Pierre Bérard

Ricardo Sa Earp

Minimal hypersurfaces in H n × R, total curvature and index

Keywords: 53C42, 58C40 Minimal hypersurfaces, stability, index

In this paper, we consider minimal hypersurfaces in the product space H n × R. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the corresponding curvature goes to zero uniformly at infinity. We show that surfaces with finite total intrinsic curvature have finite index. The converse statement is not true as shown by our examples which also serve as useful barriers.

Introduction

In this paper, we focus on complete oriented minimal hypersurfaces M immersed in H n × R equiped with the product metric.

In Section 3, we study the family {C a , a > 0} of hypersurfaces invariant under rotations about the vertical geodesic {0} × R ⊂ H n × R ("catenoids") and the family {M d , d > 0} of hypersurfaces invariant under hyperbolic translations. These examples generalize to higher dimensions some of the minimal surfaces constructed in [START_REF] Sa | Screw motion surfaces in H 2 × R and S 2 × R. Illinois[END_REF][START_REF] Nelli | Uniqueness of H-surfaces in H 2 × R, |H| ≤ 1/2, with boundary one or two parallel horizontal circles[END_REF][START_REF] Sa | Parabolic and hyperbolic screw motion surfaces in H 2 × R[END_REF].

In particular, we prove that the n-dimensional catenoids C a have vertical heights bounded from above by π/(n -1) (Proposition 3.2). In Section 3.3, we describe the maximal stable rotationally invariant domains on C a and we prove that the catenoids have index 1 (Theorem 3.5). We also give an interpretation in terms of the envelope of the family C a (Corollary 3.7). Finally, we observe that the half-catenoid C a ∩ H n × R + is not maximally stable.

We describe the minimal hypersurfaces invariant under hyperbolic translations in Theorem 3.8. In particular, we find a hypersurface M 1 which is a complete non-entire vertical graph over a half-space bounded by some hyperplane Π in H n × {0}. It takes infinite value data on Π and zero asymptotic boundary value data. When d < 1, the hypersurface M d is an entire vertical graph. When d > 1, it is a bi-graph over the exterior of an equidistant hypersurface of H n × {0}.

In Section 4, we consider the relationships between finiteness of the total curvature and finiteness of the index. In dimension 2, we consider the curvature integrals M |A M | 2 and M |K M |, where A M is the second fundamental form of the immersion and K M the Gauss curvature. Finiteness of these integrals implies that the corresponding curvatures tend to zero uniformly at infinity; finiteness of the latter implies finiteness of the index of the Jacobi (stability) operator (Theorem 4.1). The converse statements do not hold. On the one hand, the catenoids C a have finite index although they have infinite total intrinsic curvature. This is in contrast with the case of minimal surfaces in Euclidean 3-space ( [START_REF] Fischer-Colbrie | On complete minimal surfaces with finite Morse index in three-manifolds[END_REF]) and with the case of surfaces with constant mean curvature 1 in hyperbolic 3-space ( [START_REF] Manfredo | Index and total curvature of surfaces with constant mean curvature[END_REF][START_REF] Lopes | On the index of constant mean curvature 1 surfaces in hyperbolic space[END_REF]). Note that catenoids have finite total extrinsic curvature. On the other hand, the surfaces invariant under hyperbolic translations are stable graphs, their curvature goes to zero at infinity although they have infinite total curvature. The proof we give of Theorem 4.1 relies mainly on Simons' equation and the de Giorgi-Moser-Nash method which shows that finite total curvature implies that the curvature tends to zero uniformly at infinity. We point out that the finiteness of the intrinsic total curvature has deep consequences. Under this assumption on M , L. Hauswirth and H. Rosenberg ( [START_REF] Hauswirth | Minimal surfaces of finite total curvature in H × R[END_REF], Theorem 3.1) have indeed shown that the total intrinsic curvature is quantified, that the ends of M are asymptotic to Scherk type surfaces and obtained a C 2 -control on the curvature at infinity. In dimension n ≥ 3, we give an upper bound of the index in terms of the total extrinsic curvature (Theorem 4.3).

In Section 5, using the catenoids C a as barriers, we prove some symmetry and characterization results for minimal hypersurfaces in H n × R whose boundary consists of two congruent convex hypersurfaces in parallel slices (Theorem 5.1). We point out that the hypersurfaces M d (d < 1 and d = 1) have been used in [START_REF] Sa | An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H 2 × R[END_REF][START_REF] Sa | Minimal graphs in H n × R and R n+1[END_REF] as barriers for the Dirichlet problem and that they play a crucial role for some existence theorem for the vertical minimal surface equation.

Finally, we point out that most of our results may be established if the ambient space is one of the product spaces

H n × R k or H n × H k .
The authors would like to thank the Mathematics Department of PUC-Rio (PB) and the Institut Fourier -Université Joseph Fourier (RSA) for their hospitality. They gratefully acknowledge the financial support of CNPq, FAPERJ, Université Joseph Fourier and Région Rhône-Alpes.

2 General framework

Notations

We consider hypersurfaces M immersed in the space M := H n × R equiped with the product metric g = g B + dt 2 , where g B is the hyperbolic metric, (2.1)

g B := 2 1 -|x| 2 2 dx 2 1 + • • • + dx 2 n .
We have chosen the ball model B for the n-dimensional hyperbolic space H n .

Jacobi operator, Index, Jacobi fields

Let M n M n+1 be an orientable minimal hypersurface in an oriented Riemannian manifold M with metric g. Let N M be a unit normal field along M and let A M be the second fundamental form of the immersion with respect to N M . Let Ric be the normalized Ricci curvature of M . The second variation of the volume functional gives rise to the Jacobi operator (or stability operator) J M of M (see [START_REF] Simons | Minimal varieties in riemannian manifolds[END_REF][START_REF] Blaine Lawson | Lectures on minimal submanifolds[END_REF][START_REF] Colding | Estimates for parametric elliptic integrands[END_REF]), (2.2)

J M := -∆ M -|A M | 2 + Ric(N M ) ,
where ∆ M is the (non-positive) Laplacian on M (for the induced metric).

Given a relatively compact regular domain Ω on the hypersurface M , we let Ind(Ω) denote the number of negative eigenvalues of J M for the Dirichlet problem on Ω (this is well defined because Ω is compact). The index of M is defined to be the supremum (≤ +∞)

(2.3) Ind(M ) := sup{Ind(Ω) | Ω ⋐ M },
taken over all relatively compact regular domains.

Let λ 1 (Ω) be the least eigenvalue of the operator J M with Dirichlet boundary conditions in Ω.

Recall that a relatively compact regular domain Ω is said to be stable, if λ 1 (Ω) > 0; unstable, if λ 1 (Ω) < 0; stable-unstable, if λ 1 (Ω) = 0. More generally, we say that a domain Ω is stable if any relatively compact subdomain is stable.

Properties 2.1 Recall the following properties.

1.

Let Ω be a stable-unstable relatively compact domain. Then, any smaller domain is stable while any larger domain is unstable (monotonicity of Dirichlet eigenvalues).

2. Of particular interest are the solutions of the equation J M (u) = 0. We call such functions Jacobi fields on M . Let X a : M n ( M n+1 , g) be a one-parameter family of oriented minimal immersions, with variation field V a = ∂Xa ∂a and unit normal N a . Then, the function g(V a , N a ) is a Jacobi field on M ([1], Theorem 2.7).

3.

Let Ω be a relatively compact domain on a minimal manifold M . If there exists a positive function u on Ω such that J M (u) ≥ 0, then Ω is stable ([13], Theorem 1).

3 Examples of minimal hypersurfaces in H n × R

In this section we give examples of minimal hypersurfaces in H n × R. We use these examples as guidelines and counter-examples to study the relationships between index properties of the Jacobi operator and the finiteness of some total curvature of M , see Theorems 4.1 and 4.3. We also use them as barriers for a symmetry and characterization result in Section 5.

Rotation hypersurfaces in

H n × R
We first consider rotation hypersurfaces about a vertical geodesic axis in H n × R. Up to isometry, we can assume the rotation axis to be {0} × R. Recall that we take the ball model for H n .

Take the vertical plane

V := {(x 1 , . . . , x n , t) ∈ M | x 1 = • • • = x n-1 = 0}
and consider a generating curve tanh(f (t)/2), t for some positive function f which represents the hyperbolic distance to the axis R, at height t.

We define a rotation hypersurface M M by the "parametrization" (3.4)

X : R + × S n-1 → M , (t, ξ) → tanh(f (t)/2)ξ, t ,
where ξ = (ξ 1 , . . . , ξ n ) is a point in the unit sphere S n-1 and tanh(ρ/2)ξ stands for the point (tanh(ρ/2)ξ 1 , . . . , tanh(ρ/2)ξ n ) in the ball B.

The basic tangent vectors to the immersion X are

T (t, ξ) := T t,ξ X(∂ t ) = f t (t) 2 cosh 2 (f (t)/2) ξ, 1 ,
where f t is the derivative of f with respect to t, and

U (t, ξ, u) := T t,ξ X(u) = tanh(f (t)/2)u, 0 ,
where u ∈ T ξ S n-1 is a unit vector.

We collect basic formulas in the next proposition whose proof is straightforward.

Proposition 3.1 Let (M, g M ) ( M , g) be an isometric immersion. We have the following formulas in the parametrization X on R × S n-1 .

1. The induced metric g M is given by

(3.5) g M = 1 + f 2 t (t) dt 2 + sinh 2 (f (t))g S ,
where g S is the canonical metric on S n-1 .

2. The Riemannian measure dµ M for the metric g M is given by

(3.6) dµ M = 1 + f 2 t (t) 1/2 sinh n-1 (f (t)) dt dµ S ,
where dµ S is the canonical measure on the sphere.

4

3. The unit normal field to the immersion can be chosen to be

(3.7) N M (t, ξ) = (1 + f 2 t (t)) -1/2 -1 2 cosh 2 (f (t)/2) ξ, f t (t) .
In particular, the vertical component of the unit normal field is given by

(3.8) v M (t) := f t (t) 1 + f 2 t (t) -1/2 .
At the point X(t, ξ), the principal directions of curvature of M are

• the tangent to the meridian curve in the vertical 2-plane

V ξ = { tanh(ρ/2)ξ, t | (ρ, t) ∈ R 2 },
• the vectors tangent to the distance sphere X(t, S n-1 ) at ξ in the hyperbolic slice H n × {t}, where the restriction of the second fundamental form A M is a scalar multiple of the identity.

The principal curvatures with respect to N M are

• k n (t), the principal curvature in the direction tangent to the meridian curve, given by (3.9)

k n (t) = -f tt (t)(1 + f 2 t (t)) -3/2 ,
• the principal curvatures in the directions tangent to X(t, S n-1 ) at X(t, ξ),

(3.10) k 1 (t) = • • • = k n-1 (t) = coth(f (t))(1 + f 2 t (t)) -1/2 .
We conclude that the mean curvature H(t) of the rotation hypersurface M M with respect to the unit normal N M is given by (3.11)

nH(t) = -f tt (t)(1 + f 2 t (t)) -3/2 + (n -1) coth(f (t))(1 + f 2 t (t)) -1/2 , or (3.12) nf t (t) sinh n-1 (f (t)) H(t) = ∂ t sinh n-1 (f (t))(1 + f 2 t (t)) -1/2 .

Catenoids in H n × R

In this Section, we describe the minimal rotation hypersurfaces about {0}×R, in H n ×R. By analogy with the Euclidean case, we call them catenoids. They are the higher dimensional counterparts of the catenoids constructed in [START_REF] Sa | Screw motion surfaces in H 2 × R and S 2 × R. Illinois[END_REF].

Given some a > 0, let I a , f (a, •) denote the maximal solution of the Cauchy problem

(3.13)    f tt = (n -1) coth(f )(1 + f 2 t ), f (0) = a > 0, f t (0) = 0,
where f t and f tt are the first and second derivatives of f with respect to t. Proposition 3.2 For a > 0, the maximal solution I a , f (a, •) gives rise to the generating curve C a , t → tanh(f (a, t)), t ( catenary), of a complete minimal rotation hypersurface C a ( catenoid) in H n × R, with the following properties.

1. The interval I a is of the form I a =] -T (a), T (a)[ for some finite positive number T (a) and f (a, •) is an even function of the second variable.

2. For all t ∈ I a , f (a, t) ≥ a.

(3.14) λ(a, ρ) = sinh n-1 (a) ρ a sinh 2n-2 (u) -sinh 2n-2 (a) -1/2 du.
5. The catenoid C a has finite vertical height h R (a),

(3.15) h R (a) = 2 sinh n-1 (a) ∞ a sinh 2n-2 (u) -sinh 2n-2 (a) -1/2 du.
6. The function a → h R (a) increases from 0 to π (n-1) when a increases from 0 to infinity. Furthermore, given a = b, the generating catenaries C a and C b intersect at exactly two symmetric points.

Proof. Assertion 1 follows from the Cauchy-Lipschitz theorem for some positive T (a) which is finite as we will see below.

Assertion 2 follows from the fact that sinh n-1 (f (a, t))

1 + f 2 t (a, t) -1/2 = sinh n-1 (a) for all t ∈
] -T (a), T (a)[ (see (3.12)).

Assertion 3 is clear.

Assertion 4. According to Assertion 3, t → f (a, t) is increasing so that it has a limit when t tends to T (a) and this limit must be infinite because we took a maximal solution. It follows that the inverse function λ(a, •) maps [a, ∞[ onto [0, T (a)[ and that λ ρ (a, f (a, t))f t (a, t) ≡ 1. Finally, we find that λ ρ (a, ρ) = sinh n-1 (a) sinh 2n-2 (ρ)sinh 2n-2 (a) -1/2 on ]a, ∞[ and the formula for λ(a, ρ)

follows because f (a, 0) = a. Note that the integral (3.14) converges at u = a.

Assertion 5. We have that h R (a) = 2T (a), where

T (a) = lim ρ→∞ λ(a, ρ) = sinh n-1 (a) ∞ a sinh 2n-2 (u) -sinh 2n-2 (a) -1/2 du,
where the integral converges at both a and ∞.

Assertion 6. By a change of variables, we can write

T (a) = sinh(a) ∞ 1 v 2n-2 -1 -1/2 sinh 2 (a)v 2 + 1 -1/2 dv
and compute the derivative

T ′ (a) = cosh(a) ∞ 1 v 2n-2 -1 -1/2 sinh 2 (a)v 2 + 1 -3/2 dv > 0. Note that sinh(a) v 2n-2 -1 -1/2 sinh 2 (a)v 2 + 1 -1/2 ≤ v -1 v 2n-2 -1 -1/2
and that the right-hand side is in L 1 ([1, ∞[) for n ≥ 2, so that we can take the limits under the integral and obtain that lim a→0 T (a) = 0 and lim a→∞ T

(a) = ∞ 1 v -1 v 2n-2 -1 -1/2 dv. The last integral can be calculated explicitly because arctan √ v N -1 ′ = N 2 √ v N -1
. The last assertion follows by considering the function λ(a, ρ) -λ(b, ρ) and by using the monotonicity of T (a).

2

Remark.

The above proposition shows that the catenoids in H n × R have uniformly bounded finite vertical height. This is in contrast with the Euclidean catenoids (n ≥ 3) which have finite, yet unbounded, vertical heights. Notations. Let N a denote the unit normal to the catenoid C a , let A a denote its second fundamental form relative to the normal N a and let dµ a denote its Riemannian measure. When n = 2, let K a denote the Gauss curvature of C a . We state the following proposition for later purposes. Proof. We can restrict to the upper half-catenoid, C a,+ = C a ∩ (H n × R + ), which admits the parametrization Y (a, ρ, ξ) := tanh(ρ/2)ξ, λ(a, ρ) , ρ ≥ a.

Proposition 3.3 For a > 0, the n-dimensional catenoid C a in H n × R
The geometric data of C a,+ are readily calculated. In particular,

|A a | 2 (ρ) = n(n -1) sinh n-1 (a) cosh(ρ) sinh n (ρ) 2 ,
and

dµ a = sinh 2n-2 (ρ) sinh 2n-2 (ρ) -sinh 2n-2 (a) -1/2 dρ dµ S .
The first assertion follows (|A a | n dµ a tends to zero exponentially at infinity). For the second assertion, we use Gauss equation and minimality to get that

K a = K a - 1 2 |A a | 2 = -v 2 a - 1 2 |A a | 2 ,
where K a is the sectional curvature of the 2-plane tangent to C a in the ambient space H 2 × R and where v a is the vertical component of the unit normal to C a ,

v a (ρ) = g(N a , ∂ t ) = sinh 1-n (ρ) sinh 2n-2 (ρ) -sinh 2n-2 (a) 1/2 .
Assertion 2 follows because v a tends to 1 at infinity on C a,+ . 2

Catenoids in H n × R, stability properties

Recall that the catenoid C a is generated by the curve t → tanh(f (a, t)/2), t in the vertical plane V, where f (a, •) is the maximal solution of the Cauchy problem (3.13). This yields the parametrization

(3.16) X(a, t, ξ) = tanh(f (a, t)/2)ξ, t
for C a , with t ∈ R and ξ ∈ S n-1 . According to Property 2.1 (2), we have two Jacobi fields on the catenoid C a .

• The vertical Jacobi field v(a, t) comes from the vertical translations (x, t)

→ (x, t + τ ) in H n × R. It is given by v(a, t) = g(N a , ∂ t )
, where N a is the unit normal to C a . According to (3.7), it is given by the formula

(3.17) v(a, t) = f t (a, t) 1 + f 2 t (a, t) -1/2 ,
where f t stands for the derivative with respect to the variable t. Because t → f (a, t) is even, the function t → v(a, t) is odd.

• The variation Jacobi field e(a, t) comes from the variations with respect to the parameter a. It is given by e(a, t) = g(N a , ∂X ∂a ). According to (3.7) and (3.16), the function e(a, t) is given

(3.18) e(a, t) = -f a (a, t) 1 + f 2 t (a, t) -1/2 ,
where f a stands for the derivative with respect to the variable a. Because t → f (a, t) is even, the function t → e(a, t) is even.

• The Jacobi fields v(a, t) and e(a, t) have nice expressions when restricted to the upper-half

C a,+ = C a ∩ (H n × R + ) of
   v(a, t) = 1 + λ 2 ρ (a, f (a, t)) -1/2 , e(a, t) = v(a, t)λ a (a, f (a, t)).
For ρ ≥ a, define the functions v 1 (a, ρ), A 1 (a, ρ) and B 1 (a, ρ), by the following formulas

(3.20)              v 1 (a, ρ) = 1 + λ 2 ρ (a, ρ) -1 2 = sinh 2n-2 (ρ) -sinh 2n-2 (a) sinh 2n-2 (ρ) 1 2 A 1 (a, ρ) = cosh(a) cosh(ρ) sinh(a) sinh(ρ) n-2 , B 1 (a, ρ) = cosh(a) sinh(ρ) sinh(a) 1 (v 2n-2 -1) -1 2 (sinh 2 (a)v 2 + 1) -3 2 dv.
From (3.14), we can write

(3.21) λ(a, ρ) = sinh(a) sinh(ρ) sinh(a) 1 v 2n-2 -1 -1 2 sinh 2 (a)v 2 + 1 -1 2 dv
and compute λ a , λ a (a, ρ) =cosh(a) sinh n-2 (a) tanh(ρ) sinh 2n-2 (ρ)sinh 2n-2 (a)

-1 2 + + cosh(a) sinh(ρ) sinh(a) 1 (v 2n-2 -1) -1 2 (sinh 2 (a)v 2 + 1) -3 2 dv.
We obtain,

(3.22) λ a (a, ρ)v 1 (a, ρ) = -A 1 (a, ρ) + B 1 (a, ρ)v 1 (a, ρ).
We summarize the relevant properties in the following lemma whose proof is straightforward.

Lemma 3.4 Define the functions A(a, t) and B(a, t) for t ≥ 0 by

(3.23) A(a, t) = A 1 (a, f (a, t)), B(a, t) = B 1 (a, f (a, t)),
see Formulas (3.20). Then,

(3.24) e(a, t) = -A(a, t) + B(a, t)v(a, t), for t ≥ 0.
Furthermore, for t ≥ 0,

1. A(a, t) > 0, A(a, 0) = 1 and lim t→T (a) A(a, t) = 0, 2. B(a, t) > 0, B(a, 0) = 0 and lim t→T (a) B(a, t) = C(a), where C(a) = cosh(a) ∞ 1 (v 2n-2 - 1) -1 2 (sinh 2 (a)v 2 + 1) -3 2 dv.
3. v(a, t) = v 1 (a, f (a, t)) for t > 0, so that v(a, t) > 0 for t > 0, v(a, 0) = 0 and lim t→T (a) v(a, t) = 1.

Notation. For α < β ∈ [0, T (a)], let D(α, β) denote the rotationally symmetric domain

(3.25) D a (α, β) = X(a, ]α, β[, S n-1 ).
In particular, D a (0, T (a)) is the half-vertical catenoid This is a Jacobi field on C a and furthermore w(a, α, -α) = 0, because v is odd and e is even with respect to t. Note also that w(a, α, 0) = -v(a, α) < 0. Let us prove the first assertion of the Lemma, the proof of the second assertion is similar. Assume that w(a, α, •) has at least two consecutive zeroes α 1 < α 2 in the interval ] -T (a), 0[. The domain D a (α 1 , α 2 ) would then be stable-unstable because J a (w) = 0 on D a (α 1 , α 2 ) and because w vanishes on ∂D a (α 1 , α 2 ). On the other-hand, the Jacobi field v satisfies J a (v) = 0 and v < 0 in D a (α 1 , α 2 ). By Property 2.1 (3), we have that λ 1 (D a (α 1 , α 2 )) > 0 which contradicts the fact that this domain is stable-unstable. This proves the lemma.

C a,+ = C a ∩ (H n × R + ).
In order to determine whether the function w(a, α, •) vanishes on ]0, T (a)[ or not, it is sufficient to look at the behaviour of w(a, α, t) when t tends to T (a) from below. For this purpose, we use the expression (3.24) for e(a, t) and we write w(a, α, t) = -A(a, t)v(a, α) + v(a, t) e(a, α) + B(a, t)v(a, α) .

Using Lemma 3.4, we can write

W (a, α) := lim t→T (a)
w(a, α, t) = e(a, α) + C(a)v(a, α).

If W (a, α) ≤ 0, then w(a, α, t) does not vanish on ]0, T Assertion 3. Assertion 1 shows that C a has index at least 1. In order to show that the index is at most one, we use Fourier decomposition with respect to the variable ξ and an extra stability argument.

Recall that we work in the ball model for H n . Let γ be a geodesic through 0 in H n . Up to a rotation, we may assume that γ(s) = tanh(s/2), 0,

• • • , 0 . Let H n + = {(x 1 , • • • , x n ) ∈ B | x 1 > 0} and let C a,γ+ = C a ∩ (H n + × R).
We call this set a half-horizontal catenoid.

Claim 1. A half-horizontal catenoid C a,γ+ is stable.

To prove the claim, we shall find a positive Jacobi field on C a,γ+ . Let z = x + iy denote the complex coordinate in H 2 (ball model). We consider the group of hyperbolic isometries along the geodesic γ and we extend these isometries slice-wise as isometries in H 2 × R. We then have the one-parameter group of isometries

(z; t) → e τ (1 + z) -(1 -z) e τ (1 + z) + (1 -z) ; t in H 2 × R.
The associated Killing vector-field in H 2 × R is given by K γ (z; t) = 1 2 (1 -z 2 ); 0 or, in the (x, y) coordinates, K γ (x, y; t) = 1 2 (1 -x 2 + y 2 ), -xy; 0 which can be written as

K γ (x, y; t) = 1 2 1 + x 2 + y 2 ) (1, 0; 0) -x(x, y; 0)
where (1, 0; 0) and (x, y; 0) are seen as vectors in

R 2 × R = T (x,y;t) H 2 × R.
This formula can easily be generalized to higher dimensions as

K γ (x; t) = 1 2 (1 + |x| 2 (e 1 ; 0) -x 1 (x; 0), where x = (x 1 , • • • , x n ), e 1 = (1, 0, • • • , 0), |x| 2 = x 2 1 + • • • + x 2 n
, and where (e 1 ; 0) and (x; 0) are seen as vectors in R n × R = T (x,t) H n × R. Writing the point x in the parametrization X as x = tanh(f (a, t)/2)ξ, we obtain that

K γ (tanh(f (a, t)/2)ξ; t) = 1 2 1 + tanh 2 (f /2) (e 1 ; 0) -tanh 2 (f /2)ξ 1 (ξ; 0). Using the fact that (1 + f 2 t ) -1/2 = sinh(a) sinh(f )
n-1 on C a , we find that the Killing field K γ gives rise to the horizontal Jacobi field

h γ (a, t, ξ) = sinh(a) sinh(f (a, t)) n-1 ξ 1
which is positive on C a,γ+ .

Claim 2. On S n-1 equiped with the standard Riemannian metric, there exists an orthonormal basis of spherical harmonics Y k , k ≥ 0 with the property that the nodal domains of the Y k , k ≥ 1 are contained in hemispheres.

The property is clearly true on S 1 and can be proved by induction on the dimension, using polar coordinates centered at a given point on the sphere.

Claim 3. The Jacobi operator on C a can be written as

J a = L a,t -q(a, t)∆ S,ξ ,
where L a,t is a Sturm-Liouville operator on the t variable, with coefficients depending only on a and t, where q(a, t) is a positive function and where ∆ S,ξ is the Laplacian of the sphere S n-1 acting on the ξ-variable. This claim follows immediately from Formulas (3.5) and (3.6) for the metric and the Riemannian measure on a rotation hypersurface and from the expression for the quadratic form associated with J a .

Assume that the index of C a is at least 2. Then, there exists some S ∈]0, T (a)[ such that J a has at least two negative eigenvalues λ 1 (S) < λ 2 (S) < 0 in C a (-S, S) (we only consider Dirichlet boundary conditions). Because the least eigenvalue λ 1 (S) is simple, a corresponding eigenfunction u must be invariant under rotations (i.e. only depends on the variable t) and say positive. Consider an eigenfunction v associated with λ 2 (S). We claim that v cannot be invariant under rotations. Indeed, v would otherwise depend only on the variable t, it would be orthogonal to u and hence it would have to vanish on ] -S, S[. This would contradict the fact that the domains C a (-S, 0) and C a (0, S) are stable. Since v is not rotationally invariant, there exists some p ≥ 1 and some v p = 0 in the decomposition into spherical harmonics with respect to the second variable, v(t, ξ)

= ∞ k=0 v k (t)Y k (ξ). We would have J a (v p Y p ) = λ 2 (S)v p Y p .
Using Claim 2 and the fact that λ 2 (S) < 0, this would mean that any nodal domain of v p Y p is unstable, in contradiction with Claim 1.

Assuming that the index is at least 2 therefore yields a contradiction and hence the index of C a is exactly one. 2

Remark. It follows from the positivity of the Jacobi field v(a, t) for t ∈]0, T (a)[ that the upper half-catenoid C a,+ is stable (in the sense that any relatively compact domain Ω contained in C a,+ is stable, see Section 2.2). The second assertion in the preceding theorem says more. Indeed, there exists some τ (a) ∈]0, T (a)[ such that the non-compact domain D a (-τ (a), T (a)) is stable and stricly contains C a,+ . This is different from what happens for Euclidean catenoids. Indeed, the half-catenoid C a,+ in R 3 is a maximal stable domain ( [START_REF] Lindeloef | Sur les limites entre lesquelles le caténoïde est une surface minimale[END_REF]). We study this phenomenon with more details in [START_REF] Bérard | Lindeloef's theorem for catenoids, revisited[END_REF]. touches the envelope of the family of catenaries {C a } a>0 . According to [START_REF] Valiron | Équations fonctionnelles. Applications[END_REF], §58, page 127 ff, the condition defining the envelope a family Γ a , given by the parametrization x(a, t), y(a, t) , is the condition x a (a, t) x t (a, t) y a (a, t) y t (a, t) = 0.

Specializing to catenaries, we find that the envelope condition is precisely the condition that e(a, t) = 0. Therefore, the value σ(a) is precisely the value of t at which the catenary C a touches the envelope of the family, see Figure 6.

Corollary 3.7 The stable-unstable domain D a (-σ(a), σ(a)) is precisely the symmetric, rotation invariant compact domain bounded by the two spheres where the catenoid C a touches the envelope of the family.

Translation invariant hypersurfaces in

H n × R
Let γ be a complete geodesic through 0 in the ball model B of the hyperbolic space H n , parametrized by the signed distance ρ to 0. Let P be the hyperbolic hyperplane orthogonal to γ at 0. We consider the hyperbolic translations along the geodesics passing through 0 in P. The image of a point of γ under these translations is an equidistant hypersurface to P in H n . We can extend these translations "slice-wise" to give positive isometries of H n × R which we call hyperbolic translations.

A generating curve (tanh(ρ/2), µ(ρ)) in the vertical Euclidean plane γ × R gives rise, under the previous isometries, to a translation invariant hypersurface M ֒→ H n × R, whose intersection with the slice H n × {µ(ρ)} is the equidistant hypersurface to P × {µ(ρ)} in the slice, at distance ρ.

The principal directions of curvature of M are the tangent vector to the generating curve and the directions tangent to the equidistant hypersurface. The corresponding principal curvatures are given respectively by

(3.27) k G (ρ) = μ(ρ) 1 + μ2 (ρ) -3/2 ,
and

(3.28) k E (ρ) = μ(ρ) 1 + μ2 (ρ) -1/2 tanh(ρ).
It follows that the mean curvature of M is given by

nH(ρ) = μ(ρ) 1 + μ2 (ρ) -3/2 + (n -1) μ(ρ) 1 + μ2 (ρ) -1/2 tanh(ρ)
or, equivalently, by

(3.29) nH(ρ) cosh n-1 (ρ) = ∂ ρ μ(ρ) 1 + μ2 (ρ) -1/2 cosh n-1 (ρ) .
This formula allows us to study constant mean curvature hypersurfaces invariant by hyperbolic translations. In this paper we only consider the case H = 0 and we refer to [START_REF] Bérard | H-hypersurfaces in H n × R and applications[END_REF] for the case H = 0.

Translation invariant minimal hypersurfaces in H n × R

In this section, we establish the following theorem which generalizes the 2-dimensional result of [START_REF] Sa | An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H 2 × R[END_REF].

∂ ∞ H n × R.
Proof. In the minimal case, Equation (3.29) can be written

(3.30) μ(ρ) 1 + μ2 (ρ) -1/2 cosh n-1 (ρ) = d,
for some constant d which satisfies d ≤ cosh n-1 (ρ) for all ρ for which the solution exists. Changing µ to -µ if necessary, we may assume that d is non-negative and hence, μ(ρ) ≥ 0 and μ(ρ) = d cosh 2n-2 (ρ) -d 2 -1/2 whenever the square root exists. We have to consider three cases, d > 1, d = 1 and d < 1.

d > 1 Let d =: cosh n-1 (a), with a > 0. It follows from Equation (3.30) that μ(ρ) = cosh n-1 (a) cosh 2n-2 (ρ) -cosh 2n-2 (a) -1/2 .
Up to a vertical translation, the solution µ + (a, ρ) of Equation (3.30) is given by

(3.31) µ + (a, ρ) = cosh n-1 (a) ρ a cosh 2n-2 (r) -cosh 2n-2 (a) -1/2 dr
or, making cosh(r) = cosh(a) t,

(3.32) µ + (a, ρ) = cosh(a) cosh(ρ) cosh(a) 1 (t 2n-2 -1) -1 2 (cosh 2 (a)t 2 -1) -1 2 dt.
These integrals converge at ρ = a (resp. at t = 1) and at infinity and

(3.33) h T (d) := 2 cosh(a) ∞ 1 (t 2n-2 -1) -1/2 (cosh 2 (a)t 2 -1) -1/2 dt
is the height of the hypersurface M cosh n-1 (a) . The function h T (d) is decreasing in d, tends to infinity when d tends to 1 + and to π/(n -1) when d tends to infinity. (Hints. When a tends to zero, use the fact that (3.33) is bigger than some constant times the integral

2 1 (t -1)(cosh(a)t -1) -1/2 dt
which can be computed explicitly. When a tends to infinity, use the fact that (t

N -1) -1/2 t -1 dt = 2 N arctan √ t N -1.
) The assertions on the asymptotic boundary are clear.

(ρ) = cosh 2n-2 (ρ) -1 -1/2 ,
so that, when d = 1, the solution is given by

(3.34) µ 0 (ρ) = ρ b cosh 2n-2 (r) -1 -1/2 dr,
for some constant b > 0, and µ 0 (ρ) tends to -∞ when ρ tends to zero and to a finite value when ρ tends to infinity. The corresponding hypersurface is complete. It is a vertical graph so that it is stable. The assertion on the asymptotic boundary is clear. 0 < d < 1 In this case, Equation (3.30) gives the following solution (up to a vertical translation),

(3.35) µ -(d, ρ) = d ρ 0 cosh 2n-2 (r) -d 2 -1/2 dr.
The corresponding curve can be extended by symmetry and we get a complete hypersurface in a vertical slab with finite height. This surface is an entire vertical graph (hence stable). The assertion on the asymptotic boundary is clear. 2

The generating curves for translation invariant minimal hypersurfaces are given in Fig. 7 and8. Note that they cannot meet tangentially at finite distance.

Remark. Using the catenoids and the minimal translations hypersurfaces, the second author and E. Toubiana have extended the 2-dimensional results of their paper [START_REF] Sa | An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H 2 × R[END_REF] to higher dimensions, see [START_REF] Sa | Minimal graphs in H n × R and R n+1[END_REF].

4 Index and total curvature for minimal hypersurfaces in

H n × R 4.1 Dimension two, M 2 H 2 × R
For oriented minimal surfaces in H 2 × R we have the following general theorem in which we consider two possible notions of total curvature.

Remarks.

1. For complete orientable minimal surfaces in R 3 , finiteness of the index is equivalent to finiteness of the intrinsic curvature (see [START_REF] Fischer-Colbrie | On complete minimal surfaces with finite Morse index in three-manifolds[END_REF][START_REF] Manfredo | Index and total curvature of surfaces with constant mean curvature[END_REF][START_REF] Lopes | On the index of constant mean curvature 1 surfaces in hyperbolic space[END_REF]). No such statement can hold in H 2 ×R. Indeed, the surfaces M d ( [START_REF] Sa | An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H 2 × R[END_REF] and Section 3.5) are stable complete minimal surfaces, invariant under a group of hyperbolic translations. Their total curvatures are infinite, so that the converse to Assertion 3 is false.

2.

The assumption M |K M | dµ M finite is natural in view of Huber's theorem. In [START_REF] Hauswirth | Minimal surfaces of finite total curvature in H × R[END_REF], L.

Hauswirth and H. Rosenberg show that this assumption actually implies that the total intrinsic curvature is an integer multiple of 2π. There are actually many examples of such surfaces ( [START_REF] Collin | Construction of harmonic diffeomorphisms and minimal graphs[END_REF][START_REF] Hauswirth | Minimal surfaces of finite total curvature in H × R[END_REF]).

3. As pointed out in the introduction, Assertion 2 is contained in [START_REF] Hauswirth | Minimal surfaces of finite total curvature in H × R[END_REF], Theorem 3.1 whose proof actually gives a C 2 -control on the curvature at infinity. We provide a simple proof of Assertion 2 for completeness.

One can slightly improve the above theorem with the following proposition. 1. Assume that 5v 2 M ≤ 1. Then there exists a universal constant C such that if the integral

M |A M | 2 dµ M is less than C then M is a vertical plane. 2. Assume that the integral M |A M | 2 dµ M
is finite and that there exists a compact set Ω ⊂ M and a positive constant c such that

v 2 M ≤ 1 -c < 1 on M \ Ω.
Then the Jacobi operator of the immersion has finite index.

Remarks.

1. Assertion 1 generalizes the following facts : (i) A minimal surface whose intrinsic curvature K M is zero is part of a vertical plane γ × R (where γ is a geodesic in H 2 ). Indeed, we have

K M = -1 2 |A M | 2 -v 2
M and hence M is totally geodesic with horizontal normal vector. (ii) A complete minimal surface whose total intrinsic curvature is less than 2π is a vertical plane (see [START_REF] Hauswirth | Minimal surfaces of finite total curvature in H × R[END_REF]).

-u ∆ M u ≤ u 4 + (5 K M + 1)u 2 ≤ u 4 + 4u 2 ,
where K M is the sectional curvature of the 2-plane T M in H 2 × R. This formula follows from J. Simons' equation for minimal submanifolds ( [START_REF] Simons | Minimal varieties in riemannian manifolds[END_REF]), from our context (H 2 × R is locally symmetric and we work in codimension 1), and from explicit curvature computations in H 2 × R.

Fact 2. The surface M satisfies the Sobolev inequality (4.37)

f 2 ≤ S(M ) df 1
for some positive constant S(M ) and all C 1 functions f with compact support. This follows from [START_REF] Hoffman | Sobolev and isoperimetric inequalities for riemannian submanifolds[END_REF][START_REF] Hoffman | A correction to : Sobolev and isoperimetric inequalities for riemannian submanifolds[END_REF] and the fact that H 2 × R is simply-connected and non-positively curved.

Fact 3. Curvature computations in H 2 × R give (4.38) K M = -v 2 M and Ric(N M , N M ) = -(1 -v 2 M ).
The fact that v M is a Jacobi field implies that

(4.39) -∆ M v M = v 3 M + (|A M | 2 -1)v M
and a similar inequality for |v M |.

Theorem, Assertion 1. Following the general ideas of [START_REF] Schoen | Curvature estimates for minimal hypersurfaces[END_REF], we use (4.36) and (4.37), to estimate the L p -norms of u and the classical de Giorgi-Moser-Nash method to estimate u ∞ outside big balls. The details appear in the proof of Theorem 4.1, p. 282 of [START_REF] Bérard | Complete hypersurfaces with constant mean curvature and finite total curvature[END_REF], where it is observed that the proof only uses the facts that u satisfies Simons' inequality and M a Sobolev inequality. Theorem, Assertion 2. By Gauss equation, the Gauss curvature K M of M satisfies (4.40)

K M = - 1 2 |A M | 2 -v 2 M .
The assumption implies that both integrals M |A M | 2 dµ M and M v 2 M dµ M are finite. By Assertion 1, we already know that |A M | tends to zero at infinity, and hence that it is bounded. Equation (4.39) then tells us that |v M | satisfies an elliptic inequality similar to (4.36) and we can again apply the de Giorgi-Moser-Nash method to conclude.

Theorem, Assertion 3 and Proposition, Assertion 2. According to Section 2.2 and to the above curvature calculations, the Jacobi operator can be written as

J M = -∆ M + 1 -|A M | 2 -v 2
M . We now follow [START_REF] Bérard | The index of constant mean curvature surfaces in hyperbolic 3-space[END_REF], Section 2. It follows from Assertion 2 in the Theorem that J M is bounded from below, essentially self-adjoint and that its essential spectrum lies above 1. Because the eigenvalues below the essential spectrum can only accumulate at -∞ or at the bottom of the essential spectrum, it follows that J M has finite index.

Proposition, Assertion 1. It is a classical fact ( [START_REF] Do | Stable complete minimal hypersurfaces[END_REF] and [START_REF] Schoen | Curvature estimates for minimal hypersurfaces[END_REF]) that Simons' inequality (4.36) can be improved to

|du| 2 -u ∆ M u ≤ u 4 + (5 K M + 1)u 2 .

Higher dimension, M

n H n × R, n ≥ 3
Recall the formula for the Jacobi operator,

J M := -∆ M -|A M | 2 + Ric(N M )
where N M is a unit normal field along M and A M the second fundamental form of M with respect to N M (Section 2.2).

Let v M := ĝ(N M , ∂ t ) be the vertical component of the unit normal vector N M . A simple computation gives that Ric(

N M ) = -(n -1)(1 -v 2 M
). It follows that the Jacobi operator of M is given by (4.41)

J M := -∆ M + (n -1)(1 -v 2 M ) -|A M | 2 .
We have the following theorem.

Theorem 4.3 Let M n H n × R a complete oriented minimal immersion. Assume that M has finite total curvature, i.e. M |A M | n dµ M < ∞.
1. For n ≥ 2, the second fundamental form A M tends to zero uniformly at infinity. 2. For n ≥ 3, the Jacobi operator of the immersion has finite index and, more precisely, there exists a universal constant C(n) such that

(4.42) Ind(J M ) ≤ C(n) M |A M | n dµ M .

Remarks. (i)

The examples M d prove that the converse statements in the previous theorems are not true in general, see Section 3.5.

(ii) Note that we state the second assertion of Theorem 4.3 only for dim(M ) ≥ 3 (our proof does not apply in dimension 2, see [START_REF] Bérard | Number of bound states and estimates on some geometric invariants[END_REF]).

Proof. As in the proof of Theorem 4.1, the manifold M satisfies a Sobolev inequality of the form (4.37), namely f n/(n-1) ≤ S(M ) df 1 for all f ∈ C 1 0 (M ) for some constant S(M ). Furthermore, the second fundamental form A M satisfies the following Simons' equation (compare with (4.36)),

-∆|A M | ≤ |A M | 3 + C(n)|A M |,
for some constant C(n) which only depends on the dimension (this follows from the expression of the term R(A) as given in [START_REF] Simons | Minimal varieties in riemannian manifolds[END_REF].

The de Giorgi-Moser-Nash technique applies (see [START_REF] Bérard | Complete hypersurfaces with constant mean curvature and finite total curvature[END_REF], Theorem 4.1) and it follows that |A M | tends to zero uniformly at infinity. Since |v M | ≤ 1, the operator J M is bounded from below and essentially self-adjoint. Furthermore, its index is less than or equal to the index of the operator -∆ -|A M | 2 which is also bounded from below and essentially self-adjoint. The estimate (4.42) then follows by applying Theorem 39 in [START_REF] Bérard | Number of bound states and estimates on some geometric invariants[END_REF]. 2

Remark. The preceding results can be generalized to minimal hypersurfaces in H n ×R k or H n ×H k .

Applications

In this section, we use the examples constructed in Section 3 as barriers to prove some general results on minimal hypersurfaces in H n × R. These results generalize results obtained in [START_REF] Nelli | Uniqueness of H-surfaces in H 2 × R, |H| ≤ 1/2, with boundary one or two parallel horizontal circles[END_REF] for dimension 2. Similar results hold for H-hypersurfaces as well, see [START_REF] Nelli | Uniqueness of H-surfaces in H 2 × R, |H| ≤ 1/2, with boundary one or two parallel horizontal circles[END_REF][START_REF] Bérard | H-hypersurfaces in H n × R and applications[END_REF].

+ = Γ × {a} ⊂ H n × R, for some a > 0. Assume that Γ is convex. Let M ⊂ H n × R be a compact immersed minimal hypersurface such that ∂M = Γ -∪ Γ + . Then, 2a < π n -1
(the height of the family of catenoids).

Furthermore, if M is embedded, 1. M is symmetric with respect to the slice H n × {0}.

2. The parts of M above and below the slice of symmetry are vertical graphs.

3. If Γ is a horizontal graph and symmetric with respect to a hyperbolic hyperplane P , then M is a horizontal graph and symmetric with respect to the vertical hyperplane P × R. In particular, if Γ is an (n -1)-sphere, then M is part of a catenoid.

Proof. We reason ab absurdo. Suppose that the height of M is greater than or equal to π n -1

, that is 2a ≥ π n -1

. We recall that the height of the family of n-dimensional catenoids {C ρ , ρ ∈ (0, ∞)} is bounded from above by π n -1

, but each catenoid C ρ has height strictly less than π n -1

. Now as M is compact, there is a (hyperbolic) radius ρ 0 big enough such that M is strictly contained inside the vertical cylinder M ρ0 of radius ρ 0 (where M ρ0 is a cylinder over a n -1 sphere S ρ0 ⊂ H 2 × {0} of radius ρ 0 ) containing M in its mean convex side. Recall that, by the geometry of the catenoids, the catenoid C ρ0 whose distance to the t-axis is ρ 0 is contained in the closure of the non mean convex side of M ρ0 touching M ρ0 just along the n -1 sphere S ρ0 . Hence, M is strictly contained in the connected component of H 2 × R \ C ρ0 that contains the t axis of C ρ0 . Notice that the whole family of catenoids C ρ is strictly contained in the slab of H 2 × R with boundary Γ -∪ Γ + . Starting from ρ = ρ 0 , making ρ → 0, that is moving the family of catenoids {C ρ , ρ ≤ ρ 0 } towards M , we will find a first interior point of contact with some C ρ and M , since the family of catenoid cannot touch the boundary of M . We arrive at a contradiction, by the the maximum principle. The proof of the first part of the statement is completed. Now using the family of slices H × {t} ∪ H × {-t} coming from the infinity towards M we get, by maximum principle, that M is entirely contained in the closed slab whose boundary is the slices H × {a} ∪ H × {-a} and (H × {a} ∪ H × {-a}) ∩ M = ∂M.

In the same way, considering the family of vertical hyperplanes, we get that M is contained in the mean convex side of the vertical cylinder M Γ over Γ and M Γ ∩ M = ∂M. Now using Alexandrov Reflection Principle on the slices, moving the slices from t = a towards t = 0, by vertical reflections, we get that the reflection of M + = M ∩ {t 0}, the part of M above t = 0, on the horizontal slice t = 0, is above M -= M ∩ {t 0}. Moreover, we find that M + is a vertical graph. In the same way, moving the slices from t = -a towards t = 0, doing vertical reflections, we get that the reflection of M -= M ∩ {t 0}, the part of M above t = 0, on the horizontal slice t = 0, is below M + = M ∩ {t ≥ 0}. We conclude that M -= M + , hence both M + and M -are vertical graphs and M is symmetric with respect to the slice H n × {0}. Therefore, the proof of the second part of the Statement is completed.

Let us assume now that P ⊂ H × {0} is a hyperplane of symmetry of Γ. Consider the vertical hyperplane P = P × R and the family of hyperplanes P t at signed distance t from P obtained from P by horizontal translations along an oriented geodesic γ orthogonal to P at the origin. Choosing |t| big enough, we move the family P t towards P (in the two sides of H 2 × R \ P), doing Alexandrov Reflection Principle on P t , taking into account that Γ is a horizontal graph and that the symmetric of ∂M on P t stays on the slices t = ±a, so that it does not touch the interior of M . We can argue as before to conclude that P is a hyperplane of symmetry of M . Of course, is Γ is rotationally symmetry then M is a minimal hypersurface of revolution. Henceforth, by the classification theorem, M is part of a catenoid. This completes the proof of the theorem.

2

3 . 4 .

 34 The derivative f t (a, •) is positive on ]0, T (a)[, negative on ] -T (a), 0[. The function f (a, •) is a bijection from [0, T (a)[ onto [a, ∞[, with inverse function λ(a, •) given by

Figure 1 :

 1 Figure 1: Catenaries n = 2, 4

  has infinite volume and finite total extrinsic curvature Ca |A a | n dµ a . When n = 2, the catenoid C a has infinite total intrinsic curvature Ca |K a | dµ a .

Theorem 3 . 5

 35 The stability properties of the rotationally symmetric domains D a (α, β) on the catenoid C a are as follows. 1. There exists some σ(a) ∈]0, T (a)[ such that the relatively compact domain D a (-σ(a), σ(a)) is stable-unstable. Hence, for any α ∈]0, σ(a)[, the domain D a (-α, α) is stable; for any α ∈]σ(a), T (a)[, the domain D a (-α, α) is unstable. 2. There exists some τ (a) ∈]0, T (a)[ such that (a) the (non relatively compact) domain D a (-τ (a), T (a)) is stable, (b) for any α ∈]τ (a), T (a)[, there exists some β(α) ∈]τ (a), T (a)[ such that the domain D a (-α, β(α)) is stable-unstable. 3. The catenoid C a has index 1.

Figure 2 : 1 Figure 3 :Assertion 2 .

 2132 Figure 2: Case 1

Lemma 3 . 6

 36 The function w(a, α, •) vanishes only once on ] -T (a), 0[ and vanishes at most once on ]0, T (a)[.

  a)[ and in fact on ]-α, T (a)[; if W (a, α) > 0, then w(a, α, t) has one and only one zero β(α)on ]0, T (a)[. We now observe that W (a, t) := e(a, t) + C(a)v(a, t) is a Jacobi field on ]0, T (a)[ which take the value -1 at 0 and the value C(a)v(a, σ(a)) > 0 at σ(a). It follows from Lemma 3.6 that W (a, •) has one and only one positive zero τ (a) ∈]0, σ(a)[. We have that W (a, t) ≤ 0 on ]0, τ (a)[, so that for any α ∈]0, τ (a)], the function w(a, α, t) has only one zero -α on ] -T (a), T (a)[. This proves the Assertion 2(a). On the other-hand, W (a, t) > 0 on ]τ (a), T (a)[, so that for any α ∈]τ (a), T (a)[, the function w(a, α, t) has a unique positive zero β(α) ∈]0, T (a)[. This proves the Assertion 2(b).

Figure 5 : Envelope, n = 2 aFigure 6 :

 526 Figure 5: Envelope, n = 2

Figure 7 :Figure 8 :

 78 Figure 7: Generatrices translation invariant hypersurfaces, n = 2

Theorem 4 . 1

 41 Let M H 2 × R be a complete oriented minimal immersion with unit normal field N M . Let v M := ĝ(N M , ∂ t ) be the vertical component of N M , let A M be the second fundamental form of M and let K M be the intrinsic curvature of M . 1. If the integral M |A M | 2 dµ M is finite, then A M tends to zero uniformly at infinity. 2. If the integral M |K M | dµ M is finite, then A M , v M and K M tend to zero uniformly at infinity. 3. If the integral M |K M | dµ M is finite, then the Jacobi operator of M has finite index.

Proposition 4 . 2

 42 The notations are the same as in Theorem 4.1.

2 .Fact 1 .

 21 We do not know whether the sole assumption M |A M | 2 dµ M finite is sufficient to insure the finiteness of the index of the Jacobi operator of M . Proof of Theorem 4.1 and Proposition 4.2. The function u := |A M | satisfies the non-linear elliptic inequality (4.36)

Theorem 5 . 1

 51 Let Γ ⊂ H n be a compact embedded hypersurface and consider two copies of Γ in different slices, Γ -= Γ × {-a} and Γ

  Theorem 3.8 There exists a 1-parameter family {M d , d > 0} of complete embedded minimal hypersurfaces in H n × R invariant under hyperbolic translations. The hypersurfaces are M d stable (in the sense of the Jacobi operator), their principal curvatures go uniformly to zero at infinity, but they have infinite total curvatures. More precisely, 1. If d > 1, the hypersurface M d consists of the union of two symmetric vertical graphs over the exterior of an equidistant hypersurface in the slice H n × {0}. It is also a horizontal graph, and hence stable. The family M d has finite vertical height h T (d), a function which decreases from infinity to π/(n -1). In particular, it is bounded from below by π/(n -1), the upper bound of the heights of the family of catenoids. Furthermore, the asymptotic boundary of M d consists of the union of two copies of an hemisphere S n-1 + × {0} of ∂ ∞ H n × {0} in parallel slices t = ±S(d), glued with the finite cylinder ∂S n-1 If d = 1, the hypersurface M 1 is a complete stable vertical graph over a half-space in H n ×{0}, bounded by a totally geodesic hyperplane P . It takes infinite boundary value data on P and constant asymptotic boundary value data. Furthermore, the asymptotic boundary of M 1 is the union of a spherical cap S in ∂ ∞ H n × {c} with a half-vertical cylinder over ∂S.

	+	× [-S(d), S(d)].
	2.

3. If d < 1, the hypersurface M d is an entire stable vertical graph with finite vertical height.

Furthermore, its asymptotic boundary consists of a homologically non-trivial (n -1)-sphere in


Ahmad Delbani

Armine Karami

Dimitri Galayko

Naida Hodzic

Srikumar Vaidyanathan

Malal Kane

Philippe Basset

TRIBOELECTRIC-POWERED SYSTEM FOR PEDESTRIAN AND VEHICLE DETECTION WITH WIRELESS DATA TRANSMISSION

Keywords: Kinetic energy harvesters, TENG, 2-stage power management system, plasma switch, road/pavement sensor

This paper report an autonomous and wireless system for detecting the passage of a human (pedestrian) or a vehicle based on a basic and low-cost triboelectric sensor/generator made of a PTFE sheet and aluminum plates' transducer. A high-voltage conditioning circuit using a plasma switch and a 2-stage power management system is supplying a BLE module comprising 5 sensors. These sensors are being supplied solely from the harvested energy generated by the triboelectric transducer (TENG) after an initial pre-charge. In preliminary laboratory tests, up to 200 µJ where generated by a single human step, and three steps are sufficient to send the sensor data. Furthermore, we have demonstrated the data transmission/detection of each human step without any additional electronics, but only with the DC rectifier and the plasma switch using the principle of the Hertz experiment.

INTRODUCTION

We are living in an era where there is a huge demand for new power sources due to the non-sustainable source's exhaustion, accompanied with the fast and vast production of small electronics in its various kinds. Throughout the years, the roads kept evolving and adapting with the technology. The roads of tomorrow will be intelligent, automated and will communicate sensor data with the vehicles and base stations. These sensors will regularly measure and transmit various information (quantity, type, weight of vehicles, temperature, humidity, etc.…) [START_REF] Pompigna | Smart roads: a state of the art of highways innovations in the Smart age[END_REF]. A regular power supply for these sensors, which are abandoned, is necessary. Considering the traffic density in a metropolitan center, it is conceivable that a large amount of mechanical energy is spent every day and wasted. The challenge consists of recovering a part of this energy and use it to help these future roads in the communication of their sensors with their environment.

A triboelectric energy generator (TENG) is a transducer that converts mechanical energy to electricity, based on contact electrification and electrostatic induction principles that could be incorporated into the roads. To power the electronic device or sensor, a regulated DC voltage should be provided at the output. A power management system (PMS) is needed to optimize the mechanical-to-electrical energy transduction and to perform the AC-DC conversion to a low DC voltage around a few volts compatible transmission module. We have proposed two approaches for the data transmission of the pedestrian/vehicle detection. The first one is based on a BLE module connected at the output of the PMS. The other approach follows the principle of the famous Hertz experiment [START_REF] Wang | A paradigm shift fully self-powered long-distance wireless sensing solution enabled by discharge-induced displacement current[END_REF] where an electrostatic sparkle, generated when a plasma switch is activated, induces an electromagnetic wave that can be detected several meters away with a simple coil antenna.

DESCRIPTION OF THE TRANSDUCER Electrical model

A TENG is a capacitive device, which at least one electrode is covered by a dielectric material that has the tendency to trap a specific type of charges. This layer is inserted between the device's electrodes, and it acts as an electret. Charges are spread on its surface by mechanical contact. A simple model is shown in Figure 1. The charges are concentrated at the surface of a tribo-electret layer of thickness ddie and permittivity εdie, with an effective surface charge density distribution σ, over a surface S. The variable air gap is Xmax with a permittivity ε0. After a pre-charge by contact electrification, the variation of the charged capacitor due to external mechanical forces will generate electrical energy by electrostatic induction. The variable capacitance, which varies between Cmax and Cmin, is defined as:

             2    2 * cos2 (1)
The build-in triboelectric voltage VTE is the defined as:

    *     *   (2)
.

Transducer's Mechanical Design

A structure has been designed to handle the pressure of a passing vehicle (Figure 2) from which the mechanical energy is converted into electricity. The structure has the size of a car's tire and has the ability to retract to its zero position thanks to metallic springs. Two road bumps with a movable upper vertical plate have been fabricated. The transducer has a surface of up to 625 cm² and can be excited with a vertical deflection from a few mm to a few cm. The transducer is made of two aluminum electrodes, one of them being covered with a PTFE sheet of 25x23 cm² wide and 100 µm thick glued with double-sided adhesive tape (see Figure 3). A foam layer is added between the electrodes and the supporting frame for improving the transducer's reliability.

THE POWER MANAGEMENT SYSTEM Signal rectification

The full-wave (FW) diode bridge is the most used conditioning circuit for rectifying the AC signal at a TENG output. As a stable charge pump, it reaches a saturation. The saturation voltage of the half-wave bridge (HW) is higher by a factor (η+1), where η is the Cmax/Cmin capacitance ratio of the TENG. During the first cycles, the harvested energy of the FW rectifiers increases with 4 times the slope of the HW, in spite of the fact that in that case both harvested energies are far from the optimal of each circuit [START_REF] Zhang | General analysis and optimization of a two-stage power management circuit for electrostatic/triboelectric nanogenerators[END_REF]. Moreover, if the output voltage of the rectifier can be set to be half of its saturation voltage, HW surpasses the FW by a factor (η+1)/2 in terms of performance [START_REF] Ghaffarinejad | Superior performance of halfwave to full-wave rectifier as a power conditioning circuit for triboelectric nanogenerators: application to contact-separation and sliding mode TENG[END_REF].

Unstable charge pumps such as the Bennet doubler conditioning circuit have the ability to increase exponentially the charge on the TENG's electrodes during operation and to increase its bias and conversion efficiency. A minimum value of η should be satisfied, which is typically 2 for the most simple architecture, but it can vary depending on the circuit configuration [START_REF] Karami | Series-parallel charge pump conditioning circuits for electrostatic kinetic energy harvesting[END_REF]. While the output energy can be greatly improved by the unstable charge pumps, their main disadvantage is that it may take some time for reaching the conditions for a high-energy conversion [START_REF] Ghaffarinejad | Superior performance of halfwave to full-wave rectifier as a power conditioning circuit for triboelectric nanogenerators: application to contact-separation and sliding mode TENG[END_REF]. This period is mostly depending on the value of η and the ratio between the load capacitance and Cmin [START_REF] Zhang | General analysis and optimization of a two-stage power management circuit for electrostatic/triboelectric nanogenerators[END_REF]. In our application, we intend to reach quickly a very high voltage to actuate a high-voltage (HV) plasma switch and increase the energy conversion per mechanical actuation, so the HW rectifier is the better choice.

2-stage Power Management System (PMS)

A PMS should be introduced after the rectification process to adapt the high-voltage rectified signal to a compatible (low) DC voltage for the load. A practical solution is to use a 2-stage architecture to maintain a high voltage across the TENG while a DC-DC Buck downconverter lower the voltage across a large storage capacitor Cstore (see Figure 4). One of the main component in our PMS is the plasma switch, which is a fully autonomous switch that does not need any additional electronics [START_REF] Zhang | Employing a MEMS plasma switch for conditioning high-voltage kinetic energy harvesters[END_REF]. Its actuation voltage is defined by the Paschen law. When the potential across the switch terminals, which are separated by a micro-gap, reaches a certain value in the range of hundreds of volts, an electrical sparkle is formed between them and thus conduction will occur.

EXPERIMENTS

Figure 5 shows the conditioning circuit implementation. The evolution of the charges flowing through Crect is measured by an electrometer while the TENG is mechanically excited by a pedestrian. Then the energy stored inside Crect can be calculated accordingly using the formula:

!"#  $ !"# % 2 !"# (3)
Figure 6 shows the measured voltage across Crect. After one pedestrian excitation, it reaches 633 V, corresponding to an energy of 200 µJ. Choosing a value of Crect in the same order of magnitude of Cmax helps to have a high voltage across Crect. Thus, the plasma switch can actuate with a minimal amount of pedestrian steps. The TENG dynamic capacitance time variation CTENG has been measured using the technique depicted in [8], showing Cmin = 330 pF and Cmax in the range of 1800-2100 pF.

2-Stage PMS and BLE communication

The 2-stage PMS comprises a half-wave rectifier, a buck DC-DC converter with a "hand-made" plasma switch, a commercial voltage regulator (LTC-3588) and a BLE module (RIOT-001). Each diode from the rectifier has the capability to handle up to 1.6 kV, so that we insure of not having any breakdown or damage in the circuitry upon the mechanical excitations.

Figure 7 shows that 3∼4 steps are necessary to actuate the plasma switch that has ON/OFF actuation voltages of 1.25 kV / 450 V respectively. After the DC-DC buck converter, the voltage is below 10 V across Cstore = 100 µF, as shown in figure 8. At the output of the regulator, the BLE module is powered at 1.8 V. A transfer of charges from Crect to Cstore is obtained at each actuation of the plasma switch. The first voltage-drop of ~3.5 V across Cstore corresponds to the wake-up of the BLE module. In the experiment, this initial voltage drop has been obtained thanks to an external voltage source. Then 232 µJ are transferred at each switch actuation (increasing voltage), which triggers the sending of a data packet at the energy cost of 137 µJ (decreasing voltage). The sensors' data can then be read on an android app.

Alternative transmission scheme

When the plasma switch is activated, the generated sparkle induces an electromagnetic wave that can be detected several meters away with a simple coil antenna, following the principle of the famous Hertz experiment [START_REF] Pompigna | Smart roads: a state of the art of highways innovations in the Smart age[END_REF].

Figure 9: Setup for the transmission at distance "d" based on the Hertz experiment

In this experiment, the plasma switch has been modified so that the switch ON actuation voltage is reduced to 370 V, so it is activated at each step. Figure 10 shows the received signal with a 3-turn coil of 5 cm diameter at 2 meters away from the plasma switch. The maximum amplitude of the received signal gives an indication of the force applied on the transducer. This approach is very promising since almost no electronic is needed to transmit the information.

CONCLUSION

We have demonstrated electrical power generation and data transmission with a BLE module using a triboelectric power generator embedded in a road bump. During these preliminary tests, 200 μJ were obtained from scratch with a single human step. Considering that a car is much heavier and would generates four actuations (one per tire), the energy harvested in a real situation should be at least one order of magnitude higher. 232 µJ were transferred to the storage capacitor by the 2-stage PMS at each switch actuation, which is enough to send a packet of data with the BLE module. Taking benefit of the electromagnetic wave generated by the plasma switch actuation, a low-power data transmission system can be obtained. The amplitude of the received signal gives an indication of the force applied on the transducer. This approach is very promising since almost no electronic is needed to transmit the information.

Figure 1 :

 1 Figure 1: TENG electrical model.

Figure 2 :

 2 Figure 2: Fabricated Devices.

Figure 3 :

 3 Figure 3:3D view of the transducer.

Figure 4 :

 4 Figure 4: Setup for the experiment with the BLE communication.

Figure 5 :

 5 Figure 5: Conditioning circuit used.

Figure 6 :

 6 Figure 6: Measured voltage across Crect.

Figure 7 :

 7 Figure 7: Voltage across Crect showing the plasma switch hysteresis.

Figure 8 :

 8 Figure 8: Voltage across Cstore. Each down step corresponds to one packet of transmitted data with the BLE module.

Figure 10 :

 10 Figure 10: Voltage across the coil at a distance of 2 meters after a plasma switch actuation