemma 4.2. We use the estimate of Proposition 4.5 as in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] (see also [LRL12, Section 5]) to deduce an interpolation inequality. We introduce for this (as in [DF88, Beginning of Section 3]) a cutoff function χ r = χ r (ρ) such that, with 0 < r < r0 2 a small parameter (appearing in the statement of the lemma)

supp(χ r ) ⊂ r 2 < ρ < r 0 , χ r = 1 on r < ρ < r 0 2 , |∂ α χ r | ≤ C α r -|α| on r 2 < ρ < r , |∂ α χ r | ≤ C α on r 0 2 < ρ < r 0 .
We apply Proposition 4.5 to u = χ r F . The operator [P, χ r ] is a first order differential operator with supp[P, χ r ] ⊂ r 2 < ρ < r ∪ r0 2 < ρ < r 0 , being moreover of the form O(r -1 )D + O(r -2 ) on the set r 2 < ρ < r . Therefore, we obtain using (65), for all τ ≥ τ 0

|ρ -τ ∇(χ r F )| 2 + |ρ -τ χ r F | 2 ρ -1 dρdt ≤ C |ρ -τ χ r P F | 2 ρ -1 dρdt + C |ρ -τ [P, χ r ]F | 2 ρ -1 dρdt ≤ C r 2 -2τ -1 P F 2 L 2 ( Br 0 ) + C r 2 -2τ -2 F 2 H 1 r ( r 2 ≤ ρ≤r) + C r 0 2 -2τ F 2 H 1 ( r 0 2 ≤ ρ≤r0)
, where Br0 denotes the set {ρ ≤ r 0 }. Recall that the norm H 1 r is defined in (62). Concerning the left hand-side, we bound it from below by

|ρ -τ ∇(χ r F )| 2 + |ρ -τ χ r F | 2 ρ -1 dρdt ≥ 2r≤ ρ≤ r 0 4 |ρ -τ ∇(χ r F )| 2 + |ρ -τ χ r F | 2 ρ -1 dρdt ≥ r 0 4 -2τ F 2 H 1 (r≤ ρ≤ r 0
4 ) . Combining the last two estimates together with the fact that r0 4

-τ F H 1 ( Br) ≤ r 2 -τ F H 1 ( Br) yields,
for some τ 0 > 0 and all τ ≥ τ 0 and r ∈ (0, r0 10 ), r 0 4

-τ F H 1 ( B r 0 4 ) ≤ C r 2 -τ P F L 2 ( Br 0 ) + F H 1 r ( Br) + C r 0 2 -τ F H 1 ( Br 0 ) .
Multiplying by r τ 0 and recalling (65) to replace balls in ρ by real balls, we obtain, up to changing the names of the parameters r, r 0 , that

F H 1 (B r 0 4 ) ≤ C 2r 0 r τ P F L 2 (Br 0 ) + F H 1 r (Br) + C 2 τ F H 1 (Br 0 ) .
An optimization in τ ≥ τ 0 [START_REF] Robbiano | Fonction de coût et contrôle des solutions des équations hyperboliques[END_REF] (see also [LRL12, Lemma 5.2]), then implies the following interpolation inequality

F H 1 (B r 0 4 ) ≤ C P F L 2 (Br 0 ) + F H 1 r (Br) αr F 1-αr H 1 (Br 0 ) , α r = log 2 log 2r0 r + log 2 ,
and concludes the proof of the lemma.

A proof of Lemma 4.3 from Proposition A.14

In this section, we give a proof of Lemma 4.3. The latter consists in performing a scaling argument to reduce the problem to fixed-size balls. However, the scaling argument yields in these fixed balls a family of metrics (converging to a fixed metric as r → 0), and we need to use uniform interpolation/Carleman estimates for such families of metrics. These uniform estimates are proved in Appendix A (Proposition A.14).

Proof of Lemma 4.3. We first choose r 0 small enough so that B 2r0 ⊂ X S and there exist a local coordinate patch on M : Φ : {x ∈ M, dist(x, x 0 ) < 2r 0 } → U where U is a neighborhood of 0 in R n , with Φ(x 0 ) = 0. Up to a multiplication by an invertible constant matrix, we may assume that (Φ -1 ) * g (0) = Id. As a consequence, ds 2 ⊗ (Φ -1 ) * g (ry), defined on the ball of radius 2, converges uniformly in this ball towards the flat metric on the flat ball of R n+1 in the limit r → 0 + . We will thus only use the flat metric in the present proof, which behaves well with respect to scaling. The distance (hence the balls, still denoted B r or B 1 below, all centered at 0) will be defined with respect to the flat metric, as well as the Sobolev norms (still denoted H 1 r (B r ), H 1 (B 1 ) below). The final result we obtain will be formulated in terms of the flat metric, and associated balls and Sobolev spaces. Coming back to a formulation on the manifold R × M with the metric ds 2 ⊗ g only uses the uniform equivalence of norms in T * (R × M) and in L 2 (R × M) for r sufficiently small.

With this in mind, let us now proceed with the scaling argument in the coordinate chart. Denote by F r (x) = F (rx) and P r the Laplace-Beltrami operator with respect the metric ds 2 ⊗ (Φ -1 ) * g (ry) defined on the ball of radius 2, we have

F H 1 r (Br) = r (n+1)/2 F r H 1 (B1) , r 2 P F L 2 (B2r) = r (n+1)/2 P r F r L 2 (B2) , r 3/2 ∂ s F | s=0 L 2 (B2r∩{0}×M) = r 1/2 r n/2 ∂ s F r | s=0 L 2 (B2∩{0}×M) .
Note that the metric ds 2 ⊗ g(r•) defined on B 2 converges uniformly, when r tends to zero, to the flat metric ds 2 ⊗ g(0) = ds 2 ⊗ dy 2 1 ⊗ • • • ⊗ dy 2 n for the Lipschitz topology on metrics. So, the result follows if we are able to prove the following estimate: there exist , α 0 , C such that for all Lipschitz metric g with g -Id W 1,∞ < and all u ∈ H 2 (B 2 ) such that u| s=0 = 0, we have

u H 1 (B1) ≤ C (-∂ 2 s -∆ g )u L 2 (B2) + ∂ s u| s=0 L 2 (B2∩{0}×R n ) α0 F 1-α0 H 1 (B2) .
This is the object of Proposition A.14 proved in the Appendix. Note that the result of Proposition A.14 is stated with half-balls B + k but is also true with real balls B k instead by a symmetry argument.

The observability constant for positive solutions

The aim of this Section is to prove the result of Theorem 1.4 concerning observability of positive solutions to the heat equation. The main tool will be the following Li-Yau estimates.

Theorem 5.1 (Theorem 2.3 of Li-Yau [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF]). Let M be a compact manifold. Let

-K = min(0, min x∈M Ricc(x)) ≤ 0,
where Ricc(x) is the Ricci curvature at x. We assume that the boundary of M is convex, i.e. II > 0. Let u(t, x) be a positive solution on (0, +∞) of the heat equation with Neumann boundary condition. Then for any α > 1, x, y ∈ M, and 0 < t 1 < t 2 , we have

u(t 1 , x) ≤ t 2 t 1 nα/2 e nαK(t 2 -t 1 ) √ 2(α-1) e α d(x,y) 2 4(t 2 -t 1 ) u(t 2 , y).
Here, we have denoted by II(x) the second fundamental form of ∂M with respect to outward pointing normal at the point x.

Remark 5.2. The convexity assumption is not necessary to obtain a Li-Yau type estimate (if the boundary is smooth), up to a loss in the exponent. Indeed, setting -H = min(0, min x∈∂M II(x)) ≤ 0, Wang proves in [Wan97, Theorem 3.1] the estimate

u(t 1 , x) ≤ t 2 t 1 Cα e C α (t2-t1) e α d(x,y) 2 4(t 2 -t 1 ) u(t 2 , y), for all α > (1 + H) 2 .
The proof of Theorem 1.4 below shows that the result still holds without the convexity argument, but yields

u(T ) 2 L 2 (M) ≤ C ε T e (1+H+ε) 2 (L(M,ω)+ε) 2 2T T 0 u(t, •) 2 L 2 (ω) dt, u(T ) 2 L 2 (M) ≤ C ε T e (1+H+ε) 2 (L(M,z 0 )+ε) 2 2T T 0 u(t, z 0 ) 2 dt,
instead of ( 6)-( 7) (hence with a loss (1 + H) 2 in the exponent). We do not know whether this is optimal. Finally, we did not find any analogue estimate in the case of Dirichlet boundary conditions.

Proof of Theorem 1.4. Along the proof, we will need the following asymptotic constants, all depending on the chosen ε > 0. Namely, we shall use η 0 > 0 arbitrarily small, r > 1 arbitrarily large, λ ∈ (0, 1) arbitrarily close to 1, and α > 1 arbitrary close to 1. Given ε > 0, they will all be fixed at the end so that

rα (r -1)λ (L(M, ω) + 3η 0 ) 2 ≤ (1 + ε)(L(M, ω) + ε) 2 .
For any x 0 ∈ M and for any η 0 > 0, there exist η = η(x 0 , η 0 ) ∈ (0, η 0 ) and y 0 ∈ ω such that d(x 0 , y 0 ) ≤ L(M, ω) + η, and B(y 0 , η) ⊂ ω.

In particular, we have M ⊂ x0∈M B(x 0 , η) so that, the compactness of M yields the following statement: given η 0 > 0, there exist a finite set J and families (x j ) j∈J ∈ M J , (y j ) j∈J ∈ ω J and (η j ) j∈J ∈ (0, η 0 ) J such that M ⊂ j∈J B(x j , η j ), d(x j , y j ) ≤ L(M, ω) + η j , and B(y j , η j ) ⊂ ω, for all j ∈ J.

Now, fix j ∈ J, and take x ∈ B(x j , η j ) and y ∈ B(y j , η j ) ⊂ ω, and we have

d(x, y) ≤ η j + L(M, ω) + η j + η j ≤ L(M, ω) + 3η 0 =: d m .
For t ∈ [0, T /r], Theorem 5.1 with t 1 = t and t 2 = rt 1 = rt then yields

u(t, x) 2 ≤ r nα e 2nαKt(r-1) √ 2(α-1) e αd 2 m 2(r-1)t u(rt, y) 2 . Denoting γ := 2nαK(r -1) √ 2(α -1) ,
this may be rewritten as

u(t, x) 2 e -αd 2 m 2(r-1)t ≤ r nα e γt u(rt, y) 2 . ( 66
)
We may now integrate this estimate for x ∈ B(x j , η j ) and y ∈ B(y j , η j ) ⊂ ω,

e -αd 2 m 2(r-1)t u(t) 2 L 2 (B(xj ,ηj )) ≤ |B(x j , η j )| |B(y j , η j )| r nα e γt u(rt) 2 L 2 (B(xj ,ηj )) ≤ |B(x j , η j )| |B(y j , η j )| r nα e γt u(rt) 2 L 2 (ω) .
Summing all these estimates for j ∈ J yields, for a constant C(η 0 ) depending only on the geometry of (M, g), of ω, and the constant η 0 , the inequality

e -αd 2 m 2(r-1)t u(t) 2 L 2 (M) ≤ C(η 0 )r nα e γt u(rt) 2 L 2 (ω) .
Given λ ∈ (0, 1), integrating this on the interval t ∈ [λT /r, T /r] yields

T /r λT /r e -αd 2 m 2(r-1)t u(t) 2 L 2 dt ≤ C(η 0 )r nα T /r λT /r e γt u(rt) 2 L 2 (ω) dt ≤ C(η 0 )r nα e γ T r T /r λT /r u(rt) 2 L 2 (ω) dt = C(η 0 )r nα e γ T r T λT u(s) 2 L 2 (ω) ds,
after the change of variables s = rt. Concerning the left hand-side, we use the decay of the L 2 norm of solutions to the heat equation to write

u(t) L 2 (M) ≥ u(T /r) L 2 (M) ≥ u(T ) L 2 (M) , (67)
for all t ∈ [λT /r, T /r] since r > 1. Noting also that t → e -αd 2 m 2(r-1)t is increasing in t > 0, we have

T /r λT /r e -αd 2 m 2(r-1)t dt ≥ T (1 -λ) r e -rαd 2 m 2(r-1)λT .
Combining the above three estimates yields

T (1 -λ) r e -rαd 2 m 2(r-1)λT u(T ) 2 L 2 (M) ≤ C(η 0 )r nα e γ T r T λT u(s) 2 L 2 (ω) ds,
that is, for all η > 0, r > 1, λ ∈ (0, 1), and α > 1,

u(T ) 2 L 2 (M) ≤ C(η)r nα+1 T (1 -λ) e 2nαK(r-1) √ 2(α-1) T r e rα(L(M,ω)+η) 2 2(r-1)λT T λT u(s) 2 L 2 (ω) ds.
But r r-1 = 1 + 1 r-1 can be made arbitrary close to 1 + for large r, λ close to 1 -, α close to 1 + , and η to 0 + , so that rα(L(M,ω)+η) 2 2(r-1)λT ≤ L(M,ω) 2 +ε 2T . We have thus proved the first statement. To be a little more precise, we can choose α, r such that 1 r + 1 α = 1. This yields

u(T ) 2 L 2 (M) ≤ C(η) α α-1 nα+1 T (1 -λ) e 2nK √ 2(α-1) T e α 2 (L(M,ω)+η) 2 2λT T λT u(s) 2 L 2 (ω) ds,
or, with α = 1 + and λ = 1 -, we obtain for all ∈ (0, 1)

u(T ) 2 L 2 (M) ≤ C(η) 1+ (1+ )n+1 T e 2nK √ 2 T e (1+ ) 2 1- (L(M,ω)+η) 2 2T T (1-)T u(s) 2 L 2 (ω) ds ≤ C(η) T 2n+2 e 2nK √ 2 T e (1+ ) 2 1- (L(M,ω)+η) 2 2T T (1-)T u(s) 2 L 2 (ω) ds.
So we have proved the first estimate of the theorem. The second can be obtained similarly by integrating (66) in the x variable only, and not in the y variable.

Remark 5.3. In fact, remark that from (67) on, we could also put u(T /r)

2

L 2 (M) on the left hand-side of all estimates of the proof, which amounts to u(T 1+ )

2 L 2 (M)
, and, in particular, we have the stronger statement

u((1 -)T ) 2 L 2 (M) ≤ C(η)r nα+1 T e 2nK √ 2 T e (1+ ) 2 1- (L(M,ω)+η) 2 2T T (1-)T u(s) 2 L 2 (ω) ds.
Remark 5.4. All constants can be made explicit. We denote by K := min {0, -min x∈M Ricci(x)}. For instance, we have for all η > 0, all

u(T ) 2 L 2 (M) ≤ C(η)r nα+1 T (1 -λ) e 2nK √ 2(α-1) T e α 2 (L(M,ω)+η) 2 2λT T λT u(s) 2 L 2 (ω) ds,
Choosing the constants, we have, for all ∈ (0, 1), for all η > 0,

u(T 1 + ) 2 L 2 (M) ≤ C(η) T 2n+2 e 2nK √ 2 T e (1+ ) 3 (L(M,ω)+η) 2 2T T (1-)T u(s) 2 L 2 (ω) ds.
Remark that for non-negatively (Ricci) curved manifolds (this is the case of a convex domain in R n ), then K = 0 and the constant is

C(η) T 2n+2 e (1+ ) 3 (L(M,ω)+η) 2 2T
and hence decays like 1/T for T large.

A Uniform Lipschitz Carleman estimates

In this appendix, we produce Carleman estimates for a Laplace-Beltrami operator on a Riemannian manifold M with boundary ∂M . Our proof presents several advantages with respect to the existing proofs of similar results:

• it is relatively short;

• it is completely geometric and, we hope, is relatively readable;

• as we already said, it requires the minimum of regularity for the metric (in dimension ≥ 3), namely only Lipschitz regularity. Indeed, it is known that in dimension ≥ 3, local uniqueness does not hold for general elliptic operators (even in divergence form) with C 0,α coefficients for all α < 1, see [START_REF] Pliś | On non-uniqueness in Cauchy problem for an elliptic second order differential equation[END_REF] and [START_REF] Miller | Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF].

The proof, using formulae from Riemannian geometry, is inspired by Carleman estimates for the Schrödinger equation proved by the first author [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3[END_REF].

There have been several works about such Carleman estimates for Lipschitz metrics (but without boundary). The oldest result seems to be [START_REF] Aronszajn | A unique continuation theorem for exterior differential forms on Riemannian manifolds[END_REF] for elliptic operators. Another one, which actually falls short from the Lipschitz regularity is the very general result of Hörmander [Hör63, Section 8.3] which requires C 1 regularity, but applies to much more operators than elliptic ones. A proof for general elliptic operators with order 2m and Lipschitz coefficients is written by Hörmander in [Hör85, Proposition 17.2.3]. For Lipschitz regularity of the coefficients, we can also mention for instance the recent preprint [START_REF] Nakić | A quantitative Carleman estimate for secondorder elliptic operators[END_REF], with explicit dependence. One cas also mention doubling estimates directly for the parabolic equation, see [START_REF] Canuto | Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries[END_REF][START_REF] Escauriaza | Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients[END_REF] for instance.

A.1 Toolbox of Riemannian geometry

The definitions given in this section have a deep geometric meaning (see [START_REF] Gallot | Riemannian geometry. Universitext[END_REF]). We will however only use the associated calculus rules, which we recall below. Note that they are usually written for smooth metrics, but they still make sense for Lipschitz metric, as we shall see below. We follow the notations of [START_REF] Gallot | Riemannian geometry. Universitext[END_REF].

Here and in all estimates below, M is a (not necessarily compact) smooth d-dimensional manifold with boundary ∂M , so that M = ∂M Int(M ).

Given an open set U ⊂ M such that U is compact in M (note that this definition holds not only for open sets of Int(M )), we denote by L p (U ), H k (U ), W k,∞ (U ) the usual Sobolev spaces. These are defined intrinsically once U is fixed, even if the associated norms may depend on the metric or the charts chosen. The notation L p loc (M ), H k loc (M ), W k,∞ loc (M ) will be used for functions belonging to L p (U ), etc... for any open set U such that U is compact in M (and not Int(M )).

We denote by g a locally Lipschitz metric on M , (that is, x → g x (•, •) is a locally Lipschitz section of the bundle of symmetric bilinear forms on T M that is uniformly bounded from below by a positive constant on any compact set).

Given a local regularity space B as above, and U ⊂ M such that U is compact in M , we define

T 2 B (U ) = Γ B (T 2 T * M )| U
to be the space of sections of 2-tensors on T * M having regularity B on a neighborhood of U . In local charts, such a tensor t ∈ T 2 B (M ) writes t = (t ij ) with t ij having the regularity of B. Typically, a locally Lipschitz metric g satisfies g ∈ T 2 W 1,∞ loc (M ). We denote by •, • g = g(•, •) the inner product in T M . Remark that this notation omits to mention the point x ∈ M at which the inner products takes place: this allows to write X, Y g as a function on M (the dependence on x is omitted here as well) when X and Y are two vector fields on M . We also denote for a vector field X, |X| 2 g = X, X g . We recall that the Riemannian gradient ∇ g of a function f is defined by ∇ g f, X g = df (X), for any vector field X, For a function f on M , we denote by f = M f (x)d Vol g (x) its integral on M , where d Vol g (x) is the Riemannian density. We denote by div g the associated divergence, defined on a vector field X by

u div g X = - ∇ g u, X g , for all u ∈ C ∞ c (Int(M )).
We denote by ∆ g = div g ∇ g the associated (nonpositive) Laplace-Beltrami operator. We also denote by D the Levi-Civita connection associated to the metric g (see [GHL90, Chapter II Section B]).

Let us now recall how these objects write in local coordinates.

Formula 1. In coordinates, for f a smooth function and

X = i X i ∂ ∂xi , Y = i Y i ∂ ∂xi smooth vector fields on M , we have X, Y g = n i=1 g ij X i Y j , ∇ g f = n i,j=1 g ij (∂ j f ) ∂ ∂x i , f = f d Vol g = f (x) det g(x)dx, div g (X) = n i=1 1 √ det g ∂ i det gX i , ∆ g f = n i,j=1 1 √ det g ∂ i det gg ij ∂ j f , D X Y = n i=1   n j=1 X j ∂Y i ∂x j + n j,k=1 Γ i j,k X j Y k   ∂ ∂x i ,
where (g -1 ) ij = g ij and the Chritoffel symbols are defined by

Γ i j,k = 1 2 n l=1 g il (∂ j g kl + ∂ k g lj -∂ l g jk ) ,
(see for instance [START_REF] Gallot | Riemannian geometry. Universitext[END_REF]p71]). Note in particular that the Lipschitz regularity of g writes, on any local chart U with U compact, g ij ∈ W 1,∞ (U ), and implies g ij ∈ W 1,∞ (U ). This entails, if f, X, Y are smooth, that X, Y g ∈ W 1,∞ loc (M ), ∇ g f is a locally Lipschitz vector field, ∆ g f ∈ L ∞ loc (M ) and D X Y is an L ∞ loc vector field on M , since the definitions of ∆ g and D X involve one derivative of the coefficients of g.

In view of the properties of D X , it is natural to set D X f = Xf = df (X) for a function f on M . Let us now collect some properties of these objects, that we shall use below.

Formula 2. For f, h smooth functions and X = i X i ∂ ∂xi , Y = i Y i ∂ ∂xi smooth vector fields on M , we have

∇ g (f h) = (∇ g f )h + f (∇ g h), div g (f X) = ∇ g f, X g + f div g (X), D X (f Y ) = (Xf )Y + f D X Y, where Xf := df (X), D X ( Y, Z g ) = D X Y, Z g + Y, D X Z g .
That D X acts on functions as well as on vector fields suggests to extend the definition of D X to more general vector bundles (see [GHL90, Proposition 2.58]). In particular, for a one-form ω, D X ω is defined (by duality) to be the one-form acting as

(D X ω)(Y ) = X(ω(Y )) -ω(D X Y ), for all vector fields Y.
This allows to define the Hessian of a function (see [GHL90, Exercice 2.65]) Hess(f )(X, Y ) = (D X df )(Y ), for vector fields X, Y, (which only involves the values of X, Y and not their derivatives). In local charts, note that we have

Hess(f )(X, Y ) = i,j X i Y j ∂ 2 ij f -Γ k ij ∂ k f ,
which again is in L ∞ loc (M ) for a locally Lipschitz metric g and L ∞ loc vector fields X, Y . Note also that the Hessian of f is symmetric, that is Hess(f )(X, Y ) = Hess(f )(Y, X).

Lemma A.1. For any function f and any vector field X and Y , we have

Hess(f )(X, Y ) = D X ∇ g f, Y g .
Proof. According to the above calculus rules, we compute in two different ways the following quantity:

D X ( ∇ g f, Y g ) = D X df (Y ) = (D X df )(Y ) + df (D X Y ) = Hess(f )(X, Y ) + df (D X Y ).
We also have

D X ( ∇ g f, Y g ) = D X ∇ g f, Y g + ∇ g f, D X Y g = D X ∇ g f, Y g + df (D X Y ),
which, combined with the previous computation yields the result.

Finally, we recall an integration by parts formula in the present context. Formula 3 (Riemannian Stokes formula). Assume ∂M is piecewise C 1 and graph-Lipschitz. Then, for all f ∈ H 2 loc (M ) and h ∈ H 1 loc (M ) one of which being compactly supported, we have

(∆ g f )h = ∂M ∇ g f, ν g h - ∇ g f, ∇ g h g .
Here, the boundary ∂M is endowed with the Riemannian metric induced by g, and ∂M is the integral with respect to the associated surface measure (defined as in Formula 1). The vector field ν is the unit normal vector to ∂M which is outgoing. It is defined almost everywhere if ∂M is piecewise C 1 . In a local coordinate chart (x 1 , • • • , x n ) centered at 0, and in which ∂M ⊂ {x n = 0} and M ⊂ {x n ≤ 0}, we have ν = n j=1

g jn √ g nn ∂ ∂xj .
With the prescribed regularity of the boundary, the space L ∞ loc (∂M ) is defined intrinsically. We denote by ∂ ν f = ∇ g f, ν g the normal derivative at the boundary, which is only

L ∞ loc (∂M ) since ∂M is piecewise C 1 .
Note that in the above coordinate system, we have

∂ ν f = n j=1
g jn √ g nn ∂ xj f . In particular, if f satisfies Dirichlet boundary conditions, this is ∂ ν f = √ g nn ∂ xn f . Note finally the vector field X -X, ν g ν is tangential to ∂M, so that we may decompose a vector field as its normal and tangential parts. In particular, we shall decompose the gradient

∇ g f = ∂ ν f ν + ∇ T f , where ∇ T f | ∂M ∈ T ∂M.

A.2 The Carleman estimate

We stress the fact that functions u ∈ C ∞ (M ) are smooth up to the boundary of M (as opposed to functions u ∈ C ∞ (Int(M ))). We will first estimate the Carleman conjugate operator in Theorem A.2 and then give the desired estimate under appropriate assumptions in Theorem A.5.

Theorem A.2. Assume g is a Lipschitz metric on M and ∂M is piecewise C 1 and graph-Lipschitz. Let U be an open subset of M such that U is compact (in the topology of M ⊃ ∂M ) and denote Σ = ∂M ∩ U . Then, for any f ∈ W 1,∞ (U ), ϕ ∈ W 2,∞ (U ), u ∈ H 2 comp (U ) and τ ≥ 0, we have

e τ ϕ ∆ g (e -τ ϕ u) 2 + R(u) ≥ τ 3 2Hess(ϕ)(∇ g ϕ, ∇ g ϕ) + (∆ g ϕ) |∇ g ϕ| 2 g -f |∇ g ϕ| 2 g |u| 2 + τ 2Hess(ϕ)(∇ g u, ∇ g u) -(∆ g ϕ) |∇ g u| 2 g + f |∇ g u| 2 g + BT (u),
with boundary terms

BT (u) = -2τ Σ ∇ g u, ν g ∇ g ϕ, ∇ g u g + τ Σ ∇ g ϕ, ν g |∇ g u| 2 g -τ 3 Σ ∇ g ϕ, ν g |u| 2 |∇ g ϕ| 2 g + τ Σ ∇ g u, ν g f u, (68)
and remainder R(u) satisfying

|R(u)| ≤ f -∆ g ϕ 2 L ∞ (U ) + 1 2 ∇ g f L ∞ (U ) τ 2 u 2 L 2 + 1 2 ∇ g f L ∞ (U ) ∇ g u 2 L 2 . (69)
Note that the last term in (68) is actually of lower order. We keep it here since it vanishes in case of Dirichlet Boundary conditions. Remark A.3. It is very important for our purpose to notice that all terms in this identity only involve derivatives of order 0 or 1 of the metric. This will be important when we will consider stability issues with respect to Lipschitz perturbations of the metric. This identity suggests to introduce and study the following two important quantities, given X a smooth vector field on M :

B g,ϕ,f (X) = 2Hess(ϕ)(X, X) -(∆ g ϕ) |X| 2 g + f |X| 2 g , E g,ϕ,f = 2Hess(ϕ)(∇ g ϕ, ∇ g ϕ) + (∆ g ϕ) |∇ g ϕ| 2 g -f |∇ g ϕ| 2 g .
Note that for a Lipschitz metric g, we have E g,ϕ,f ∈ L ∞ loc (M ) and B g,ϕ,f (X) ∈ L ∞ loc (M ) for any locally bounded vector field X.

Remark A.4. At this level, it would be very tempting to set F = -∆ g ϕ + f and work with the associated simplified expressions of B g,ϕ,f (X) and E g,ϕ,f . From a conceptual point of view, this is completely fine, see Remark A.8 below. However, since we consider the limiting Lipschitz regularity of the metric, this change of additional function is not admissible. Indeed, the remainder term R(u) in Theorem A.2 requires the regularity ∇ g f ∈ L ∞ and f = F + ∆ g ϕ is already in L ∞ and consumes one derivative of the metric g. Having ∇ g F ∈ L ∞ would then require g to be W 2,∞ . We define w 2 L 2 = |w| 2 (see Formula 1 for the notation ) for a function w and X 2 L 2 = |X| 2 g for a vector field X.

We can now state the Carleman estimate.

Theorem A.5. Let U be an open subset of M such that U is compact (in the topology of M ⊃ ∂M ) and denote Σ = ∂M ∩ U . Assume that the functions (ϕ, f ) satisfy: f ∈ W 1,∞ (U ), ϕ ∈ W 2,∞ (U ), |∇ g ϕ| 2 g > 0 on U , and there exists C 0 > 0 such that for any vector field X, we have almost everywhere on U :

B g,ϕ,f (X) ≥ 2C 0 |X| 2 g , (70)
E g,ϕ,f ≥ 2C 0 |∇ g ϕ| 2 g . (71)
Then, denoting c(ϕ) = max 1, min U |∇ g ϕ| 2 g

-1

, we have the following statements.

1. For all τ ≥ c(ϕ

) C0 f -∆ g ϕ 2 L ∞ (U ) + 1 2 ∇ g f L ∞ (U ) and all v ∈ C ∞ c (U ) we have the estimate C 0 3 τ 3 e τ ϕ v∇ g ϕ 2 L 2 (U ) + τ e τ ϕ ∇ g v 2 L 2 (U ) ≤ e τ ϕ ∆ g v 2 L 2 (U ) + τ e τ ϕ ∇ g v 2 L 2 (Σ) + τ 2 e τ ϕ v∇ g ϕ 2 L 2 (Σ) K f,ϕ , (72) with K f,ϕ = 3 c(ϕ) τ f L ∞ (Σ) + 3 ∇ g ϕ L ∞ (Σ) . 2. For all τ ≥ c(ϕ) C0 f -∆ g ϕ 2 L ∞ (U ) + 1 2 ∇ g f L ∞ (U ) and all v ∈ C ∞ c (U ) such that v = 0 on Σ, we have C 0 3 τ 3 e τ ϕ v∇ g ϕ 2 L 2 (U ) + τ e τ ϕ ∇ g v 2 L 2 (U ) ≤ e τ ϕ ∆ g v 2 L 2 (U ) + τ Σ e 2τ ϕ ∂ ν ϕ|∂ ν v| 2 . ( 73
) 3. If ϕ| Σ is constant and -m(ϕ) := max Σ ∂ ν ϕ < 0, then setting M (ϕ) := max Σ (-∂ ν ϕ) > 0, we have for all τ ≥ max c(ϕ) C0 f -∆ g ϕ 2 L ∞ (U ) + 1 2 ∇ g f L ∞ (U ) , √ f L ∞ (Σ) m(ϕ) and all v ∈ C ∞ c (U ), e τ ϕ ∆ g v 2 L 2 + M (ϕ)τ Σ e 2τ ϕ |∇ T v| 2 g ≥ C 0 3 τ 3 e τ ϕ v∇ g ϕ 2 L 2 (U ) + τ e τ ϕ ∇ g v 2 L 2 (U ) + τ 8 m(ϕ) 3 M (ϕ) 2 Σ e 2τ ϕ |∂ ν v| 2 + τ 3 m(ϕ) 3 4 Σ |v| 2 . ( 74
)
Remark A.6. In the last two statements of this result, we assume boundary conditions (either for v or for ϕ) on the whole boundary Σ. Since the integrals involved are local, we could also assume different conditions on parts of the boundary, obtaining the associated terms in the estimates.

For simplicity, in the proof, we shall denote by

u 2 H 1 τ = τ 2 u∇ g ϕ 2 L 2 + ∇ g u 2 L 2
the semiclassical norm (recall that |∇ g ϕ| 2 g > 0 here).

Proof of Theorem A.5. We first let v = e -τ ϕ u, and apply the estimate of Theorem A.2. The latter, together with our assumption (70)-(71) (applied almost everywhere in M to X = ∇ g u) implies for all τ ≥ 0 and u ∈ C ∞ c (U )

e τ ϕ ∆ g (e -τ ϕ u) 2 L 2 + R(u) ≥ 2C 0 τ 3 u∇ g ϕ 2 L 2 + 2C 0 τ ∇ g u 2 L 2 + BT (u) = 2C 0 τ u 2 H 1 τ + BT (u),
where BT (u) is defined in (68) and R(u) estimated in (69). Now, we have

|R(u)| ≤ c(ϕ) f -∆ g ϕ 2 L ∞ + 1 2 ∇ g f L ∞ u 2 H 1 τ , which implies that if τ C 0 ≥ c(ϕ) f -∆ g ϕ 2 L ∞ + 1 2 ∇ g f L ∞ , we obtain e τ ϕ ∆ g (e -τ ϕ u) 2 L 2 ≥ C 0 τ u 2 H 1 τ + BT (u). (75)
We now consider the boundary terms. Without any assumption on the boundary, we have

|BT (u)| ≤ 3τ ∇ g ϕ L ∞ (Σ) ∇ g u 2 L 2 (Σ) + τ 2 u∇ g ϕ 2 L 2 (Σ) + 1 2 f L ∞ (Σ) ∂ ν u 2 L 2 (Σ) + τ 2 u 2 L 2 (Σ) ,
and hence obtain in this case

C 0 τ u 2 H 1 τ ≤ e τ ϕ ∆ g (e -τ ϕ u) 2 L 2 + c(ϕ) f L ∞ (Σ) + 3τ ∇ g ϕ L ∞ (Σ) ∇ g u 2 L 2 (Σ) + τ 2 u∇ g ϕ 2 L 2 (Σ) .
Recalling that u = e τ ϕ v, this implies ∇ g u = e τ ϕ ∇ g v + τ u∇ g ϕ, and hence

e τ ϕ ∇ g v 2 L 2 ≤ 2 ∇ g u 2 L 2 + 2τ 2 u∇ g ϕ 2 L 2 = 2 u 2 H 1 τ , (76)
∇ g u 2 L 2 ≤ 2 e τ ϕ ∇ g v 2 L 2 + 2τ 2 e τ ϕ v∇ g ϕ 2 L 2 . The last four estimates imply C 0 3 τ 3 e τ ϕ v 2 L 2 + τ e τ ϕ ∇ g v 2 L 2 ≤ C 0 τ u 2 H 1 τ ≤ e τ ϕ ∆ g v 2 L 2 + 3 c(ϕ) f L ∞ (Σ) + 3τ ∇ g ϕ L ∞ (Σ) × e τ ϕ ∇ g v 2 L 2 (Σ) + τ 2 e τ ϕ v∇ g ϕ 2 L 2 (Σ) ,
and hence (72). Second, we assume the Dirichlet boundary condition v| Σ = 0. This implies u| Σ = 0 and ∇ g u| Σ = ∂ ν u| Σ ∂ ν , so that we obtain

BT (u) = -τ Σ ∂ ν ϕ|∂ ν u| 2 = -τ Σ ∂ ν ϕe 2τ ϕ |∂ ν v| 2 . Estimate (75) then reads e τ ϕ ∆ g (e -τ ϕ u) 2 L 2 + Σ ∂ ν ϕe 2τ ϕ |∂ ν v| 2 ≥ C 0 τ u 2 H 1 τ .
Using again (76) to come back to the variable v yields (73).

Finally, we consider the case where ϕ| Σ is constant and ∂ ν ϕ ≤ -m(ϕ) < 0, in which case we obtain from (68):

BT (u) = -2τ Σ ∂ ν ϕ|∂ ν u| 2 + τ Σ ∂ ν ϕ |∇ g u| 2 g -τ 3 Σ (∂ ν ϕ) 3 |u| 2 + τ Σ ∂ ν uf u = -τ Σ ∂ ν ϕ|∂ ν u| 2 + τ Σ ∂ ν ϕ |∇ T u| 2 g -τ 3 Σ (∂ ν ϕ) 3 |u| 2 + τ Σ ∂ ν uf u. Estimate (75) then reads e τ ϕ ∆ g (e -τ ϕ u) 2 L 2 + τ Σ ∂ ν ϕ|∂ ν u| 2 -τ Σ ∂ ν ϕ |∇ T u| 2 g + τ 3 Σ (∂ ν ϕ) 3 |u| 2 -τ Σ ∂ ν uf u ≥ C 0 τ u 2 H 1 τ , and hence, using -M (ϕ) ≤ ∂ ν ϕ ≤ -m(ϕ) < 0, e τ ϕ ∆ g (e -τ ϕ u) 2 L 2 + M (ϕ)τ Σ |∇ T u| 2 g -τ Σ ∂ ν uf u ≥ C 0 τ u 2 H 1 τ + m(ϕ)τ Σ |∂ ν u| 2 + τ 3 m(ϕ) 3 Σ |u| 2 . Now, we estimate Σ ∂ ν uf u ≤ f L ∞ (Σ) u L 2 (Σ) ∂ ν u L 2 (Σ) ≤ f L ∞ (Σ) 2m(ϕ)τ ∂ ν u 2 L 2 (Σ) + f L ∞ (Σ) m(ϕ)τ 2 u 2 L 2 (Σ) , so that, for τ ≥ √ f L ∞ (Σ) m(ϕ)
this term is absorbed in the right handside, and we obtain e τ ϕ ∆ g (e -τ ϕ u)

2 L 2 + M (ϕ)τ Σ |∇ T u| 2 g ≥ C 0 τ u 2 H 1 τ + m(ϕ) 2 τ Σ |∂ ν u| 2 + τ 3 m(ϕ) 3 2 Σ |u| 2 .
Recalling that u = e τ ϕ v and ∇ T ϕ| Σ = 0, this implies ∇ T u = e τ ϕ ∇ T v and e

2τ ϕ |∂ ν v| 2 ≤ 2|∂ ν u| 2 + 2τ 2 |∂ ν ϕ| 2 |u| 2 , hence 1 4 m(ϕ) 2 M (ϕ) 2 Σ e 2τ ϕ |∂ ν v| 2 ≤ 1 2 Σ |∂ ν u| 2 + τ 2 m(ϕ) 2 2 Σ |u| 2 .
Finally using again (76) with the last two estimates implies (74), which concludes the proof of Theorem A.5.

Proof of Theorem A.2. The statement of the theorem is a lower bound for the L 2 norm of the quantity e τ ϕ ∆ g (e -τ ϕ u), which we may compute as

P ϕ u := e τ ϕ ∆ g (e -τ ϕ u) = ∆ g u -2τ ∇ g ϕ, ∇ g u g -τ (∆ g ϕ)u + τ 2 |∇ g ϕ| 2 g u.
We then decompose the conjugated operator P ϕ as

P ϕ = Q 2 + Q 1 , with Q 1 u := -2τ ∇ g ϕ, ∇ g u g -τ f u Q 2 u := ∆ g u + τ 2 |∇ g ϕ| 2 g u -τ (∆ g ϕ)u + τ f u = Q 2 u + R 2 u where Q 2 is the principal part of Q 2 , that is Q 2 u = ∆ g u + τ 2 |∇ g ϕ| 2 g u, and R 2 u = τ (-∆ g ϕ + f )u.
Now, we write ( • denotes the L 2 norm for short)

2 P ϕ u 2 + 2 R 2 u 2 ≥ P ϕ u -R 2 u 2 = Q 1 u + Q 2 u 2 ,
where we estimate the remainder as

R 2 u 2 ≤ τ 2 f -∆ g ϕ 2 L ∞ u 2 L 2 . (77)
Hence, we are left to produce a lower bound for

Q 1 u + Q 2 u 2 = Q 1 u 2 + Q 2 u 2 + 2 Re(Q 1 u, Q 2 u).
Now, remark that the all differential operators P ϕ , Q 1 , Q 2 involved have real coefficients. Hence, if we consider complex valued functions u = u R + iu I , we have

P ϕ u 2 = P ϕ u R 2 + P ϕ u I 2 , u 2 = u R 2 + u I
2 so that proving P ϕ u 2 ≥ c 0 u 2 for real valued functions u implies the same inequality for complex valued ones. As a consequence, we only prove the result for a real valued function u, and associated real inner product. We now provide an explicit computation for (Q 1 u, Q 2 u), which is the key step in the proof.

Lemma A.7. For all functions ϕ ∈ W 2,∞ loc (M ), f ∈ W 1,∞ loc (M ) and u ∈ H 2 comp (M ), we have

(Q 1 u, Q 2 u) = τ 3 2Hess(ϕ)(∇ g ϕ, ∇ g ϕ) + (∆ g ϕ) |∇ g ϕ| 2 g -f |∇ g ϕ| 2 g |u| 2 + τ 2Hess(ϕ)(∇ g u, ∇ g u) -(∆ g ϕ) |∇ g u| 2 g + f |∇ g u| 2 g + τ u ∇ g u, ∇ g f g + BT (u), with BT (u) = -2τ Σ ∇ g u, ν g ∇ g ϕ, ∇ g u g + τ Σ ∇ g ϕ, ν g |∇ g u| 2 g -τ 3 Σ ∇ g ϕ, ν g |u| 2 |∇ g ϕ| 2 g -τ Σ ∇ g u, ν g f u.
To conclude the proof of the theorem, we now simply write

2 P ϕ u 2 + 2 R 2 u 2 ≥ Q 1 u + Q 2 u 2 ≥ 2(Q 1 u, Q 2 u). (78)
In the estimates of Lemma A.7, the remainder term is

R 3 (u) = -τ u ∇ g u, ∇ g f g , |R 3 (u)| ≤ ∇ g f L ∞ 2 ∇ g u 2 L 2 + τ 2 u 2 L 2 ,
which, combined with (78), (77) and Lemma A.7, concludes the proof of the Theorem A.2 with

R(u) = R 2 u 2 + |R 3 (u)|.
For the proof of Theorem A.2 to be complete, it only remains to prove Lemma A.7.

Proof of Lemma A.7. We have

(Q 1 u, Q 2 u) = (-2τ ∇ g ϕ, ∇ g u g -τ f u)(∆ g u + τ 2 |∇ g ϕ| 2 g u) = -τ (2J + 2τ 2 K + L), (79) with J
= ∇ g ϕ, ∇ g u g ∆ g u, K = ∇ g ϕ, ∇ g u g |∇ g ϕ| 2 g u, L = f u(∆ g u + τ 2 |∇ g ϕ| 2 g u).
We now perform one (and only one, which is the more we can do with the Lipschitz regularity of g) integration by parts in each of these integrals. Firstly, we compute J as

J = Σ ∇ g u, ν g ∇ g ϕ, ∇ g u g - ∇ g u, ∇ g ( ∇ g ϕ, ∇ g u g ) g .
But, we also have

∇ g u, ∇ g ( ∇ g ϕ, ∇ g u g ) g = D ∇gu ( ∇ g ϕ, ∇ g u g ) = D ∇gu ∇ g ϕ, ∇ g u g + ∇ g ϕ, D ∇gu ∇ g u g = Hess(ϕ)(∇ g u, ∇ g u) + Hess(u)(∇ g u, ∇ g ϕ), so that J = Σ ∇ g u, ν g ∇ g ϕ, ∇ g u g -Hess(ϕ)(∇ g u, ∇ g u) -Hess(u)(∇ g u, ∇ g ϕ).
We then notice that

∇ g ϕ, ∇ g |∇ g u| 2 g g = d(|∇ g u| 2 g )(∇ g ϕ) = D ∇gϕ ( ∇ g u, ∇ g u g ) = D ∇gϕ ∇ g u, ∇ g u g + ∇ g u, D ∇gϕ ∇ g u g = 2 D ∇gϕ ∇ g u, ∇ g u g = 2Hess(u)(∇ g ϕ, ∇ g u), (80)
so that we have in particular

2 Hess(u)(∇ g ϕ, ∇ g u) = ∇ g ϕ, ∇ g |∇ g u| 2 g g = -(∆ g ϕ) |∇ g u| 2 g + Σ ∇ g ϕ, ν g |∇ g u| 2 g .
Coming back to J, this finally implies the expression

2J = 2 Σ ∇ g u, ν g ∇ g ϕ, ∇ g u g -2 Hess(ϕ)(∇ g u, ∇ g u) + (∆ g ϕ) |∇ g u| 2 g - Σ ∇ g ϕ, ν g |∇ g u| 2 g . (81)
Secondly, remarking that ∇ g |u| 2 = 2u∇ g u, we write K as

K = ∇ g ϕ, ∇ g u g |∇ g ϕ| 2 g u = 1 2 |∇ g ϕ| 2 g ∇ g ϕ, ∇ g |u| 2 g .
An integration by parts yields

(∆ g ϕ)|u| 2 |∇ g ϕ| 2 g = Σ ∇ g ϕ, ν g |u| 2 |∇ g ϕ| 2 g - ∇ g ϕ, ∇ g |u| 2 |∇ g ϕ| 2 g g = Σ ∇ g ϕ, ν g |u| 2 |∇ g ϕ| 2 g -|∇ g ϕ| 2 g ∇ g ϕ, ∇ g |u| 2 g -|u| 2 ∇ g ϕ, ∇ g |∇ g ϕ| 2 g g .
Combining these two formulas, we obtain

2K = -(∆ g ϕ)|u| 2 |∇ g ϕ| 2 g + Σ ∇ g ϕ, ν g |u| 2 |∇ g ϕ| 2 g - ∇ g ϕ, ∇ g |∇ g ϕ| 2 g g |u| 2 = -(∆ g ϕ)|u| 2 |∇ g ϕ| 2 g + Σ ∇ g ϕ, ν g |u| 2 |∇ g ϕ| 2 g -2 Hess(ϕ)(∇ g ϕ, ∇ g ϕ)|u| 2 , (82)
where we have used as in (80) that

∇ g ϕ, ∇ g |∇ g ϕ| 2 g g = D ∇gϕ ∇ g ϕ, ∇ g ϕ g = 2 D ∇gϕ ∇ g ϕ, ∇ g ϕ g = 2Hess(ϕ)(∇ g ϕ, ∇ g ϕ).
Thirdly, let us compute L with one integration by parts as

L = f u(∆ g u + τ 2 |∇ g ϕ| 2 g u) = Σ ∇ g u, ν g f u - ∇ g u, ∇ g (f u) g + τ 2 |∇ g ϕ| 2 g f |u| 2 = Σ ∇ g u, ν g f u -f |∇ g u| 2 g + τ 2 |∇ g ϕ| 2 g f |u| 2 -u ∇ g u, ∇ g f g . (83)
Coming back to (79) and combining the computations of J, K, L in (81)-( 82)-(83), we have obtained the statement of Lemma A.7.

Remark A.8. We wish to compare the above proof with the more usual proofs of Carleman estimates [Hör94, Chapter 23], [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]. Note first that the fact that operators and functions are real-valued implies, for u

∈ C ∞ c (Int(M )) that (Q 1 u, Q 2 u) = (Q 2 Q 1 u, u) = -(u, Q 1 Q 2 u) = 1 2 ([Q 2 , Q 1 ]u, u
). Note also that the principal symbol of the conjugated operator P ϕ is given by

p ϕ (x, ξ) = σ(P ϕ )(x, ξ) = ξ 2 g -τ 2 |∇ g ϕ| 2 g + 2iτ ∇ g ϕ, ξ g = |ξ| 2 g * -τ 2 |dϕ| 2 g * + 2iτ dϕ, ξ g * ,
where g * is the dual metric on T * M , i.e. g * = (g ij ), and ξ is defined by ξ , X g = ξ(X).

Here, a computation shows that we have {Re p ϕ , Im p ϕ }(x, ξ) = 4τ Hess(ϕ)(ξ , ξ ) + 4τ 3 Hess(ϕ)(∇ g ϕ, ∇ g ϕ).

As a consequence, the important quantity in the Carleman estimate of Theorem A.5 is

B g,ϕ,f (ξ ) + τ 2 E g,ϕ,f = (f -∆ g ϕ) ξ 2 g -τ 2 |∇ g ϕ| 2 g + 2Hess(ϕ)(ξ , ξ ) + 2τ 2 Hess(ϕ)(∇ g ϕ, ∇ g ϕ) = (f -∆ g ϕ) Re p ϕ + 1 2τ {Re p ϕ , Im p ϕ }.
The main assumption under which the Carleman estimate of Theorem A.5 holds is hence the existence of a function F = F (x) (of the position variable only) so that

F Re p ϕ + 1 2τ {Re p ϕ , Im p ϕ } ≥ C(|ξ| 2 + τ 2 ). (84)
The choice of F under the form F = f -∆ϕ is only made in order not to consume regularity of the metric g, see above Remark A.4. Of course, Assumption (84) is stronger than the usual subellipticity of the Hörmander theorem [Hör94, Chapter 23]:

{Re p ϕ , Im p ϕ } > 0 on the set {Re p ϕ = 0, Im p ϕ = 0} .

The proof of the Hörmander theorem [Hör94, Section 23.3] then uses a symbol F (x, ξ) instead of just a function F (x), for instance having the form F (x, ξ) = Re pϕ ξ 2 +τ 2 . We refer to [LRL12, Section 3.1] for a related discussion regarding the Fursikov-Imanuvilov approach to Carleman estimates.

A.3 Constructing weight functions via convexification

In this section, we explain how to construct weight functions (ϕ, f ) that satisfy the Assumption of Theorem A.5, via the usual convexification procedure. In the present context (as opposed to the usual situation), this also requires a very specific choice of the function f . Lemma A.9 (Explicit convexification). Let Ψ ∈ W 2,∞ loc (M ; R) and G ∈ W 2,∞ loc (R), and choose

ϕ = G(Ψ) and f = 2G (Ψ) |∇ g Ψ| 2 g . (85)
Then we have

B g,ϕ,f (X) = 2G (Ψ)Hess(Ψ)(X, X) + 2G (Ψ) ∇ g Ψ, X g 2 + G (Ψ) |∇ g Ψ| 2 g -G (Ψ)∆ g Ψ |X| 2 g , E g,ϕ,f = G (Ψ) 2 2G (Ψ)Hess(Ψ)(∇ g Ψ, ∇ g Ψ) + G (Ψ) |∇ g Ψ| 4 g + G (Ψ)∆ g Ψ |∇ g Ψ| 2 g .
To state the next corollary, for

B ∈ T 2 L ∞ loc (M ) a L ∞ loc section of bilinear forms on T M , we define |B| g (x) = sup X∈TxM \0 |B(x,X,X)| |X| 2 g which yields a L ∞ loc function on M .
Corollary A.10. Let Ψ ∈ W 2,∞ loc (M ; R), λ > 0, and define ϕ, f as in (85) with G(t) = e λt . Then, for any λ > 0 and any vector field X, we have almost everywhere on M

B g,ϕ,f (X) ≥ λe λΨ |X| 2 g λ |∇ g Ψ| 2 g -2|Hess(Ψ)| g -∆ g Ψ , E g,ϕ,f ≥ λe λΨ |∇ g ϕ| 2 g λ |∇ g Ψ| 2 g -2|Hess(Ψ)| g + ∆ g Ψ .
Proof of Corollary A.10. With this choice of G, Lemma A.9 gives

B g,ϕ,f (X) = λe λΨ 2Hess(Ψ)(X, X) + 2λ ∇ g Ψ, X g 2 -∆ g Ψ |X| 2 g + λ |∇ g Ψ| 2 g |X| 2 g , Lemma A.12. Let U be an open subset of M such that U is compact (in the topology of M ⊃ ∂M ) and denote Σ = ∂M ∩ U . Given a metric g 0 ∈ T 2 W 1,∞ loc (M ), D ≥ ε > 0, and a function Ψ ∈ W 2,∞ (U ) such that |∇ g0 Ψ| 2
g0 > 0 on U , there exists C 0 > 0 and λ > 0 such that for any g ∈ Γ ε,D (U , g 0 ), the functions

ϕ = e λΨ , f = 2λ 2 |∇ g Ψ| 2 g satisfy B g,ϕ,f (X) ≥ 2C 0 |X| 2 g , for all vector fields X, (86)
E g,ϕ,f ≥ 2C 0 |∇ g ϕ| 2 g , (87)
almost everywhere in U .

Note that the constant C 0 involved is explicitely computable in terms of D and ε, which we do not write for the sake of readability. Yet, if one is interested in obtaining explicit constants, the choice G(t) = e λt of convexifying function is probably not the best one.

Proof. Denote by g * = (g ij ) the metric on T * M induced by g. For g ∈ Γ ε,D (U , g 0 ), we have 1

D g * 0 ≤ g * ≤ 1 ε g * 0 .
With this notation, we have

1 D |∇ g0 Ψ| 2 g0 = 1 D |dΨ| 2 g * 0 ≤ |∇ g Ψ| 2 g = |dΨ| 2 g * ≤ 1 ε |dΨ| 2 g * 0 = 1 ε |∇ g0 Ψ| 2 g0 , (88)
where |ω| 2 g * = ω, ω g * is the cotangent squared norm. Next, using the uniform W 1,∞ (U ) bound in Γ ε,D (U , g 0 ), we have

|∆ g Ψ| ≤ C(ε, D) Ψ W 2,∞ (U ) , |Hess(Ψ)| g ≤ C(ε, D) Ψ W 2,∞ (U ) .
Now, the compactness of U with the assumption yields c 0 > 0 such that |∇ g0 Ψ| 2 g0 ≥ c 0 everywhere on U . According to Corollary A.10 and the above two estimates, we obtain for any λ > 0 and any vector field X We directly deduce the following uniform Carleman estimate in the class Γ ε,D (U , g 0 ). We only state it with the Dirichlet boundary condition here for conciseness (the case without boundary condition writes the same). in the left handside, which does no longer depend on the metric g. Hence, the sole dependence on the metric g in ( 90) is through ∆ g and ∂ ν .

Proof. We choose f = 2λ 2 |∇ g Ψ| 2 g and according to Lemma A.12, the bounds (70)-(71) with constant C 0 are satisfied for λ large enough uniformly in the class g ∈ Γ ε,D (U , g 0 ). According to Theorem A.5, this implies (89) with C 1 = C0 3 c(ϕ) for all τ ≥ τ 0 (g) with τ 0 (g) = c(ϕ

) C0 f -∆ g ϕ 2 L ∞ (U ) + 1 2 ∇ g f L ∞ (U )
with c(ϕ) = max 1, min U |∇ g ϕ| uniformly for g ∈ Γ ε,D (U , g 0 ). This concludes the proof of (89). The proof of (90) follows again from (88) (applied to v) and the fact that d Vol g0 ≤ ε -d/2 d Vol g (recall that d = dim M ).

Note that for the application that we have in Proposition A.14 below, it is sufficient to have some stability results in the following sense: if an interpolation inequality or a Carleman inequality is true for some metric g 0 , it is still true for any metric in a suitable neighborhood. This is of course a byproduct of our results.

A.5 Uniform interpolation estimate at the boundary

In this section, we consider a very particular case of the above Carleman estimate to prove a local interpolation estimate in a neighborhood of a boundary point for metrics g in the neighborhood of the constant flat metric. The manifold M considered is R n+1 + = R n × R + (that is, d = n + 1) and the reference metric is g 0 = Id. The proof follows [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF][START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]. Note that the above sections prove much more than what is actually needed for this argument.

Below, we denote B r = B(0, r) ⊂ R n+1 and B + r = B(0, r) ∩ R n+1 + .

Proposition A.14. There exists ε > 0, C > 0 and α 0 ∈ (0, 1) so that for any metric g ∈ Γ ε,D (B R n (0, 2), Id), we have

v H 1 (B + 1 ) ≤ C (-∂ 2 s -∆ g )v L 2 (B + 2 ) + ∂ s v| s=0 L 2 (B2∩{0}×R n ) α0 v 1-α0 H 1 (B +
2 ) , for any v ∈ H 2 (B + 2 ) such that v| s=0 = 0. Proof. In the proof, we shall denote (with a slight abuse of notation) by x = (s, x) ∈ R + × R n the overall variable and recall that all balls are centered at zero. We choose a point x a = (-a, 0, • • • , 0) / ∈ R n+1 + . We define the weight function Ψ(x) = -|x -x a |, which is smooth and satisfies Ψ < 0 and dΨ = 0 in B + 2 . For a sufficiently small, there exist 0 < ρ 1 < ρ 2 such that we have

B + 1 ⊂ W 1 ⊂ W 1 ⊂ W 2 ⊂ W 2 ⊂ B + 2 , with W j = {Ψ > -ρ j } ∩ R n+1 + , j = 1, 2. ( 91
)
As a consequence of Theorem A.13, there exist λ > 0, C 1 > 0, τ 0 > 0 such that for ϕ = e λΨ and for any g = Id ⊗g ∈ Γ ε,D (B + 2 , Id), for all τ ≥ τ 0 and all u ∈ C ∞ c (B + 2 ) such that u = 0 on {s = 0}, we have Here, the ball, the gradient and the volume element are taken w.r.t. the Euclidean metric. Moreover, the normal vector-field ∂ ν is that associated to the metric g = Id ⊗g, and hence ∂ ν = -∂ s (and does not depend on g). The sole dependence on the metric in (92) is thus in ∆ g = ∂ 2 s + ∆ g . Note that levelsets of ϕ are those of Ψ i.e. pieces of spheres. Note also that we have ϕ ≤ ϕ(0) on B + ≥ τ e 2τ ϕ1 v 2 H 1 (B + 1 ) . Finally, we have ∆ g χv = χ∆ g v + [∆ g , χ]v, where [∆ g , χ] (recall ∆ g = ∂ 2 s + ∆ g ) is a first order differential operator with L ∞ coefficients supported in W 2 \ W 1 , and such that [∆ g , χ] H 1 →L 2 ≤ CD on that set uniformly for g ∈ Γ ε,D (B R n (0, 2), Id). Moreover, we have ϕ ≤ ϕ 1 on W 2 \ W 1 . Thus, we have

e τ ϕ ∆ g u 2 L 2 (B + 2 ) ≤ e τ ϕ χ∆ g v 2 L 2 (B + 2 ) + e τ ϕ [∆ g , χ]v 2 L 2 (B + 2 )
≤ e 2τ ϕ(0) ∆ g v 2

L 2 (B + 2 ) + CDe 2τ ϕ 1 v 2 H 1 (B +
2 ) . Combining the last three estimates with (92), we find that there are C, τ 0 > 0 such that for all g = Id ⊗g ∈ Γ ε,D (B + 2 , Id), for all τ ≥ τ 0 and all v ∈ C ∞ (B + 2 ) such that v = 0 on {s = 0}, we have

e 2τ ϕ1 v 2 H 1 (B + 1 ) ≤ Ce 2τ ϕ(0) ∂ s v| s=0 2 L 2 (B + 2 ∩{s=0}) + ∆ g v 2 L 2 (B + 2 ) + Ce 2τ ϕ 1 v 2 H 1 (B +
2 ) . Recalling that ϕ(0) > ϕ 1 > ϕ 1 and after an optimization in the parameter τ (see [START_REF] Robbiano | Fonction de coût et contrôle des solutions des équations hyperboliques[END_REF]), this yields the result of the lemma.

A.6 A uniform Lebeau-Robbiano spectral inequality

In this section, we give a proof of Theorem 1.16. For this, we follow the strategy of proof of [BHLR10, Section 2] with our uniform Carleman estimates (Theorem A.13). The original proof of [LR95] also works (see the above Section 4) but is less straightforward in the present setting. We recall that M is the ambient compact manifold with boundary ∂M, and set M = [0, S 0 ] × M, having piecewise C 1 and graph-Lipschitz boundary ∂M = {0} × M ∪ {S 0 } × M ∪ [0, S 0 ] × ∂M. We denote by (s, x) the variable in M . the metric is g = Id ⊗g. Note finally that ∂ ν = ∂ νx on [0, S 0 ] × ∂M, were ν x denotes here the outward unit normal to M at ∂M, that ∂ ν = ∂ s on {S 0 } × M, and that ∂ ν = -∂ s on {0} × M. Finally, we are left with the existence of C, τ 0 > 0 such that for all v ∈ H 2 ([0, S 0 ] × M), g ∈ Γ ε,D (M, g 0 ), and τ ≥ τ 0 , we have (93).

From Theorem A.16, we now deduce a proof of Theorem 1.16, following closely (and carefully) [BHLR10, Proof of Theorem 1.1].

Proof of Theorem 1.16. Given w ∈ E g ≤λ take the function

v(s) = sinh(s -∆ g ) -∆ g Π g + w + sΠ g 0 w,
where ∆ g is the Dirichlet Laplacian, Π g 0 the orthogonal projector on ker(∆ g ) (in case ∂M = ∅, otherwise Π g 0 = 0) and Π g + = Id -Π g 0 , that is v is the unique solution to (-∂ 2 s -∆ g )v = 0, v| (0,S0)×∂M = 0, (v, ∂ s v)| s=0 = (0, w).

We may now apply (93), keeping only the penultimate term in the left hand-side: Finally, using sinh(S0 ) ≥ S 0 and the orthogonality of the eigenfunctions, we also have

M |v(S 0 , •)| 2 = sinh 2 (S 0 -∆ g ) -∆ g Π g + w, Π g + w L 2 (M,d Volg) + S 2 0 Π g 0 w 2 L 2 (M,d Volg) ≥ S 2 0 w 2 L 2 .
The last two inequalities conclude the proof of the theorem.

Contents1

  Introduction and main results 1.1 The control cost for the heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Tunneling constants in control theory, and their links . . . . . . . . . . . . . . . . . . . . . . 1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries: links between the different constants 2.1 Different definitions of K wave (ω, T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The constant K eig (ω) as a lower bound for K heat (ω), K ∞ (ω), K wave (ω, T ): Proof of Proposition 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Link between K heat (ω) and K wave (ω, T ): Proof of Proposition 1.7 . . . . . . . . . . . . . . . 2.4 Link between K wave (ω, T ) and analytic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Reformulation of the definition of the constants in terms of localization functions . . . . . . 3 Construction of maximally vanishing eigenfunctions 3.1 The sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 General surfaces of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t∆g u 2 L 2

 22 g. [Cor07, Section 2.5.2]): For any non-empty open set ω and T > 0, there exist C T,ω such that we have e T ∆g u 2 L 2 (M) ≤ C 2 (ω) dt, for all T > 0 and all u ∈ L 2 (M).

e T ∆g y 0 2 L 0 e t∆g y 0 2 L 2

 2022 2 (M) ≤ C e 2 c * T T (ω) dt, for all 0 < T < T 0 and all y 0 ∈ L 2 (M), with c * = a + √ b + a 2 + 2a √ b 2 and C a constant depending only on T 0 , a, b, C.

B

  g,ϕ,f (X) ≥ λe λΨ |X| 2 g λ |∇ g Ψ| 2 g -2|Hess(Ψ)| g -∆ g Ψ , ≥ λe λ min K Ψ |X| 2 g λ c 0 D -3C(ε, D) Ψ W 2,∞ (U ) ,which yields (86) when taking λ large enough. Similarly, (87) follows from taking λ large enough inE g,ϕ,f ≥ λe λ min K Ψ |∇ g ϕ| ε, D) Ψ W 2,∞ (U ) .

WDC 1 τ 3 e τ ϕ v 2 L 2

 22 Theorem A.13 (Uniform Lipschitz Carleman estimate). Let U be an open subset of M such that U is compact (in the topology of M ⊃ ∂M ) and denote Σ = ∂M ∩ U . Given a metric g 0 ∈ T 2 ≥ ε > 0, and a function Ψ ∈ W 2,∞ (U ) such that |∇ g0 Ψ| 2 g0 > 0 on U , there exist λ > 0, C 1 > 0, τ 0 > 0 such that for ϕ = e λΨ and for any g ∈ Γ ε,D (U , g 0 ), for all τ ≥ τ 0 and all v ∈ C ∞ c (U ) such that v = 0 on Σ, we haveC 1 τ 3 e τ ϕ v 2 L 2 (U ) + τ e τ ϕ ∇ g v 2 L 2 (U ) ≤ e τ ϕ ∆ g v 2 L 2 (U ) + τ Σ e 2τ ϕ ∂ ν ϕ|∂ ν v| 2 , (89) (U ) + τ e τ ϕ ∇ g0 v 2 L 2 (U ) ≤ e τ ϕ ∆ g v 2 L 2 (U ) + τ Σ e 2τ ϕ ∂ ν ϕ|∂ ν v| 2 . (90)Note that in the second inequality (90), we implicitely wrote e τ ϕ ∇ g0 v 2 L 2 (U ) = U e 2τ ϕ |∇ g0 v| 2 g0 d Vol g0

≤

  c(ϕ) ≤ max 1, D min U |e λΨ ∇ g0 Ψ| 2 g0 -1uniformly for g ∈ Γ ε,D (U , g 0 ), and, similarly τ 0 (g) ≤ C(ε, D, Ψ, g 0 ),

C 1 τ 3 e τ ϕ u 2 e

 2 2τ ϕ ∂ ν ϕ|∂ ν u| 2 . (92)

  2and define ϕ(0) > ϕ 1 := minB + 1 ϕ > ϕ 1 := min W1 ϕ = e -λρ1 = max W2\W1 ϕ,which only depend the geometric setting (not on the metric). We let χ ∈ C ∞ c (R n+1 ) such that, with W j as in (91), χ = 1 on W 1 and χ = 0 on B + 2 \ W 2 , and apply (92) to u = χv ∈ C ∞ c (B + 2 ) with v ∈ C ∞ (B +2 ) satisfies v| s=0 = 0. We have ∂ ν u| s=0 = -χ| s=0 ∂ s v| s=0 since v| s=0 = 0 and hence{s=0} e 2τ ϕ ∂ ν ϕ|∂ ν u| 2 ≤ Ce 2τ ϕ(0) χ| s=0 ∂ s v| s=0 2 L 2 (W2∩{s=0}) .Using that χ = 1 on W 1 ⊃ B + 1 , we have thatτ 3 e τ ϕ u 2 L 2 (B + 2 ) + τ e τ ϕ ∇u 2 L 2 (B + 2 ) ≥ τ 3 e τ ϕ u 2 L 2 (B + 1 ) + τ e τ ϕ ∇u 2 L 2 (B + 1 )

Lemma A. 15 .

 15 Let g 0 ∈ T 2 W 1,∞ (M) be a metric on M and write g 0 = Id ⊗g 0 . Then, there exists a function ψ ∈ C 2 (M ; R) and c > 0 such that|∇ g0 ψ| g0 ≥ c in M, ∂ νx ψ < 0 on [0, S 0 ] × ∂M, ∂ s ψ ≥ c on {0} × (M \ ω), ∇ g0 ψ = 0 and ∂ s ψ ≤ -c on {S 0 } × M.We refer to [BHLR10, Appendix C] for the proof of this result in case M ⊂ R n , and [LR11, Section 5] for the adaptation to the case of a manifold. With this weight function in hand, we obtain the following global uniform Carleman estimate.

Theorem A. 16 ( 4 Σ|v| 2 ≥

 1642 Global uniform Lipschitz Carleman estimate). Given a metric g 0 ∈ T 2 W 1,∞ (M) , and Ψ as in Lemma A.15, for any D ≥ ε > 0, there exist λ > 0, C 1 > 0, τ 0 > 0 such that for ϕ = e λΨ and for any g ∈ Γ ε,D (M, g 0 ), for all τ ≥ τ 0 and all v ∈ H 2 ([0, S 0 ] × M) such that v = 0 on {0} × M ∪ [0, S 0 ] × ∂M, we have with M = [0, S 0 ] × M and g = Id ⊗g,τ 3 e τ ϕ v 2 L 2 (M ) + τ e τ ϕ ∇ g v 2 L 2 (M ) + τ e 2τ ϕ(S0) M |∂ s v(S 0 , •)| 2 + τ 2 M |v(S 0 , •)| 2 + τ M\ω e 2τ ϕ(0,•) |∂ s v(0, •)| 2 ≤ C e τ ϕ (-∂ 2 s -∆ g )v 2 L 2 (M ) + τ ω e 2τ ϕ(0,•) |∂ s v(0, •)| 2 + τ e 2τ ϕ(S0) M |∇ g v(S 0 , •)| 2 g .(93)Proof. We use the Carleman estimates (73)-(74) together with Remark A.6 and Lemma A.12 for the uniformity in the metric. More precisely, on the boundary {0} × M ∪ [0, S 0 ] × ∂M, the Dirichlet boundary condition is prescribed and the only boundary term is +τ Σ e 2τ ϕ ∂ ν ϕ|∂ ν v| 2 , according to (73). That∂ ν ϕ ≤ -c < 0 on {0} × (M \ ω) ∪ [0, S 0 ] × ∂M impliesthat the associated integral is dominated on that set, whereas the only observation term on that part of the boundary is -τ ω e 2τ ϕ(0,•) ∂ s ϕ(0, •)|∂ s v(0, •)| 2 . Now, on the part {S 0 } × M of the boundary, we have the observation term τ Σ e 2τ ϕ |∇ T v| 2 g = τ e 2τ ϕ(S0) M |∇ g v(S 0 , •)| 2 g . On the other side of the inequality, we have the two observed terms τ 8 m(ϕ) 3 M (ϕ) 2 Σ e 2τ ϕ |∂ ν v| 2 + τ 3 m(ϕ) 3 Cτ e 2τ ϕ(S0) M |∂ s v(S 0 , •)| 2 + τ 2 M |v(S 0 , •)| 2 .

e

  2τ ϕ(S0) τ 3 M |v(S 0 , •)| 2 ≤ C τ ω e 2τ ϕ(0,•) |∂ s v(0, •)| 2 + τ e 2τ ϕ(S0) M |∇ g v(S 0 , •)| 2 g .Now, we haveω e 2τ ϕ(0,•) |∂ s v(0, •)| 2 ≤ e 2τ sup M ϕ(0,•) w 2 L 2 (ω) ,together with (using an integration by parts, together with w ∈ E g ≤λ ),M |∇ g v(S 0 , •)| 2 g = (-∆ g v(S 0 , •), v(S 0 , •)) L 2 (M,d Volg) ≤ λ (v(S 0 , •), v(S 0 , •)) L 2 (M,d Volg) .The last three inequalities imply for all τ ≥ τ 0τ 2 v(S 0 , •) 2 L 2 ≤ C e 2τ (sup M ϕ(0,•)-ϕ(S0)) w 2 L 2 (ω) + λ v(S 0 , •) 2 L 2 ,and hence, when choosing τ = max{2 √ λ, τ 0 }, we obtainv(S 0 , •) 2 L 2 ≤ Ce 4 √ λ(sup M ϕ(0,•)-ϕ(S0)) w 2 L 2 (ω) .

  )4.3 A proof of Lemma 4.2 from Aronszajn estimatesIn this section, we give a proof of Lemma 4.2 starting from Carleman-Aronszajn estimates as stated in [DF88, Proposition 2.10] and [DF90, Proposition 2.10] (and slightly modified according to the remarks in [JL99, Beginning of Section 14.3]), which we now state. An alternative proof of a closely related estimate is given by Hörmander in [Hör85, Inequality (17.2.11), Chapter XVII.2].

Preliminaries: links between the different constants 2.1 Different definitions of K wave (ω, T )Let us start by proving equality (14). This is a consequence of the following lemma.

Acknowledgements. The first author is partially supported by the Agence Nationale de la Recherche under grants SRGI ANR-15-CE40-0018 and IPROBLEMS ANR-13-JS01-0006. The second author is partially supported by the Agence Nationale de la Recherche under grant GERASIC ANR-13-BS01-0007-01. Both authors are partially supported by the Agence Nationale de la Recherche under grant ISDEEC ANR-16-CE40-0013. Part of this research was done when the second author was in CRM, CNRS UMI 3457, Université de Montréal, in Université Paris Diderot, IMJ-PRG, UMR 7586, and in École polytechnique, Centre de Mathématiques Laurent Schwartz UMR7640, 91128 Palaiseau cedex France.

Recalling the definition of α r , this is the sought result of Theorem 1.15 (up to changing 2r into r, and the names of the constants accordingly) with the restriction r ∈ (0, r0 4 ). To conclude for all r > 0, it suffices to notice that (64) remains true with α r 0 16 on the r.h.s. uniformly for observation terms ψ L 2 (B M (x0,2r)) with r ≥ r0 8 (the constants are non-increasing functions of the observation set).

To conclude the proof in the general case, we need to consider the situation ∂M = ∅ in full generality. We again follow [START_REF] Donnelly | Nodal sets of eigenfunctions on Riemannian manifolds[END_REF][START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF]. In this case, we define the double manifold M = M M, consisting in gluing two copies of M, endowed with a smooth structure of compact manifold, as in [Lee13, Theorem 9.29-Example 9.32]. Then, the procedure is very well explained in [Ant08, Section 3] and we only sketch the proof. We extend the metric g on M by symmetry/parity with respect to the boundary ∂M as a metric g on M. Note that even if g is smooth, the extended metric g is only Lipschitz on M. This is not an issue since the three lemmata 4.1, 4.2 and 4.3 remain valid for Lipschitz metrics (as a consequence of Appendix A, [AKS62, DF90], and Appendix A, respectively). In the case of Dirichlet boundary condition on ∂M, and given ψ ∈ E ≤λ we take its anti-symmetric/odd extension on M, yielding a function ψ ∈ Ẽ≤λ . Here, Ẽ≤λ is the counterpart of E ≤λ defined for the Laplace-Beltrami operator ∆ g on M. The above computations are then made for ∆ g on M and the estimate (64) is proved for ψ. The same estimate for ψ follows. Similarly, in the case of Neumann boundary condition, we take the symmetric/even extension of functions, yielding the sought result.

together with

, which yields the sought result.

Proof of Lemma A.9. We first have dϕ = G (Ψ)dΨ and ∇ g ϕ = G (Ψ)∇ g Ψ. We then compute the Hessian and the Laplacian as

In particular, we have

As a consequence, we obtain

as well as

Now, recalling the choice f = 2G (Ψ) |∇ g Ψ| 2 g concludes the proof of the lemma.

Remark A.11. Note that in this proof, the choice f = αG (Ψ) |∇ g Ψ| 2 g yields a useful lower bound only if α ∈ (1, 3). See also [LRL12, Section 3.1] for a related discussion.

A.4 Uniformity with respect to the metric

Until this point, all calculations are exact for a fixed metric. In the present section, we prove uniform estimates in a class of metrics. For this, even though the manifold with boundary M is not assumed compact, we will consider only open subsets U of M such that U is compact in M (not in Int(M )). On the compact set K, the spaces W k,∞ (K) are defined intrinsically, even if the associated norms may depend on the metric or the charts chosen. We fix one of these norms • W 1,∞ (K) for functions on M , as well as for forms on M (still denoted • W 1,∞ (K) ). Now, given a reference metric g 0 and two constants D ≥ ε > 0, we consider the class

B Local behavior of vanishing functions

In this appendix, we give an explicit link between the different definitions of the vanishing rate of a function.

Lemma B.1. Let f ∈ C ∞ (B R n (0, 1)) and assume that there are C, D > 0 such that we have uniformly for 0 < r < 1 the estimate

Then, we have