Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold

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Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold

Amine Aribi, Sorin Dragomir, Ahmad El Soufi

To cite this version:

Amine Aribi, Sorin Dragomir, Ahmad El Soufi. Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold: Eigenvalues of the sub-Laplacian. Differential Geometry and its Applications, Elsevier, 2015, 39, pp.113–128. �10.1016/j.difgeo.2015.01.005�. �hal-01208082�

EIGENVALUES OF THE SUB-LAPLACIAN AND DEFORMATIONS OF CONTACT STRUCTURES ON A

COMPACT CR MANIFOLD

AMINE ARIBI, SORIN DRAGOMIR, AND AHMAD EL SOUFI

Abstract. Given a compact strictly pseudoconvex CR manifoldM, we study the differentiability of the eigenvalues of the sub-Laplacian∆b,θas- sociated with a compatible contact form (i.e. a pseudo-Hermitian struc- ture)θonM, under conformal deformations ofθ. As a first application, we show that the property of having only simple eigenvalues is generic with respect toθ, i.e. the set of structuresθsuch that all the eigenvalues of∆b,θ are simple, is residual (and hence dense) in the set of all com- patible positively oriented contact forms on M. In the last part of the paper, we introduce a natural notion of critical pseudo-Hermitian struc- ture of the functionalθ7→λk(θ), whereλk(θ) is thek-th eigenvalue of the sub-Laplacian∆b,θ, and obtain necessary and sufficient conditions for a pseudo-Hermitian structure to be critical.

1. Introduction

LetMbe a compact strictly pseudoconvex CR manifold of real dimension 2n+1. A pseudo-Hermitian structure on M is a contact formθ ∈Γ(T^∗M) whose kernel coincides with the horizontal distribution of M. The strict pseudoconvexity ofMmeans that the Levi form associated to such a contact form is either positive definite or negative definite. We denote byP₊(M) the set of all pseudo-Hermitian structures with positive definite Levi form onM.

To every pseudo-Hermitian structure θ ∈ P₊(M) we associate its sub- Laplacian ∆b,θ (or simply ∆b if there is no risk of confusion) which is a sub-elliptic operator of order 1/2, and denote by

0=λ₀(θ)< λ₁(θ)≤λ₂(θ)≤ · · · ≤λ_k(θ)≤ · · · → ∞ the nondecreasing sequence of eigenvalues of∆b,θ.

Several works published in recent years are devoted to the study of the sub-Laplacian and the investigation of its spectral properties, see for in- stance [3, 5, 4, 6, 8, 9, 10, 14, 18, 19, 21, 25, 26, 27, 28, 30]. The aim of most of them is to extend to the CR context some of the spectral geomet- ric results established in the Riemannian setting for the Laplace-Beltrami operator.

In our previous paper [5], we discussed the continuity of the eigenvalues λ_k(θ), as functions on the setP₊(M) that we have endowed with a natural

2000Mathematics Subject Classification. 32V20, 35H20, 58J50.

Key words and phrases. CR manifold, sub-Laplacian, eigenvalue, generic property.

1

metric topology. In the present paper, we start by studying the differentia- bility of the spectrum of the sub-Laplacian∆b,θ under one-parameter defor- mations of the contact structureθ. We apply classical perturbation theory of selfadjoint operators to get a differentiability result (Theorem 3.2). More- over, we prove that ifθ(t) ∈ P₊(M) is an analytic deformation of a contact structureθ, then the functiont7→ λ_k(θ(t)), which is not differentiable ifλ_k(θ) is not simple, admits left-sided and right-sided derivatives att = 0, and re- late these derivatives to the eigenvalues of an explicit symmetric operator acting on theλ_k(θ)-eigenspace (Theorem 3.3).

In the second part of the paper we use these facts to show that the property of having only simple eigenvalues is generic for the sub-Laplacians on a given compact strictly pseudoconvex CR manifold M. Indeed, we prove that the set of contact structures θ ∈ P₊(M) such that all the eigenvalues of ∆b,θ are simple, is a residual set in the complete metric space P₊(M) (see Theorem 4.1). Our proof relies on an eigenvalue splitting technique (Proposition 4.1) used by many authors in the Riemannian setting (see [1, 7, 13]; see also [31] for a different approach).

The last section is devoted to the notion of critical pseudo-Hermitian structure. Despite the lack of differentiability of the eigenvaluesλ_k(θ) upon analytic deformations θ(t) ∈ P₊(M) of the pseudo-Hermitian structure, a natural notion of criticality can be defined using the existence of left- sided and right-sided derivatives of λ_k(θ(t)) at t = 0 (see Definition 5.1).

Since λ_k(θ) is not invariant under scaling of the pseudo-Hermitian struc- ture, we restrict ourselves to the deformations that preserve the global vol- ume vol(θ) = R

Mθ ∧(dθ)ⁿ. We give necessary and sufficient conditions for a pseudo-Hermitian structure to be a critical point of the functional θ ∈ P₊(M) 7→ λk(θ), under the volume-preserving constraint. In par- ticular, we will see that the criticality condition is strongly related to the existence of a finite family of λ_k(θ)-eigenfunctions v₁,· · ·,v_d, satisfying v²₁ +· · ·+ v²_d = 1 (Corollary 5.1). This last condition is satisfied for in- stance by the first positive eigenvalue of the standard CR sphereS²ⁿ⁺¹ (see [30, Proposition 4.4]).

2. Preliminaries

LetMbe a compact connected orientable CR manifold of CR dimension n (and real dimension 2n+1). Such a manifold M is equipped with a pair (H,J), whereH is a sub-bundle of the tangent bundleT M of real rank 2n (often called Levi distribution) and J is an integrable complex structure on Hwhich means that,∀X,Y ∈Γ(H),

[X,Y]−[JX,JY]∈Γ(H) and

[JX,Y]+[X,JY]= J([X,Y]−[JX,JY]).

SinceMis orientable, there exists a nonzero 1-formθ∈Γ(T^∗M) whose ker- nel coincides with H. Such a 1-form, calledpseudo-Hermitianstructure on M, is of course not unique. Actually, the set of pseudo-Hermitian structures on Mconsists in all the forms±e^uθ,u∈C^∞(M).

To each pseudo-Hermitian structureθ we associate itsLevi formG_θ de- fined on Hby

G_θ(X,Y)= −dθ(JX,Y)= θ([JX,Y]).

The integrability of J implies that G_θ is symmetric and J-invariant. The CRmanifoldM is said to bestrictly pseudoconvexif the Levi formGθof a pseudo-Hermitian structureθis either positive definite or negative definite.

Of course, this condition does not depend on the choice of θ. In all the sequel, we assume that M is strictly pseudoconvex and denote by P₊(M) the set of all pseudo-Hermitian structures with positive definite Levi form on M. Everyθ ∈ P₊(M) is in fact a contact form which induces on M the following volume form

ψ_θ = 1

2ⁿn! θ∧(dθ)ⁿ.

The associated divergence div_θ is defined, for every smooth vector field Z on M, by

L_Zψ_θ = divθ(Z)ψ_θ.

We denote by L²(M) the set of squared integrable functions on M with respect toψ_θ. A functionu∈L²(M) isweakly differentiable(w.d.) alongH if there isY_u∈Γ(H) such that|Y_u|_G

θ =G_θ(Y_u,Y_u)¹² ∈L¹_loc(M) and Z

M

Gθ(Y_u,X)ψ_θ = − Z

M

udiv_θ(X)ψ_θ

for every X ∈ Γ^∞(H). SuchY_u is unique up to a set of measure zero and is denoted byY_u =∇^Huand calledweak horizontal gradientofu.It is easy to check that ifuis differentiable, then∀X ∈Γ^∞(H),du(X)=Gθ(X,∇^Hu). Let

D(∇^H)= n

u∈L²(M) : u is(w.d.)along H and∇^Hu∈L²(H)o , where L²(H) stands for the set of squared integrable sections of H with respect to the inner productG_θ and the volume elementψ_θ. Then we may regard the weak horizontal gradient as a linear operator

∇^H :D(∇^H)⊂L²(M)→L²(H).

As C^∞(M) ⊂ D(∇^H) it follows thatD(∇^H) is a dense subspace of L²(M).

Let

(∇^H)^∗:D[(∇^H)^∗]⊂ L²(H)→ L²(M)

be the adjoint of∇^H.ThenΓ^∞(H)⊂ D[(∇^H)^∗] and, for allX ∈Γ^∞(H), one has

(∇^H)^∗X =−div_θ(X).

In particular, (∇^H)^∗ is densely defined in L²(H). The sub-Laplacian ∆^b, or

∆b,θ if it is necessary to avoid confusion, is given by D(∆b)=

u∈ D(∇^H) :∇^Hu∈ D[(∇^H)^∗] ,

∆b =(∇^H)^∗◦ ∇^H= −div_θ◦ ∇^H. Note that

(∆bu,u)_L²_(M)= k∇^Huk²_L2(H) ≥0

for anyu∈ D(∆^b).Moreover, the sub-Laplacian is symmetric, i.e.

(1) D(∆b) is dense inL²(M).

(2) D(∆b)⊂ D(∆^∗_b) and (∆bu,v)_L2(M) =(u,∆bv)_L2(M) ∀u,v∈ D(∆b).

The operator ∆b is also known to be subelliptic of orderε = 1/2. Indeed, one has (cf. [15, Theorem 2.1]) for anyu∈C^∞(M),

kuk²_H1/2(M) ≤C

(∆^bu,u)_L²_(M)+kuk²_L2(M)

, (2.1)

for some constantC independent of u. It is worth noticing that∆b can be seen as the real part of the Kohn Laplacian acting on functionsb = ∂¯^∗_b∂¯_b, where ¯∂_bu is the projection ofduonto T_(0,1)^∗ M. Indeed, we have (cf. [24, Theorem 2.3])b = ∆b+ inT, whereT is the unique vector field satisfying T⌋θ= 1 andT⌋dθ=0.

Lemma 2.1. The space H^1/2(M)= W^1/2,2(M)admits a compact embedding into L²(M).

The proof of this Lemma uses standard arguments (see [4]).

Lemma 2.2. The operator(∆b+I)⁻¹ :D((∆b+I)⁻¹)⊂L²(M)→ L²(M)is compact.

Proof. Based on the estimate (2.1) one has Ker(∆^b+I) ={0}. Consequently,

∆b+I :C^∞(M)→ R(∆b+I)⊂C^∞(M)

is invertible, whereR(A) denotes the range of the operatorA. Therefore, we may consider the inverse

(∆b+I)⁻¹:D

(∆b+I)⁻¹

=R(∆b+I)⊂ L²(M)→ H^1/2(M).

Letv∈ D

(∆b+I)⁻¹

and let us apply (2.1) to the functionu=(∆b+I)⁻¹(v) followed by the Cauchy-Schwartz inequality

k(∆b+I)⁻¹vk²_H1/2(M) ≤C

v, (∆b+I)⁻¹v

L²(M) ≤Ckvk_L2(M)k(∆b+I)⁻¹vk_L2(M). Moreover, there is a continuous embeddingH^1/2(M)→ L²(M) so that

kuk_L2(M) ≤C^′kuk_H1/2(M), u∈H^1/2(M), for some constantC^′ >0 independent ofu. Thus,

k(∆b+I)⁻¹vk²_H1/2(M) ≤C^′′kvk_L2(M)k(∆b+I)⁻¹vk_H1/2(M)

(withC^′′ =CC^′) or

k(∆b+I)⁻¹vk_H1/2(M) ≤C^′′kvk_L2(M)

which proves the continuity of the operator (∆^b+ I)⁻¹. Finally, by Lemma 2.1, the embedding H^1/2(M) → L²(M) is compact. Hence, (∆b + I)⁻¹ : D((∆b + I)⁻¹) ⊂ L²(M) → L²(M) is compact (as the composition of a

compact operator with a continuous operator).

Corollary 2.1. The spectrum σ(∆b) of the sub-Laplacian is discrete and consists of eigenvalues of finite multiplicity.

3. Differentiability of eigenvalues with respect to1-parameter deformations of the pseudo-Hermitian structure

We start by recalling the needed notions of functional analysis, cf. e.g. A.

Kriegl & P.W. Michor [22, 23] and T. Kato [20]. LetH be a Hilbert space and{A(t)}_t∈Ra family of linear operatorsA(t) :D(A(t))⊂ H → H. We say that A(t) is a real analytic (respectivelyC^∞, orC^k,α) family of selfadjoint operators if there is a dense subspaceV ⊂ H such that

i)D(A(t))=V andA(t) is selfadjoint for anyt∈Rand

ii) the function t ∈R 7−→ (A(t)u,v)_H ∈Cis real analytic (respectivelyC^∞, orC^k,α) for everyu∈Vandv∈ H.

If this is the case then (by a result in [22]) the (vector valued) function t ∈R7−→ A(t)u∈ H,

is of the same class for everyu∈V.

A sequence {λ_ν}_ν≥1 of scalar functions λ_ν : R → C is said to parame- trize the eigenvalues of {A(t)}_t∈R if for any t ∈ R and anyλ ∈ σ(A(t)), the cardinality of the set{ν≥1 :λν(t)= λ}equals the multiplicity ofλ.

We shall make use of the following result, which is referred hereafter as the Rellich-Alekseevsky-Kriegl-Losik-Michor theorem (cf. F. Rellich [29]

for statement (i), D. Alekseevski & A. Kriegl & M. Losik & P.W. Michor [2] for statement (ii), and A. Kriegl & P.W. Michor [23] for statements (iii)- (iv)).

Theorem 3.1. Let t ∈R7→ A(t)be a curve of unbounded selfadjoint oper- ators in a Hilbert spaceH, with common domain of definition and compact resolvent. Then

(i)If A(t)is real analytic in t ∈R, then the eigenvalues and the eigenvec- tors of A(t)may be parameterized real analytically in t.

(ii)If A(t)is C^∞ in t∈Rand if no two unequal continuously parameter- ized eigenvalues meet of infinite order at any t ∈R, then the eigenvalues and eigenvectors can be parameterized C^∞in t on the whole parameter domain.

(iii) If A(t) is C^∞ in t ∈ R, then the eigenvalues of A(t) may be parame- terized C²in t.

(iv)If A(t)is C^k,α in t ∈ Rfor some α > 0, then the eigenvalues of A(t) may be parameterized C¹ in t.

Among the applications to statements (i) and (iii) in Theorem 3.1 as pro- posed in [23], one may consider a compact manifoldMand a smooth curve

t 7→gtof smooth Riemannian metrics onM. If moreovert7→∆^gt is the cor- responding smooth curve of Laplace-Beltrami operators onL²(M) then (by (iii) in Theorem 3.1) the eigenvalues may be parameterizedC²int. This was exploited by A. El Soufi & S. Ilias, [16]-[17], who discussed an array of re- lated questions such as critical points of the functionalg∈ M(M)7→λ_k(g), or suitable deformations ofg∈ M(M) producing quantitative variations of λ_k. HereM(M) is the set of all Riemannian metrics onM.

Let M be a compact strictly pseudoconvex CR manifold and let θ be a pseudo-Hermitian structure on Mwith positive definite Levi form. Let

θ(t)=e^utθ, t∈R,

be an analytic deformationof θ, i.e. {u_t}_t∈R is a family of real valued C^∞ functions which is analytic with respect to t andu0 = 0. Here C^∞(M,R) is thought of as organized as a real Fr´echet space and the vector valued function

u:R→C^∞(M,R), u(t)=ut, t∈R,

is assumed to be of classC^ω. For a theory of power series in Fr´echet spaces we shall use Appendix B in [11].

Let∆bbe the sub-Laplacian onM associated withθand denote for each t, by∆b,tthe sub-Laplacian associated withθ(t).

Theorem 3.2. Letθ(t) = e^utθ be an analytic deformation ofθand let λ ∈ σ(∆b)be an eigenvalue of multiplicity m. There exist a positive real number ε, a family of m real analytic functions {Λi}_1≤i≤m ⊂ C^ω((−ε, ε),R), and m families of C^∞ functions {v_i(t)}_|t|<ε ∈C^∞(M,R), 1 ≤ i ≤ m, such that each v_i : (−ε, ε)→C^∞(M,R)is real analytic in t and

(1)Λi(0)=λ,1≤ i≤m,

(2)∆b,tv_i(t)= Λi(t)v_i(t),1≤i≤m, t ∈(−ε, ε)

(3){v_i(t) : 1≤i≤ m}is orthonormal in L²(M, ψ_θ(t)), t∈(−ε, ε).

Proof. The proof relies on the Rellich-Alekseevsky-Kriegl-Losik-Michor theorem (cf. Theorem 3.1 above). To this end we introduce the family of operatorsU_t : L²(M, ψ_θ)→L²(M, ψ_θ(t)),

U_tv=e^−(n+1)ut/2v, v∈L²(M, ψ_θ).

The family{U_t}_t∈R is a real analytic family of unitary operators , i.e.

kU_tvk_L²_(M,ψθ(t))= kvk_L²_(M,ψθ),

andU_t⁻¹v= e^(n+1)ut/2v. Moreover, letA(t) be the family of operators A(t)=U⁻¹_t ◦∆b,t ◦U_t :L²(M, ψ_θ)→ L²(M, ψ_θ).

Then

∆b,tv(t)=λv(t)⇐⇒A(t)

U_t⁻¹v(t)

=λU_t⁻¹v(t).

In particular, the spectrum of ∆b,t coincides with that of A(t). Let us show that the family{A(t)}_t∈Ris analytic int. Indeed, the dense subspaceD(∆b)⊂ L²(M, ψ_θ) is the domain ofA(t) and, as we shall check in a moment,A(t)⊂

A(t)^∗. Indeed, the sub-Laplacians∆^band∆^b,t = ∆^b,θ(t)are related by (see [8, Proposition 5] or [30, Lemma 1.8])

∆^b,tv=e^−ut

∆^bv−nGθ(∇^Hut,∇^Hv)

, v∈C²(M). (3.1) Then, for eachv∈ D(∆b),

A(t)v=(U⁻¹_t ◦∆b,t ◦U_t)v=· · ·=

=e^−ut

"

∆bv+Gθ(∇^Hu_t,∇^Hv)− n+1

2 ∆bu_t− (n−1) 2 |∇^Hu_t|²

! v

# . Finally, the family{A(t)}_t∈R is an analytic curve of self-adjoint operators in L²(M, ψ_θ) with common domain of definition and with compact resol- vent. Therefore, we can apply Theorem 3.1 (i) to deduce that the eigenval- ues and the eigenvectors ofA(t) depend analytically int, i.e., there existsm analytic families of vectorsv_i(t) and mreal analytic valued functionsΛi(t) intsatisfying (1), (2) and (3) of Theorem 3.2.

For anyθ∈ P₊(M), the set of all pseudo-Hermitian structures with posi- tive definite Levi form on M, let

0=λ₀(θ)< λ₁(θ)≤λ₂(θ) ≤ · · · ≤λ_k(θ)≤ · · ·

be the spectrum of the sub-Laplacian∆b = ∆b,θ of (M, θ). For everyk ∈N, let

E_k(θ)= Ker (∆b−λ_k(θ)I)

be the eigenspace of ∆^b corresponding to the eigenvalue λ_k(θ). Also let Π^k : L²(M, ψθ) → Ek(θ) be the orthogonal projection on Ek(θ). Let us fix k ∈ N and consider the functional θ ∈ P₊(M) 7−→ λ_k(θ) ∈ R. This functional is continuous (with respect to an appropriate metric topology on P₊(M), as shown in [5]) but not differentiable in general. However, one has the following

Theorem 3.3. Let M be a compact strictly pseudoconvex CR manifold and letθ∈ P₊(M). Letθ(t)= e^utθ, t ∈(−ε, ε), be an analytic deformation ofθ.

Then, for every positive k ∈N,

(1)The function t ∈(−ε, ε) 7−→λ_k(θ(t))admits left and right derivatives at t = 0.

(2)The derivatives _dt^dλ_k(θ(t))

_t=0−and_dt^dλ_k(θ(t))

_t=0+are eigenvalues of the operatorΠk ◦∆^′_b :E_k(θ)→E_k(θ)where,∀v∈C^∞(M),

∆^′bv= d dt∆^b,tv

_t=0 =−f∆bv−n Gθ

∇^Hf,∇^Hv with f = _dt^du_t

_t=0.

(3)Ifλk(θ)> λk−1(θ), then _dt^dλk(θ(t))

_t=0− and _dt^dλk(θ(t))

_t=0+ are the great- est and the least eigenvalues ofΠk◦∆^′b on E_k(θ), respectively.

(4) If λ_k(θ) < λ_k+1(θ) then _dt^dλ_k(θ(t))

_t=0− and _dt^dλ_k(θ(t))

_t=0+ ∈ Rare the smallest and the greatest eigenvalue ofΠk◦∆^′_b on E_k(θ), respectively.

Proof. Let us denote by m the dimension of Ek(θ). We apply Theorem 3.2 with λ = λ_k(θ) to derive the existence of m real analytic functions {Λi}_1≤i≤m ⊂ C^ω((−ε, ε),R) andmanalytic families of functions{v_i(t)}_|t|<ε ∈ C^∞(M,R), 1≤ i≤m, satisfying (1)-(3) of Theorem 3.2. Sincet 7→λ_k(θ(t)) andt7→ Λi(t) are continuous andΛ1(0)=... = Λm(0)=λ_k(θ), one deduces that λ_k(θ(t)) ∈ {Λ1(t),· · ·,Λm(t)}for sufficiently smallt. Since, moreover,

∀i ≤ m, t 7→ Λi(t) is analytic, there existδ > 0 and two integers p, q ≤ m such that

λ_k(θ(t))=

( Λp(t) fort∈(−δ,0) Λq(t) fort∈(0, δ).

Therefore, the function t 7−→ λ_k(θ(t)) admits left and right derivatives at t =0 with

d

dtλ_k(θ(t))

_t=0− = Λ^′p(0) and d

dtλ_k(θ(t))

_t=0+ = Λ^′q(0).

Now, one has for alli≤mandt ∈(−δ, δ),∆b,tv_i(t)= Λi(t)v_i(t). Differen- tiating att= 0, we get

∆^′bvi+ ∆^bv^′_i = Λ^′i(0)vi+λk(θ)v^′_i (3.2) wherevi = vi(0) andv^′_i = _dt^dvi(t)

_t=0. Multiplication byvj and integration by parts yield

Z

M

v_j∆^′bv_iψθ =

( Λ^′i(0) if j= i 0 otherwise.

Since{v₁,· · ·,v_m}is an orthonormal basis ofE_k(θ) with respect to the inner product of L²(M, ψθ), we deduce that

(Πk◦∆^′b)v_i = Λ^′i(0)v_i.

That isΛ^′₁(0),· · ·,Λ^′m(0) are the eigenvalues ofΠk ◦∆^′_b : E_k(θ) → E_k(θ).

Differentiating the identity (3.1) att =0 we get

∆^′bv=−f∆bv−nG_θ

∇^Hv,∇^Hf .

Assume now λ_k(θ) > λk−1(θ). For any i ≤ m, one then has Λi(0) = λk(θ) > λk−1(θ). By continuity, we necessarily haveΛⁱ(t) > λk−1(θ(t)) for sufficiently smallt. Hence, there existsη >0 such that,∀ |t|< ηand∀i≤m, Λⁱ(t) ≥ λ_k(θ(t)), which means thatλ_k(θ(t)) = min{Λ¹(t),· · · ,Λ^m(t)}. This implies that

d

dtλ_k(θ(t))

_t=0− =max

Λ^′1(0),· · ·,Λ^′m(0)

and d

dtλ_k(θ(t))

_t=0+ = min

Λ^′1(0),· · · ,Λ^′m(0) which proves (3).

Smilarily, if λk(θ) < λ_k+1(θ), one has, for sufficiently small t, Λⁱ(t) ≤ λ_k(θ(t)) which means thatλ_k(θ(t))= max{Λ¹(t),· · ·,Λ^m(t)}and, then,

d

dtλ_k(θ(t))

_t=0+ =max

Λ^′1(0),· · ·,Λ^′m(0)

and

d

dtλ_k(θ(t))

_t=0− =min

Λ^′1(0),· · ·,Λ^′m(0) .

Corollary 3.1. Let M be a compact strictly pseudoconvex CR manifold and letθ ∈ P₊(M). Letθ(t) = e^utθ, t ∈(−ε, ε), be an analytic deformation ofθ and set f = _dt^du_t

_t=0. For every positive integer k, let Qf,k : E_k(θ) → R be the quadratic form given by

Q_f,k(v)= − Z

M

λ_k(θ)v²+ n 2∆bv²

fψ_θ.

(1) If Qf,k is positive definite on E_k(θ), then there existsε > 0 such that λk(θ(−t))< λk(θ)< λk(θ(t))for all t∈(0, ε).

(2)Assume thatλ_k(θ)> λk−1(θ). If Qf,k takes negative values somewhere in Ek(θ), thenλk(θ(t))< λk(θ)for all t∈(0, ε), for someε >0.

(3)Assume thatλ_k(θ) < λ_k+1(θ). If Q_f,k takes positive values somewhere in E_k(θ), thenλ_k(θ(t))> λ_k(θ)for all t∈(0, ε), for someε >0.

Proof. First, we have with the notations of Theorem 3.3,∀v∈E_k(θ), Q_f,k(v)=

Z

M

v∆^′bvψ_θ. (3.3)

Indeed,∀v∈E_k(θ), Z

M

v∆^′bvψ_θ = − Z

M

v

f∆bv+nG_θ(∇^Hv,∇^Hf) ψ_θ

= − Z

M

fλ_k(θ)v²+ n

2G_θ(∇^Hv²,∇^Hf)

ψ_θ

= − Z

M

λ_k(θ)v²+ n 2∆bv²

f ψ_θ = Q_f,k(v).

Now, if Q_f,k is positive definite on E_k(θ), then, thanks to (3.3), all the eigenvalues of the operatorΠk◦∆^′_b :E_k(θ)→ E_k(θ) are positive. Applying Theorem 3.3 (2), it follows that both _dt^dλ_k(θ(t))

_t=0+ and _dt^dλ_k(θ(t))

_t=0− are positive and that there existsε >0 such thatλ_k(θ(−t))< λ_k(θ)< λ_k(θ(t)) for allt∈(0, ε).

Assume that λ_k(θ) > λk−1(θ) and that there exists v ∈ E_k(θ) such that Q_f,k(v) < 0. This implies that the operatorΠk◦∆^′_b : E_k(θ) → E_k(θ) has at least one negative eigenvalue. Applying Theorem 3.3 (3), we deduce that

d

dtλ_k(θ(t))

_t=0+ is negative and that there exists ε > 0 such that λ_k(θ(t)) <

λ_k(θ) for allt∈(0, ε).

The last part of the corollary can be proved using similar arguments.

4. Generic simplicity of sub-Laplacian eigenvalues

Let M be a compact stirctly pseudoconvex CR manifold and denote by P₊(M) the set of all pseudo-Hermitian structures with positive definite Levi form on M. In [5], we defined a complete distance on P₊(M) so that the eigenvalues of the sub-Laplacianθ ∈ P₊(M)7→ λk(θ) are continuous. This distance is defined as follows : We fix a form θ ∈ P₊(M). Givenθ₁ = e^u1θ andθ₂= e^u2θinP₊(M), we set

d(θ₁, θ₂)=d_C^∞(u₁,u₂)+ρ(G_θ1,G_θ2)

where d_C∞ is the distance function associated with the canonical Frechet structure ofC^∞(M) and

ρ(G_θ1,G_θ2)= inf{δ >0 : e^−δG_θ1(X,X)≤G_θ2(X,X)≤e^δG_θ1(X,X), ∀X ∈H}.

In [5], we proved that (P₊(M),d) is a complete metric space and that if ρ(Gθ1,Gθ2)< ε, then,∀k≥1,

e^−ε ≤ λ_k(θ₁) λ_k(θ₂) ≤e^ε.

In the sequel, we denote byJthe set of all elementsθ∈ P₊(M) such that all the eigenvalues of the sub-Laplacian∆b,θhave multiplicity one, that is,

J ={θ∈ P₊(M) : 0< λ₁(θ)< λ₂(θ)<· · ·< λ_k(θ)<· · · } Our main aim in this section is to prove the following

Theorem 4.1. The setJ is a residual set in (P₊(M),d), i.e., a countable intersection of open dense subsets. In particular,J is dense in(P₊(M),d).

The proof of this theorem relies on the following proposition which is a consequence of Theorem 3.3.

Proposition 4.1. Let M be a compact strictly pseudoconvex CR manifold and letθ ∈ P₊(M). Letλ∈σ ∆^b,θ

be an eigenvalue of multiplicity m ≥ 2 and let k∈Nbe such that

λ=λ_k(θ)= λ_k+1(θ)=· · ·=λ_k+m−1(θ).

There exist f ∈ C^∞(M) and ε > 0 such that θ(t) = e^{t f}θ satisfies for all t ∈(0, ε),

λ_k(θ(t))< λ_k+m−1(θ(t)).

Proof. LetE = E_k(θ) = E_k+m−1(θ) be the eigenspace of∆b,θ corresponding to the eigenvalueλand letΠ: L²(M)→ E be the orthogonal projection on E. For every f ∈C^∞(M), we denote byL_f :E →Ethe operator defined by

Lfv= Π◦∆^′bv= −Πh

λf v+nGθ

∇^Hv,∇^Hfi .

From the definition of the integers k and m, one has λk(θ) > λk−1(θ) and λ_k+m−1(θ) < λ_k+m(θ). Therefore, Theorem 3.3 tells us that, for any

f ∈ C^∞(M), _dt^dλ_k(e^{t f}θ)

_t=0+ and _dt^dλ_k+m−1(e^{t f}θ)

_t=0+ represent the smallest and the largest eigenvalues of the operator L_f : E → E, respectively.

Therefore, it suffices to prove the existence of a function f ∈ C^∞(M) so that the operator L_f has at least two distinct eigenvalues (i.e. L_f is not proportional to the identity of E). Indeed, in this case, we would have _dt^dλ_k(e^{t f}θ)

_t=0+ < _dt^dλ_k+m−1(e^{t f}θ)

_t=0+ which implies the conclusion of the proposition.

Thanks to (3.3), one has,∀v,w∈E L_fv,w

L²(M) =· · ·=− Z

M

n

2∆b(vw)+λvw

f ψ_θ.

Let{u₁,u2} ⊂ Ebe a pair of functions withku₁k_L²_(M)= ku₂k_L²_(M)andu²₁, u²₂ (recall thatE is of dimension at least 2) and setv=u1−u2andw= u1+u2

so that (v,w)_L2(M)= 0. The function f₀ = n

2∆b(vw)+λvw= n

2∆b(u²₁−u²₂)+λ(u²₁−u²₂) (4.1) is such that

L_f0v,w

L²(M) =− Z

M

n

2∆b(u²₁−u²₂)+λ(u²₁−u²₂) 2

ψ_θ

which does not vanish since ∆b has no negative eigenvalues. Thus, L_f0 cannot be proportional to the identity ofE.

Proof of Theorem 4.1. For every positive integerk, letJ_k be the subset of P₊(M) defined by

J_k = {θ∈ P₊(M) : 0< λ₁(θ)< λ₂(θ)<· · ·< λ_k(θ)}. We haveP₊(M)=J₁ ⊃ J₂ ⊃... ⊃ J_k ⊃...and

J =

∞

\

k=1

J_k.

According to Baire’s category theorem, it suffices to prove that eachJ_k is an open dense subset ofP₊(M).

The fact that J_k is open follows immediately from the continuity of the eigenvaluesθ∈ P₊(M)7→λ_i(θ),i≤ k.

Let us prove that, for anyk ≥1,J_k+1is a dense subset ofJ_k. An obvious recursion would then imply that eachJ_k is a dense subset ofP₊(M). So, let θ ∈ J_k\ J_k+1and letηbe any positive real number. Thus, one has

λ₁(θ)< λ₂(θ)< · · ·< λ_k−1(θ)< λ_k(θ)= λ_k+1(θ)=· · ·=λ_k+m−1(θ)< λ_k+m(θ), wheremis the multiplicity ofλ_k(θ). Using Proposition 4.1 and the continu- ity of the eigenvalues, one can find f ∈C^∞(M) andε >0 such that the form θ(t)=e^{t f}θsatisfies, for everyt∈(0, ε),

λ₁(θ(t))< λ₂(θ(t))<· · ·< λk−1(θ(t))< λ_k(θ(t))< λ_k+m−1(θ(t))

which means thatθ(t) belongs toJ_k and the multiplicity ofλ_k(θ) is at most m−1. Choosingt₁ ∈(0, ε) sufficiently small, one gets a formθ₁= θ(t₁)∈ J_k such that the multiplicity of λ_k(θ₁) is at most m− 1 and d(θ₁, θ) < η/m.

Repeating this argument at most m−1 times, we prove the existence of a 1-form ˆθ∈ J_k+1 such thatd(ˆθ, θ)< η.

5. Critical pseudo-Hermitian structures

The content of this section is patterned after the article [17] by Ilias and the third author dealing with Laplacian eigenvalues in the Riemannian set- ting. For the sake of completeness, we shall give self-contained proofs of the results we obtain in the CR context.

Let M be a compact strictly pseudoconvex CR manifold. For every pos- itive integer k we consider the map θ ∈ P₊(M) 7→ λ_k(θ) ∈ R, where, as before,P₊(M) denotes the set of all pseudo-Hermitian structures with pos- itive definite Levi form on M, and λ_k(θ) is the k-th eigenvalue of the sub- Laplacian associated to θ. Since the eigenvalues are not invariant under scaling, we restrictλ_kto the subset

P_+,0(M)={θ∈ P₊(M) : vol(θ)=1}

where vol(θ)= R

Mψ_θis the volume ofMwith respect toψ_θ. Thanks to Theorem 3.3, one can introduce the following

Definition 5.1. A pseudo-Hermitian structureθis said to be critical for the functionalλ_krestricted toP_+,0(M)if, for any analytic deformation{θ(t)=e^utθ} ⊂ P_+,0(M)ofθ, we have

d

dtλ_k(θ(t))

_t=0−× d

dtλ_k(θ(t))

_t=0+ ≤0.

It is easy to see that d

dtλ_k(θ(t))

_t=0+ ≤0≤ d

dtλ_k(θ(t))

_t=0− ⇐⇒λ_k(θ(t))≤ λ_k(θ)+o(t) as t→0 and

d

dtλ_k(θ(t))

_t=0− ≤ 0≤ d

dtλ_k(θ(t))

_t=0+ ⇐⇒λ_k(θ(t))≥λ_k(θ)+o(t) as t→0.

Of course, if θ is a local maximizer or a local minimizer of λk, then θ is critical in the sense of the previous definition. We set

A₀(M, θ)= (

f ∈C^∞(M) : Z

M

f ψ_θ =0 )

and recall the definition of the quadratic form Qf,k : Ek(θ) → Rassociated to a pair (f,k)∈C^∞(M)×N^∗(see Corollary 3.1):

Q_f,k(v)= − Z

M

λ_k(θ)v²+ n 2∆bv²

fψ_θ.

Theorem 5.1. Let M be a compact strictly pseudoconvex CR manifold. Let θ ∈ P_+,0(M)be a pseudo-Hermitian structure and k∈N^∗.

1) If θ is a critical pseudo-Hermitian structure of the functionalλ_k re- stricted toP_+,0(M), then,∀f ∈ A₀(M, θ), the quadratic form Q_f,k is indefi- nite on E_k(θ).

2) Assume thatλ_k(θ)> λk−1(θ)orλ_k(θ)< λ_k+1(θ). The pseudo-Hermitian structureθis critical for the functionalλ_k restricted toP_+,0(M)if and only if,∀f ∈ A₀(M, θ), the quadratic form Q_f,k is indefinite on Ek(θ).

Proof. Let f ∈ A₀(M, θ) and letθ(t) be the analytic deformation ofθgiven by

θ(t)= vol(θ) vol(e^{t f}θ)

!_n+1¹

e^{t f}θ= e^utθ, t ∈R, withu_t =t f − ¹

n+1ln(vol(e^{t f}θ)). Since

ψ_e^{t f}_θ =e^{(n+1)t f}ψ_θ,

it is easy to check that vol(θ(t))=1, that isθ(t) belongs toP_+,0(M) for every t ∈R. One has

d

dtvol(e^{t f}θ(t))

_t=0 = d dt

Z

M

e^{(n+1)t f}ψ_θ

_t=0= (n+1) Z

M

fψ_θ =0.

Therefore,

d dtu_t

_t=0 =· · ·= f and, then,

∆^′bv= d dt∆^b,t

_t=0 = −f∆^bv−nGθ(∇^Hv,∇^Hf).

Thus, we have (see (3.3)),

Q_f,k(v)= Z

M

v∆^′bvψ_θ. (5.1)

Now, assuming that θ is a critical pseudo-Hermitian structure of λ_k re- stricted toP_+,0(M), we obtain, using the definition of criticality and Theo- rem 3.3 (2), that the operatorΠk◦∆^′_badmits both nonnegative and nonpos- itive eigenvalues in E_k(θ), which means (thanks to (5.1) that the quadratic formQ_f,k is indefinite onE_k(θ). This proves the first part of the theorem.

The last part of the theorem follows from Theorem 3.3 (3) and (4), and (5.1).

Proposition 5.1. Let M be a compact strictly pseudoconvex CR manifold and let θ ∈ P_+,0(M) be a pseudo-Hermitian structure. For any positive integer k, the two following conditions are equivalent:

(1) For all f ∈ A₀(M, θ),the quadratic form Q_f,k is indefinite on E_k(θ).

(2) There exists a finite family {v₁,· · ·,v_d} ⊂ E_k(θ) of eigenfunctions associated withλ_k(θ)such thatPd

i=1v²_i =1.

Proof. Assume first that there existv₁,· · ·,v_d ∈E_k(θ) such thatPd

i v²_i = 1.

Therefore,∀f ∈ A₀(M, θ), Xd

i=1

Q_f,k(v_i)=· · ·=−λ_k(θ) Z

M

fψ_θ =0 which implies thatQ_f,k is indefinite onE_k(θ).

Conversely, assume that Q_f,k is indefinite onE_k(θ) for all f ∈ A₀(M, θ) and consider the convex set

K =









 X

i∈J

λ_k(θ)v²_i + n 2∆^bv²_i

;vi ∈Ek(θ), J ⊂N, Jfinite











⊂ L²(M).

Let us prove that the constant function 1 belongs to K. Indeed, if 1 <

K, then, applying classical separation theorem in the finite dimensional subspace of L²(M, θ) generated by K and 1, we deduce the existence of h ∈ L²(M) such that (h,1)_L²_(M) = R

Mhψθ > 0 and, ∀ w ∈ K, (h,w)_L² = R

Mhwψ_θ ≤ 0.Let f = h− ¹ vol(θ)

R

Mhψ_θ ∈ A₀(M, θ).Then ,∀v∈E_k(θ) Qf,k(v) = −

Z

M

λk(θ)v²+ n 2∆^bv²

fψθ

= − Z

M

λ_k(θ)v²+ n 2∆^bv²

hψ_θ+

R

Mhψ_θ vol(θ)λ_k(θ)

Z

M

v²ψ_θ sinceR

M∆^bv²ψ_θ = 0. Moreover,∀v∈ Ek(θ), the function

λ_k(θ)v²+ ⁿ₂∆^bv² belongs to Kwhich implies thatR

M

λ_k(θ)v²+ ⁿ₂∆bv²

hψ_θ ≤ 0 and, then Qf,k(v)≥ λ_k(θ)R

Mhψ_θ vol(θ)

Z

M

v²ψ_θ.

Therefore, the quadratic form Q_f,k is positive definite on E_k(θ) which con- tradicts the assumptions.

Now, since 1 ∈K, there existv₁,· · ·,v_d ∈E_k(θ) such that

d

X

i=1

λ_k(θ)v²_i + n 2∆bv²_i

=λ_k(θ) (5.2)

which leads to

∆^b







 X

i≤d

v²_i −1







=−2 nλ_k(θ)







 X

i≤d

v²_i −1









This implies thatP

i≤dv²_i −1 =0 since the sub-Laplacian admits no negative eigenvalues.

Theorem 5.1 and Proposition 5.1 lead to the following