Jean-François Grosjean
email: grosjean@iecn.u-nancy.fr

Jean-Francois Grosjean

EXTRINSIC UPPER BOUNDS FOR THE FIRST EIGENVALUE OF ELLIPTIC OPERATORS

Keywords:

We consider operators defined on a Riemannian manifold M m by L T (u) = -div(T ∇u) where T is a positive definite (1, 1)-tensor such that div(T ) = 0. We give an upper bound for the first nonzero eigenvalue λ 1,T of L T in terms of the second fundamental form of an immersion φ of M m into a Riemannian manifold of bounded sectional curvature. We apply these results to a particular family of operators defined on hypersurfaces of space forms and we prove a stability result.

Introduction

Let (M m , g) be a compact, connected m-dimensional Riemannian manifold. In this paper, we are interested in extrinsic upper bounds for the first nonzero eigenvalue of elliptic operators defined on (M m , g) (i.e. in terms of the second fundamental form of an isometric immersion of (M m , g) into an n-dimensional Riemannian manifold (N n , h)). The elliptic second order differential operators L T , which we are interested in, are of the form

L T u = -div M (T ∇ M u)
where u ∈ C ∞ (M ), T is a (1, 1)-tensor on M (which will be divergence-free and symmetric), and div M and ∇ M denote respectively the divergence and the gradient of the metric g. In the sequel, we will denote by λ 1,T , the first nonzero eigenvalue of such operator L T .

When T is the identity, L T = L Id is nothing but the Laplace operator of (M m , g). In this case, it is well known that if (M m , g) is isometrically immersed in a simply connected space form N n (c) (c = 0, 1, -1 respectively for the Euclidean space IR n , the sphere I S n or the hyperbolic space IH n ), then we have the following estimate of λ 1 = λ 1,Id in terms of the square of the length of the mean curvature

λ 1 V (M ) ≤ m M |H| 2 + c dv g (1)
where dv g and V (M ) denote respectively the Riemannian volume element and the volume of (M m , g) and where H denotes the mean curvature of the immersion of (M m , g) into N n (c). Furthermore, the equality in (1) occurs if and only if (M m , g) is immersed as a minimal submanifold of some geodesic hypersphere of N n (c). For c = 0, this inequality was proved by Reilly ([17]) and can easily be extended to the spherical case c = 1 by considering the canonical embedding of I S n in IR n+1 and by applying the inequality (1) for c = 0 to the obtained immersion of (M m , g) in IR n+1 . For immersions of (M m , g) in the hyperbolic space IH n , Heintze ([14]) first proved an L ∞ equivalent of (1) and conjectured [START_REF] Aithal | Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces[END_REF] which was finally obtained by El Soufi and Ilias ( [START_REF] Soufi | Une inegalité du type "Reilly" pour les sous-variétés de l'espace hyperbolique[END_REF]). Note that, the estimates shown in [START_REF] Heintze | Extrinsic upper bound for λ 1[END_REF] and [START_REF] Soufi | Une inegalité du type "Reilly" pour les sous-variétés de l'espace hyperbolique[END_REF] are given for immersions of (M m , g) in a space which is not necessarly of constant sectional curvature.

Later, these estimates were extended to more general operators called L r (0 ≤ r ≤ n) defined on hypersurfaces (M m , g) of N m+1 (c). Let us first define these operators. Let φ be an isometric immersion of (M m , g) into N m+1 (c) and denote by A its shape (or Weingarten) operator. For any integer r ∈ {0, ..., n}, the (1, 1)-tensors T r of Newton are defined inductively by: T 0 = Id and T r = S r Id -AT r-1 , where S r is the r-th elementary symmetric function of the eigenvalues of A (i.e. the principal curvatures). Note that T r is a free divergence tensor because the ambient space is of constant curvature (see for instance [START_REF] Rosenberg | Hypersurfaces of constant curvature in space forms[END_REF]). The r-th mean curvature of φ is H r = 1/ m r S r . Now, the operator L r is defined by L r = L Tr which is the linearized operator of the first variation of S r+1 ( [START_REF] Reilly | Variational properties of functions of the mean curvatures for hypersurfaces in space forms[END_REF]). It is important for our paper to know when L r is elliptic. Walter proved in [START_REF] Walter | Compact hypersurfaces with a constant higher meancurvature function[END_REF] that if H r+1 > 0 and if the immersion φ is convex (i.e. the second fundamental form is semidefinite), then T r is positively definite (i.e. L r is elliptic). This result was strengthened by Barbosa and Colares ([6]). They proved without any convexity assumption that if H r+1 > 0 and if, in the case c = 1, φ(M ) is contained in a hemisphere, then L r is elliptic. For simplicity the first nonzero eigenvalue of L r will be denoted λ 1,r (which is λ 1,Tr ). The first extension of the Reilly inequality (1) to such operators L r was obtained by Alencar, do Carmo and Rosenberg ([4] and [START_REF] Alencar | Erratum to On the first eigenvalue of the linearized operator of the r-th mean curvature of ahypersurface[END_REF]). They proved that if (M m , g) is an m-dimensional compact immersed hypersurface of the Euclidean space IR m+1 and if H r+1 > 0 then

λ 1,r M H r dv g ≤ (m -r) m r M H 2 r+1 dv g
and equality holds if and only if (M m , g) is a geodesic sphere of IR m+1 . In our paper [START_REF] Grosjean | A Reilly inequality for some natural elliptic operators on hypersurfaces[END_REF] (theorem 1.1, see also [START_REF] Grosjean | Majoration de la première valeur propre de certains opérateurs elliptiques naturels sur les hypersurfaces des espaces formes et applications[END_REF]), we obtained a similar optimal upper bound for λ 1,r of hypersurfaces of any space form N m+1 (c). We proved for all 0 ≤ r ≤ m -2, that if H r+1 > 0 and if φ is convex (i.e. the second fundamental form is semi-definite) then

λ 1,r V (M ) ≤ (m -r) m r M H 2 r+1 + cH 2 r H r dv g (2)
and equality holds if and only if φ immerses M as a geodesic sphere of N m+1 (c).

Our approach to obtain such estimates was a generalization of the conformal technic used by El Soufi and Ilias and in this approach the convexity assumption was essential to obtain the estimate [START_REF] Alencar | Stable hypersurfaces with constant scalar curvature[END_REF]. Nevertheless, it is natural to ask if such estimates still valid without the convexity assumption. In this paper, to answer this purpose, we use a different approach inspired by the method of Heintze ([14]). In fact, an L ∞ estimate similar to (2) will be a consequence of an estimate (theorem 1.1) obtained in a more general setting: for the operators L T defined above and for ambient spaces not necessarly of constant sectional curvature.

Before stating the results, we need to define the following normal vector field H T . If φ is an isometric immersion of (M m , g) in (N n , h) and B is its second fundamental form then we define H T at a point x ∈ M , by

H T (x) = i≤m B(T e i , e i )
where (e i ) 1≤i≤m is an orthonormal basis of the tangent space of M at x.

The main result of our paper is the Theorem 1.1 Let (M m , g) be a compact, connected, m-dimensional Riemannian manifold (m ≥ 2) and let φ be an isometric immersion of (M m , g) in an n-dimensional Riemannian manifold (N n , h) of sectional curvature bounded above by δ. If δ ≤ 0 we assume that (N n , h) is simply connected and if δ > 0 we assume that φ(M ) is contained in a convex ball of radius less or equal to π/4 √ δ. Let L T be an elliptic operator defined on (M m , g) as above. Then, we have

λ 1,T ≤ sup M |H T | 2 + sup M δ(tr(T )) 2 inf M tr(T )
and if equality holds then φ(M ) is contained in a geodesic sphere.

When (N n , h) is a simply connected space form and T = T r , we deduce from this theorem an estimate of λ 1,r without the convexity assumption. In fact, we have Corollary 1.1 Let (M m , g) be a compact, connected and orientable m-dimensional Riemannian manifold (m ≥ 2), immersed in a space form (N m+1 (c), h) (c = 0, -1, +1). Assume, if c = 1, that φ(M ) is contained in a ball of radius π/4. If H r+1 > 0 for r ∈ {0, ..., m -1}, then we have

λ 1,r ≤ (m -r) m r sup M H 2 r+1 + sup M (cH 2 r ) inf M H r
and equality holds if and only if φ(M ) is a geodesic sphere.

This last corollary has just been obtained independently by Alencar, do Carmo and Marques ([3]).

When |H T | is constant, we show a different estimate which is usefull in the proof of stability results; indeed, we have the Theorem 1.2 Let (M m , g) be a compact, connected, m-dimensional Riemannian manifold (m ≥ 2) and let φ be an isometric immersion of (M m , g) in an n-dimensional Riemannian manifold (N n , h) of sectional curvature bounded above by δ. If δ ≤ 0 we assume that (N n , h) is simply connected and if δ > 0 we assume that φ(M ) is contained in a convex ball of radius less or equal to π/4 √ δ. Let L T be an elliptic operator defined on (M m , g) as above. Then, we have

λ 1,T ≤ sup M (|H T ||H| + δtr(T ))
and if equality holds then φ(M ) is contained in a geodesic sphere.

As a consequence, we have the Corollary 1.2 Let (M m , g) be a compact, connected and orientable n-dimensional Riemannian manifold (n ≥ 2), immersed in a space form (spf mnpi, h) (c = 0, -1, +1). Assume, if c = 1 that φ(M ) is contained in a ball of radius π/4. If for r ∈ {0, ..., m -1}, H r+1 is a positive constant, then we have

λ 1,r ≤ sup M (m -r) m r (H r+1 H 1 + cH r )
and equality holds if and only if φ(M ) is a geodesic sphere.

This paper is structured as follows: the first part deals with the proofs of these theorems and corollaries and in a second part we give an application of our results to the stability problem of hypersurfaces of constant r-th mean curvature in a space form. The results of this paper were announced in the note [START_REF] Grosjean | Estimations extrinsèques de la première valeur propre d'opérateurs elliptiques définis sur des sous variétés et applications[END_REF].

Proofs of the results

Let (M m , g) be a compact, connected m-dimensional Riemannian manifold isometrically immersed by φ in an n-dimensional Riemannian manifold (N n , h) which sectional curvature is bounded by δ. The manifold M is endowed with a symmetric definite positive (1, 1)-tensor T of free divergence. The associated operator L T defined by L T (u) = -div(T ∇ M u) is self adjoint and elliptic and we denote by λ 1,T its first nonzero eigenvalue.

Let p 0 ∈ N and exp p 0 the exponential map at this point. We consider (x i ) 1≤i≤n the normal coordinates of N centered at p 0 and for all x ∈ N , we denote by r(x) = d(p 0 , x), the geodesic distance between p 0 and x on (N n , h). If δ > 0 we assume that φ(M ) lies in a convex ball around p 0 of radius less or equal to π/2 √ δ. Let s δ and c δ be functions defined by

s δ (r) =      1 √ δ sin √ δr if δ > 0 r if δ = 0 1 √ |δ| sinh |δ|r if δ < 0 and c δ (r) =    cos √ δr if δ > 0 1 if δ = 0 cosh |δ|r if δ < 0 We remark that c 2 δ + δs 2 δ = 1, s δ = c δ and c δ = -δs δ .
In the sequel, we denote respectively by ∇ M and ∇ N the gradients associated to (M m , g) and (N n , h). It is easy to see that the coordinates of Z = s δ (r)∇ N r in the normal local frame are s δ (r) r x i 1≤i≤n

. Furthermore, the tangential and normal projection of a vector field X respectively on the tangent bundle and the normal bundle to φ(M ) will be denoted respectively by X t and X n . We recall now some facts and properties of the exponential map. Let U, V ∈ T p 0 N and x ∈ N . If we set X = exp -1 p 0 (x). Then, we have

i≤n h x (∇ N x i , d exp p 0 X (U ))h x (∇ N x i , d exp p 0 X (V )) = h p 0 (U, V ) (3)
On the other hand, exp p 0 is a radial isometry (Gauss lemma), that is for each x of N , we have

h x ( d exp p 0 X (X), d exp p 0 X (U )) = h p 0 (X, U ) (4)
First, we begin by proving some lemmas Lemma 2.1 For each x of M , we have

1≤i≤n g x T ∇ M s δ (r) r x i , ∇ M s δ (r) r x i ≤ tr(T ) -δg x (T Z t , Z t ) (5)
and equality holds if (N n , h) has a constant sectional curvature equal to δ.

Proof: We compute the left hand side of (5)

∇ M s δ (r) r x i = rc δ (r) -s δ (r) r 2 (∇ M r)x i + s δ (r) r ∇ M x i thus 1≤i≤n g x T ∇ M s δ (r) r x i , ∇ M s δ (r) r x i = 1≤i≤n rc δ (r) -s δ (r) r 2 x i 2 g x (T ∇ M r, ∇ M r) + 2 1≤i≤n rc δ (r) -s δ (r) r 2 s δ (r) r x i g x (T ∇ M r, ∇ M x i ) + 1≤i≤n s 2 δ (r) r 2 g x (T ∇ M x i , ∇ M x i )
and using the fact that 1≤i≤n x i ∇ M x i = r∇ M r, we deduce

1≤i≤n g x T ∇ M s δ (r) r x i , ∇ M s δ (r) r x i = s 2 δ (r) r 2 1≤i≤n g x (T ∇ M x i , ∇ M x i ) + (rc δ (r) -s δ (r)) 2 r 2 + 2 rc δ (r) -s δ (r) r 2 s δ (r) g x (T ∇ M r, ∇ M r)
After an easy computation and noting that Z t = s δ (r)∇ M r, we obtain

1≤i≤n g x T ∇ M s δ (r) r x i , ∇ M s δ (r) r x i = s 2 δ (r) r 2 1≤i≤n g x (T ∇ M x i , ∇ M x i ) + 1 - s 2 δ (r) r 2 g x (T ∇ M r, ∇ M r) -δg x (T Z t , Z t ) ( 6
)
Since T is a positive symmetric (1, 1)-tensor, we can define a natural positive symmetric (1, 1)-tensor √ T . Indeed, if (e i ) 1≤i≤m is an orthonormal basis at x which diagonalizes T such that

T = diag(µ 1 , ..., µ m ), then √ T is defined at x by √ T = diag( √ µ 1 , ..., √ µ m ).
Now let (e i ) 1≤i≤m be an orthonormal frame in x, such that √ T e m lies in the direction of ∇ M r and let e * m be a unit vector orthogonal to ∇ N r in order to have:

√ T e m = λ∇ N r + µe * m . Then (6) becomes 1≤i≤n g x T ∇ M s δ (r) r x i , ∇ M s δ (r) r x i = s 2 δ (r) r 2 i ≤ n j ≤ m h x (∇ N x i , √ T e j ) 2 + 1 - s 2 δ (r) r 2 g x (T ∇ M r, ∇ M r) -δg x (T Z t , Z t ) = s 2 δ (r) r 2 i ≤ n j ≤ m -1 h x (∇ N x i , √ T e j ) 2 + s 2 δ (r) r 2 1≤i≤n h x (∇ N x i , λ∇ N r) + h x (∇ N x i , µe * m ) 2 + 1 - s 2 δ (r) r 2 | √ T ∇ M r| 2 x -δg x (T Z t , Z t ) (7)
Now, setting v j = √ T e j -h( √ T e j , ∇ M r)∇ N r, for all j ≤ m -1, we rewrite the first term of the right hand side of (7)

s 2 δ (r) r 2 i ≤ n j ≤ m -1 h x (∇ N x i , √ T e j ) 2 = s 2 δ (r) r 2 i ≤ n j ≤ m -1 h x (∇ N x i , v j ) + h x ( √ T e j , ∇ M r)h x (∇ N x i , ∇ N r) 2 = s 2 δ (r) r 2 i ≤ n j ≤ m -1 h x (∇ N x i , v j ) 2 + s 2 δ (r) r 2 i ≤ n j ≤ m -1 h x ( √ T e j , ∇ M r) 2 h x (∇ N x i , ∇ N r) 2 +2 s 2 δ (r) r 2 i ≤ n j ≤ m -1 h x ( √ T e j , ∇ M r)h x (∇ N x i , v j )h x (∇ N x i , ∇ N r) (8)
We compute each term of the right hand side of [START_REF] Chavel | Riemannian geometry-A modern introduction[END_REF]. Using the standard Jacobi field estimates (cf for instance corollary 2.8, p 153 of [START_REF] Sakai | Riemannian geometry[END_REF]), we have for all v orthogonal to

∇ N r s 2 δ (r) r 2 d(exp -1 p 0 ) x (v) 2 p 0 ≤ |v| 2 x (9)
with equality if N has a constant sectional curvature equal to δ. Now v j is orthogonal to ∇ N r and then applying successively (3) and ( 9), we obtain

s 2 δ (r) r 2 i ≤ n j ≤ m -1 h x (∇ N x i , v j ) 2 = s 2 δ (r) r 2 j≤m-1 d exp -1 p 0 x (v j ) 2 p 0 ≤ j≤m-1 |v j | 2 x = 1≤j≤m-1 | √ T e j | 2 x - 1≤j≤m-1 h x ( √ T e j , ∇ M r) 2 (10)
Moreover, from (3) and (4), we have for all v orthogonal to

∇ N r 1≤i≤n h x (∇ N x i , v)h x (∇ N x i , ∇ N r) = h p 0 d(exp -1 p 0 x (v), d exp -1 p 0 x (∇ N r) = h p 0 d exp -1 p 0 x (v), X/r = h x (v, ∇ N (r)) = 0 (11)
Hence, the last term of the right hand side of (8) vanishes identically. Now, reporting [START_REF] Soufi | Majoration de la seconde valeur propre d'un opérateur de Schrödinger sur une variété compacte et applications[END_REF] in [START_REF] Chavel | Riemannian geometry-A modern introduction[END_REF], and noting that 1≤i≤n h x (∇ N x i , ∇ N r) 2 = 1 by (3), we find

s 2 δ (r) r 2 i ≤ n j ≤ m -1 h x (∇ N x i , √ T e j ) 2 ≤ 1≤j≤m-1 | √ T e j | 2 x + s 2 δ (r) r 2 -1 1≤j≤m-1 h x ( √ T e j , ∇ M r) 2 = tr(T ) -| √ T e m | 2 x + s 2 δ (r) r 2 -1 1≤j≤m-1 h x ( √ T e j , ∇ M r) 2 (12)
Furthermore, from ( 9) and ( 11), we deduce that

s 2 δ (r) r 2 1≤i≤n h x (∇ N x i , λ∇ N r) + h x (∇ N x i , µe * m ) 2 = s 2 δ (r) r 2 λ 2 1≤i≤n h x (∇ N x i , ∇ N r) 2 + s 2 δ (r) r 2 µ 2 1≤i≤n h x (∇ N x i , e * m ) 2 ≤ λ 2 s 2 δ (r) r 2 + µ 2 (13)
Finally, by reporting ( 12) and ( 13) in ( 7), we get

1≤i≤n g x T ∇ M s δ (r) r x i , ∇ M s δ (r) r x i ≤ tr(T ) -| √ T e m | 2 x + s 2 δ (r) r 2 -1 j≤m-1 h x ( √ T e j , ∇ M r) 2 + λ 2 s 2 δ (r) r 2 + µ 2 + 1 - s 2 δ (r) r 2 g x ( √ T ∇ M r, e m ) 2 + 1 - s 2 δ (r) r 2 1≤i≤m-1 g x ( √ T ∇ M r, e i ) 2 -δg x (T Z t , Z t ) = tr(T ) -| √ T e m | 2 x + λ 2 s 2 δ (r) r 2 + µ 2 + 1 - s 2 δ (r) r 2 g x ( √ T ∇ M r, e m ) 2 -δg x (T Z t , Z t ) now g x ( √ T ∇ M r, e m ) = h x ( √ T e m , ∇ N r) = λ and λ 2 + µ 2 = | √ T e m | 2
x And after simplification, this gives the desired inequality and if (N n , h) is of constant sectional curvature all the inequalities above are in fact equalities. 2

Now, we will prove the Lemma 2.2 For all symmetric divergence-free positive definite (1, 1)-tensors T on M , we have

div M (T Z t ) ≥ (tr(T ))c δ + h(Z, H T )
and if T is the identity and (N n , h) has a constant sectional curvature equal to δ, then equality holds.

Proof: We use the same local frame as in the proof of lemma (2.1) and we compute div M (T Z t ) in this frame by using the fact that T is a free divergence tensor (i.e. 1≤i≤m ∇ M e i T e i = 0)

div M (T Z t ) = 1≤i≤m g x (∇ M e i (T Z t ), e i ) = 1≤i≤m g x ( ∇ M e i T Z t , e i ) = 1≤i≤m g x (∇ M e i Z t , T e i ) = 1≤i≤m h x (∇ N e i Z t , T e i ) = 1≤i≤m h x (∇ N e i Z, T e i ) - 1≤i≤m h x (∇ N e i Z n , T e i ) = 1≤i≤m h x (∇ N e i Z, T e i ) + 1≤i≤m h x (Z, B(T e i , e i )) = 1≤i≤m h x (∇ N e i Z, T e i ) + h x (Z, H T ) (14)
Now, we want to estimate 1≤i≤m h x (∇ N e i Z, T e i ). We first have

1≤i≤m h x (∇ N e i Z, T e i ) = 1≤i≤m h x (∇ N e i (s δ ∇ N r), T e i ) = c δ h x (∇ N r, T (∇ N r) t ) + s δ 1≤i≤m h x (∇ N e i ∇ N r, T e i ) = c δ h x (T (∇ N r) t , (∇ N r) t ) + s δ 1≤i≤m h x (∇ N √ T e i ∇ N r, √ T e i ) ( 15
)
And using the standard Jacobi field estimates (see lemma 2.9 p 153 of [START_REF] Sakai | Riemannian geometry[END_REF]) we can find a lower bound of the last term of [START_REF] Korevaar | Sphere theorem via Alexandrov for constant Weingarten curvature hypersurfaces,Appendix to a note of A.Ros[END_REF]. Indeed, we have for all vector ξ orthogonal to ∇ N r at x, the inequality

h x (∇ N ξ ∇ N r, ξ) ≥ c δ s δ |ξ| 2
x and equality holds if N has a constant sectional curvature equal to δ. Thus,

1≤i≤m h x (∇ N √ T e i ∇ N r, √ T e i ) = 1≤i≤m-1 h x (∇ N √ T e i ∇ N r, √ T e i ) + h x (∇ N √ T em ∇ N r, √ T e m ) ≥ c δ s δ 1≤i≤m-1 | √ T e i | 2 x + µ 2 h x (∇ N e * m ∇ N r, e * m ) ≥ c δ s δ 1≤i≤m-1 | √ T e i | 2 x + µ 2 c δ s δ
and reporting this inequality in [START_REF] Korevaar | Sphere theorem via Alexandrov for constant Weingarten curvature hypersurfaces,Appendix to a note of A.Ros[END_REF], we obtain

1≤i≤m h x (∇ N e i Z, T e i ) ≥ c δ | √ T (∇ N r) t | 2 x + c δ 1≤i≤m-1 | √ T e i | 2 x + µ 2 c δ ( 16
)
now

λ 2 = h x ( √ T e m , ∇ N r) 2 = h x ( √ T e m , (∇ N r) t ) 2 = h x (e m , √ T (∇ N r) t ) 2 ≤ | √ T (∇ N r) t | 2
x and if T is the identity, this last inequality is an equality. Furthermore, it is easy to verify that

λ 2 + µ 2 = | √ T e m | 2
x thus, inequality (15) becomes

1≤i≤m h x (∇ N e i Z, T e i ) ≥ c δ λ 2 + c δ 1≤i≤m-1 | √ T e i | 2 x + µ 2 c δ = tr(T )c δ
and inserting this last inequality in [START_REF] Montiel | Compact hypersurfaces:the Alexandrov theorem for higher order mean curvatures, Differential Geometry , Blaine Lawson and Keti Tonenblat[END_REF] we complete the proof of lemma 2.2.

2 Lemma 2.3 We have δ M g x (T Z t , Z t )dv g ≥ M tr(T )c 2 δ dv g - M |H T |s δ c δ dv g Proof: If δ = 0, then δ M g(T Z t , Z t )dv g = 1 δ M g(T ∇ M c δ (r), ∇ M c δ (r))dv g = - 1 δ M div M (T ∇ M c δ (r))c δ (r)dv g = M div M (T Z t )c δ dv g ≥ M c 2 δ tr(T )dv g - M |H T |s δ c δ dv g
where the last inequality is proceeding from the previous lemma 2.2. Moreover, if δ = 0, then c δ (r) = 1 and we have

0 = M div M (T Z t )c δ dv g ≥ M c 2 δ tr(T )dv g - M |H T |s δ c δ dv g
This concludes the proof. 2

We can now give the proof of our results.

Proof of theorem 1.1: Let p 0 ∈ N and r(x) = d(p 0 , x), where r(x) is the geodesic distance between p 0 and x. We will use s δ (r) r x i as test functions in the variational characterization of λ 1,T but the mean of these functions must be zero. For this purpose, we use a standard argument used by Chavel and Heintze before ( [START_REF] Heintze | Extrinsic upper bound for λ 1[END_REF] and [START_REF] Chavel | Riemannian geometry-A modern introduction[END_REF]). Indeed, let Y be a vector field defined by

Y q = M s δ (d(q, p)) d(q, p) exp -1 q (p)dv g (p) ∈ T q N
From the theorem of fixed point of Brouwer, there exists a point p 0 ∈ N such that Y p 0 = 0 and consequently, for a such p 0 , the mean of s δ (r) r x i will be zero. But for δ > 0, we must assume φ(M ) is contained in a ball of radius π/4 √ δ. Indeed, in this case φ(M ) lies in a ball of center p 0 (the point p 0 such that Y p 0 = 0) with a radius less or equal to π/2 √ δ (this hypothesis is necessary in the proof of the preceding lemmas). It follows from above and the variational characterization of λ 1,T , that

λ 1,T M s 2 δ (r)dv g = λ 1,T M |Z| 2 dv g = λ 1,T M 1≤i≤n s δ (r) r x i 2 dv g ≤ M 1≤i≤n L T s δ (r) r x i s δ (r) r x i dv g
Proof of corollaries 1.1 and 1.2: (M m , g) is a compact, connected and orientable n-dimensional Riemannian manifold (n ≥ 2) isometrically immersed by φ in a simply connected space form N m+1 (c) (c = 0, 1 or -1 respectively for IR n+1 , I S n+1 or IH n+1 ) and A is the Weingarten operator associated to the second fundamental form of the immersion. When c ≤ 0, assumptions of theorems 1.1 and 1.2 are trivially verified. For c = 1, we assume that φ(M ) lies in a ball of radius π/4. Since H r+1 > 0 with φ(M ) contained in a hemisphere when c = 1, then L r is elliptic ( [START_REF] Barbosa | Stability of hypersurfaces with constant r-mean curvature[END_REF]). Finally, under these hypotheses, the corollaries follow from the theorems by applying them to the special (1, 1)-tensors T r defined in the introduction and by using the following relations: tr(T r ) = (n -r) m r H r and tr(AT r ) = (n -r) m r H r+1 ( [START_REF] Reilly | On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space[END_REF]). Furthermore, from theorems, if inequalities expressed in corollaries are equalities, then φ(M ) is a geodesic sphere. Conversely, if φ(M ) is a geodesic sphere, then M is totally umbilical and we have:

H r = H r 1 and L r = m r H r 1 ∆
, where ∆ is the usual Laplacian, and we have equality.

2

Remark 2.1: A generalization of these results to Schrödinger type operators L = L T + q, where q ∈ C ∞ (M ) can be easily obtained. Denoting by λ 2 (L T + q) the second eigenvalue of L, and by u a first positive eigenfunction of L, we consider the vector field Y defined by

(|H T ||H| + δtr(T ) + q)s 2 δ dv g (17)
and finally,

λ 2 (L T + q) ≤ sup M (|H T ||H| + δ tr(T ) + q)
This last relation [START_REF] Reilly | On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space[END_REF] will be very useful for the proof of stability results.

Applications to stability

Consider an m-dimensional hypersurface with constant r + 1-th mean curvature immersed in a space form N m+1 (c) whose sectional curvature is constant equal to c (c = 0, -1, 1). First, recall briefly the variational problem associated to these hypersurfaces (for more details see for instance [START_REF] Alencar | On the first eigenvalue of the linearized operator of the r-th mean curvature of a hypersurface[END_REF] and [START_REF] Barbosa | Stability of hypersurfaces with constant r-mean curvature[END_REF]) Let φ : (M m , g) → (N m+1 (c), h) be an orientable compact hypersurface oriented by the global normal field η, and let us define the functionals

A 0 = M dv g , A 1 = M S 1 dv g and for each r, 2 ≤ r ≤ m, A r = M S r dv g + k(m -r + 1) r -1 A r-2
Where we put k(s) = (m -s) m s . Now, let F :] -ε, ε[×M → (N m+1 (c), h) be a variation of the immersion φ for all t ∈]-ε, ε[. The immersion F (t, .), its r-th elementary symmetric function and the associated functional A r will be denoted respectively by F t , S r (t) and A r (t). Moreover, we set f = dF dt | t=0 , η . To formulate the variational problem we need to determine the derivative of S r (t) with respect to t ( cf [START_REF] Reilly | Variational properties of functions of the mean curvatures for hypersurfaces in space forms[END_REF])

d dt t=0 M S r+1 (t)dv F * t h = M (c(m -r)S r -(r + 2)S r+2 )f dv g
and an easy calculation shows that

A r (0) = M (-(r + 1)S r+1 + κ)f dv g ( 18
)
where κ is a constant. On the other hand, the balance volume is the function

V :] -ε, ε[→ IR defined by V (t) = [0,t[×M F * dv h
for which, we have

d dt t=0 V (t) = M f dv g (19)
The isometric immersion φ is a critical point of the functional A r , with constant balance volume (i.e. A r (0) = 0 for all variations such that V (t) ≡ 0) if and only if for all variations, we have

A r (0) + λV (0) = 0
where λ is a Lagrange's multiplier. It follows from ( 18) and ( 19) that for all variations and for a constant κ M (-H r+1 + κ)f dv g = 0 Thus, M is a constant r + 1-th mean curvature hypersurface if and only if, φ is a critical point of A r , with constant balance volume and in this case

A r (0) = M (L r (f ) + qf ) f dv g (20)
where we put q = k(r + 1)H r+2 -m k(r) r+1 H 1 H r+1 -ck(r)H r . We give now a definition for the stability of hypersurfaces with constant r-th mean curvature H r+1 following [START_REF] Alencar | On the first eigenvalue of the linearized operator of the r-th mean curvature of a hypersurface[END_REF] and [START_REF] Barbosa | Stability of hypersurfaces with constant r-mean curvature[END_REF]. Definition 3.1 Let (M m , g) be an orientable compacte hypersurface of (N m+1 (c), h) with H r+1 constant. Then (M m , g) is H r+1 -stable if A r (0) ≥ 0 for all variations such that V (t) = 0.

From theorem 1.2, we deduce the following theorem Theorem 3.1 Let (M m , g) be an orientable compact riemannian manifold of dimension m ≥ 2 and φ an isometric immersion of (M m , g) in IH m+1 . Then if (M m , g) is a nonegative constant, M is H r+1 -stable if and only if φ(M ) is a geodesic sphere. Remark 3.1: Note that Alencar, do Carmo and Rosenberg have proved this stability result for hypersurfaces of IR m+1 ([4] and [START_REF] Alencar | Erratum to On the first eigenvalue of the linearized operator of the r-th mean curvature of ahypersurface[END_REF]). Barbosa and Colares extend it to hypersurfaces of IH m+1 and of an open hemisphere of I S m+1 , but without using estimates of the eigenvalues of the second variation operator ( [START_REF] Barbosa | Stability of hypersurfaces with constant r-mean curvature[END_REF]). In [START_REF] Grosjean | Majoration de la première valeur propre de certains opérateurs elliptiques naturels sur les hypersurfaces des espaces formes et applications[END_REF] and [START_REF] Grosjean | A Reilly inequality for some natural elliptic operators on hypersurfaces[END_REF], we proved independently a stability result for convex hypersurfaces of IH m+1 and I S m+1 , by using an upper bound of the second eigenvalue of the second variation operator (of A r ).

Proof of theorem 3.1: A straightforward computation shows that geodesic spheres are H r+1 -stable. In fact such spheres are totally umbilical. This implies that H r = H r 1 and

L r = m -1 r -1 H r 1 ∆.
Variations (F t ) t for which V (t) ≡ 0 are variations such that M f dv g = 0 ( [START_REF] Barbosa | Stability of hypersurfaces with constant r-mean curvature[END_REF]). For such variations we have from [START_REF] Sakai | Riemannian geometry[END_REF]:

A r (0) = H r 1 M f ∆f -m(H 2 1 + c)f 2 dv g ≥ H r 1 M λ 1 -m(H 2 1 + c) f 2 dv g = 0
where λ 1 denotes the first nonzero eigenvalue of the Laplacian. This proves the stability of the geodesic spheres. Conversely, suppose that φ is H r+1 -stable. This implies that A r (0) ≥ 0 for all variations (F t ) such that V (t) ≡ 0, and from (20), we have M (L r + q)(f )f dv g ≥ 0 for any function f such that M f dv g = 0. Hence, by the min-max principle, we deduce that λ 2 (L r + q) ≥ 0 and from the inequality [START_REF] Reilly | On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space[END_REF]

  of the remark 2.1, we have λ 2 (L r + q) H 1 H r+1 s 2 δ dv g now, using the fact that H r+2 ≤ H 1 H r+1 , with equality at umbilical points ([4]), we obtain k(r)H r+1 H 1 + k(r + 1)H r+2 -m

	M	s 2 δ dv g ≤	M	k(r)H r+1 H 1 + k(r + 1)H r+2 -m	k(r) r + 1	H 1 H r+1 s 2 δ dv g
	and consequently
	0 ≤	M	k(r)H r+1 H 1 + k(r + 1)H r+2 -m	k(r) r + 1
							k(r) r + 1	H 1 H r+1 ≤
					k(r) + k(r + 1) -m	k(r) r + 1	H 1 H r+1
	and it is easy to verify that
					k(r) + k(r + 1) -m	k(r) r + 1	= 0
	thus finally, we get

M

(H r+2 -H r+1 H 1 )s 2 δ dv g = 0 hence M is totally umbilical and then it is a geodesic sphere. 2

Acknowledgements

The author would like to express his gratefulness to his advisor Professor S.Ilias.