Small eigenvalues of the Witten Laplacian acting on p-forms on a surface

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Small eigenvalues of the Witten Laplacian acting on p-forms on a surface

Dorian Le Peutrec

To cite this version:

Dorian Le Peutrec. Small eigenvalues of the Witten Laplacian acting on p-forms on a surface. Asymp- totic Analysis, IOS Press, 2011, 73 (4), pp.187 - 201. �10.3233/ASY-2011-1036�. �hal-00473114�

Small eigenvalues of the Witten Laplacian acting on p -forms on a surface

D. Le Peutrec

Institut f¨ur Angewandte Mathematik, Universit¨at Bonn, Endenicher Allee 60, 53115 Bonn, Germany.

email: dorian.lepeutrec@uni-bonn.de

March 8, 2010

Abstract

In this article, we are interested in the exponentially small eigenval- ues of the self adjoint realization of the semiclassical Witten Laplacian

∆^(p)_f,h, in the general framework of p-forms, on a connected compact Riemannian manifold without boundary. Our purpose is to notice that the knowledge of (the asymptotic formulas for) the smallest non zero eigenvalues of the self adjoint realization of ∆⁽⁰⁾_f,h(acting on functions), presented in [HeKlNi], essentially contains all the necessary informa- tion to the treatment of the case of oriented surfaces, forp-forms. The functionf is assumed to be a Morse function on Ω.

MSC 2010: 58J37, 58J10, 81Q10, 58A10, 15A18.

Key words and phrases: Witten complex, exponentially small eigenvalues, dif- ferential p-forms on surfaces.

1 Introduction and main result

The study of the small eigenvalues of some self adjoint realization of the Wit- ten Laplacian acting on functions (i.e. on 0-forms), ∆⁽⁰⁾_f,h, wheref is a Morse function, relies closely on the study of the metastability in reversible diffusion processes: the inverses of these small eigenvalues can indeed be interpreted as the mean exit times of local wells for a particle in a gradient field deriv- ing from the function f. Similar problems were considered by many authors

via a probabilistic approach in [FrWe], [HoKuSt], [Mic], and [Kol]. More recently, in the case of Rⁿ, accurate asymptotic forms of the exponentially small eigenvalues were obtained in [BoEcGaKl] and [BoGaKl]. These results were improved and extended to the cases of boundaryless compact manifolds in [HeKlNi] and of compact manifolds with boundaries for the Dirichlet and Neumann realizations of the Witten Laplacian in [HeNi] and [Lep1].

Nevertheless, the only case of the 0-forms has been fully treated. A nat- ural question then remains the accurate study of the small eigenvalues of

∆^(p)_f,h, the Witten Laplacian acting onp-forms, for generalp-forms. Note also that, if the general case of p-forms has not been fully understood yet, the number of small eigenvalues of the self adjoint realizations of ∆^(p)_f,h is well known for every p: in the case of a compact manifold without boundary, the number m_p of eigenvalues of the self adjoint realization of the Witten Lapla- cian ∆^(p)_f,h in some interval [0, Ch³²] (for h > 0 small enough) is the number of critical points of f with index p (see e.g. [Wit][Hen][HeSj4]).

In this connection, in the above references [BoEcGaKl, BoGaKl, HeKlNi, HeNi, Lep1], the accurate computation of the m0 smallest eigenvalues of

∆⁽⁰⁾_f,h is done after coupling the m0 local minima with some relevant critical points with index 1. This coupling relies on some topological aspects. The constant C in the exponential terms e^−Ch of these small eigenvalues are then on the form f(U¹)−f(U⁰), where U⁰ is a local minimum and U¹ the asso- ciated saddle point. Since all the local minima are considered, this gives in particular a canonical one to one correspondence between the small eigenval- ues of ∆⁽⁰⁾_f,h and the local minima of f.

According to the result of Witten which ensures that the number of small eigenvalues of ∆^(p)_f,h is mp, the same kind of correspondence between these eigenvalues and the critical points of f with index p is expected for any p.

As for the 0-forms, this correspondence means an association between the critical points with index p and attached relevant critical points (probably with index p+ 1 or p−1 as it will be seen further).

The aim of this short paper is to notice that in the case of an oriented compact surface Ω without boundary, it is possible to obtain all the small eigenvalues of the self adjoint realization of ∆^(p)_f,h, for p ∈ {0,1,2}. More- over, as it is the case for the 0-forms, there is actually a canonical association between the small eigenvalues of ∆^(p)_f,hand the critical points off with indexp.

Before stating the main result, let us recall that ∆^(p)_f,h and ∆^(n−p)_−f,h are es-

sentially dual operators under the action of the Hodge operator (see Propo- sition 4.1 further). The knowledge of the small eigenvalues of ∆⁽⁰⁾_f,hthen gives the small eigenvalues of ∆⁽ⁿ⁾_−f,h. The complete treatment of the case of a man- ifold with dimension 1 is therefore obvious.

The analysis done in [HeKlNi] requires an assumption which ensures that the exponentially small eigenvalues of ∆⁽⁰⁾_f,h are simple, with different loga- rithmic equivalents as h tends to 0¹. As in [Nie], we are going to work here with a generic assumption which is slightly stronger in order to avoid some technical and unnecessary considerations.

Let us denote by U^(p), for p ∈ {0, . . . , n}, the set of the critical points of f with index p. Our generic assumption is:

Assumption 1.1. The function f ∈ C^∞(Ω,R) is a Morse function whose critical values are distinct. Moreover, the quantities f(U)−f(V), where U and V are two different critical points of f, are also distinct.

Following this assumption, a one to one mappingj01(resp. j21) can be defined from U⁽⁰⁾\n

U₁⁽⁰⁾o

(resp. U⁽²⁾\n U₁⁽²⁾o

), where U₁⁽⁰⁾ is the global minimum (resp. maximum) of f, into the set U⁽¹⁾.

The minima (resp. maxima) are denoted byU_k⁽⁰⁾₀ (resp. U_k⁽²⁾₂ ),k₀ ∈ {1, . . . , m₀} (resp. k2 ∈ {1, . . . , m2}), and the saddle points byU_j⁽¹⁾,j ∈ {1, . . . , m1}. The ordering of these extrema as well as the construction of the one to one map- pings j01 and j21 will be specified further, in Section 3.

Moreover, according to the following proposition, which is proved at the beginning of Section 4,

Proposition 1.2. The ranges of j01:U⁽⁰⁾\n U₁⁽⁰⁾o

→ U⁽¹⁾ and j21:U⁽²⁾\n

U₁⁽²⁾o

→ U⁽¹⁾ are disjoint.

a one to one mappingkfromU⁽¹⁾\{U_j⁽¹⁾s.t.U_j⁽¹⁾ ∈/ Ran (j01), U_j⁽¹⁾ ∈/ Ran (j21)}

intoU⁽⁰⁾∪ U⁽²⁾ can be defined in the same spirit asj01and j21. The labelling of the saddle points as well as the construction of k are also specified in Sec- tion 3.

Our main theorem, involving the definition of the previous injective maps, can now be stated:

1More precisely, such that the quantities hlnλ(h) andhlnλ^′(h) have different limits as h→0, whereλ(h) andλ^′(h) are two distinct small eigenvalues of ∆⁽⁰⁾_f,h.

Theorem 1.3. Under Assumption 1.1, there exists h0 such that, for h ∈ (0, h₀] andp∈ {0,1,2}, the spectrum in[0, h²³)of ∆^(p)_f,h consists ofm_p eigen- values,

0 =λ^(p)₁ (h) =· · ·=λ^(p)_bp (h)< λ^(p)_bp+1(h)< . . . < λ^(p)_mp(h).

The above mp −bp non zero eigenvalues are moreover simple, exponentially small, and admit a complete asymptotic expansion.

For p∈ {0,2}, the Betti number bp is 1, and, for every kp in {2, . . . , mp},

λ^(p)_kp(h) = h π|bλ^(p)₁ |

det Hessf(U_k^(p)_p )

1 2

det Hessf(U_j⁽¹⁾_p1(kp))

1 2

e⁻²

˛

˛

˛

˛

˛ f(U(1)

jp1(kp))−f(U(p) kp)

˛

˛

˛

˛

˛ h

1 +hc^(p)_kp(h) ,

where c^(p)_kp(h) ∼ P∞

m=0h^mc^(p)_kp,m and bλ⁽⁰⁾₁ (resp. bλ⁽²⁾₁ ) is the negative (resp.

positive) eigenvalue of Hessf(U_j⁽¹⁾_01(k0)) (resp. Hessf(U_j⁽¹⁾_21(k2))).

For p = 1, the eigenvalue 0 has multiplicity b1 and for every j in {b1 + 1, . . . , m₁},

λ⁽¹⁾_j (h) = h π|bλ₁|

det Hessf(U_k(j)^(p))

1 2

det Hessf(U_j⁽¹⁾)

1 2

e⁻²|^f(Uk(j)^(p))−f(U(1) j )|

h

1 +hc⁽¹⁾_j (h) ,

where p ∈ {0,2}, c⁽¹⁾_j (h) ∼ P∞

m=0h^mc⁽¹⁾_j,m, and, if k(j) ∈ U⁽⁰⁾ (resp. k(j) ∈ U⁽²⁾), bλ1 is the negative (resp. positive) eigenvalue of Hessf(U_j⁽¹⁾).

Let us make a few comments about this theorem. First, the result concerning the small eigenvalues of ∆⁽⁰⁾_f,his nothing but the main result of [HeKlNi]. The simplicity of the non zero eigenvalues of ∆⁽⁰⁾_f,h and ∆⁽²⁾_f,h is moreover obvious, according to the theorem and to Assumption 1.1 (look indeed at the quanti- ties involved in the exponential term). The proof of the result for ∆⁽²⁾_f,h only lies on a simple trick involving the Hodge operator: the small eigenvalues of

∆⁽²⁾_f,h are the small eigenvalues of ∆⁽⁰⁾_−f,h.

Using some elementary results of linear algebra, it is then quite easy to show that the set of the m1 −b1 non zero eigenvalues of ∆⁽¹⁾_f,h is

nλ⁽⁰⁾_k0(h), λ⁽²⁾_k2(h), with (k₀, k₂)∈ {2, . . . , m₀} × {2, . . . , m₂}o .

The proper writing of its non zero eigenvalues is therefore a simple conse- quence of the definition of the one to one mappingk. Owing to the writing of

λ^(p)_kp(h) in the theorem, it suffices indeed to defineλ⁽¹⁾_jp1(kp)asλ^(p)_kp, forp∈ {0,2}

andkp ∈ {2, . . . , mp}. In other words, forj > b1,λ_j⁽¹⁾ =λ^(p)_kp, wherekp =k(j).

The proof of Theorem 1.3 relies on the main result of [HeKlNi], Hodge theory and some primary results of linear algebra and homology theory. In partic- ular, the finest part of the proof is the one to one correspondence between the saddle points of f and the small eigenvalues of ∆⁽¹⁾_f,h (i.e. the proof of Proposition 1.2).

This result will be proved in the fourth section. The second one is devoted to the recalling of some generalities about Witten Laplacians and the third one to the ordering of the critical points as well as to the construction of the one to one mappings j01,j21 and k.

Remark 1.4. According to the above comments, we do not need Proposi- tion 1.2 and the introduction of the one to one mapping k if we only want to show that all the small eigenvalues of ∆⁽¹⁾_f,h are obtained.

Nevertheless, without Proposition 1.2, it is not possible to ensure that the canonical pairing between these small eigenvalues and the saddle points (with index 1) is a one to one correspondence. By canonical pairing, we mean the pair (λ⁽¹⁾_j , U_j⁽¹⁾), where U_j⁽¹⁾ is the critical point involved in the writing of λ⁽¹⁾_j (in the exponential term). It would indeed be a priori possible that λ⁽¹⁾_j >0 and λ⁽¹⁾_j′ >0, j 6=j^′, involve in their writing the same saddle point U_j⁽¹⁾.

2 Generalities about Witten Laplacians

Let (Ω, g0) be a C^∞ connected compact oriented Riemannian manifold with- out boundary and with dimensionn∈N^∗,g₀ denoting the given Riemannian metric on Ω.

The cotangent (resp. tangent) bundle on Ω is denoted byT^∗Ω (resp. TΩ) and the exterior fiber bundle by ΛT^∗Ω =⊕ⁿ_p=0Λ^pT^∗Ω (resp. ΛTΩ =⊕ⁿ_p=0Λ^pTΩ).

The space of the C^∞, L², H^s sections in ΛT^∗Ω are respectively denoted by C^∞(Ω; ΛT^∗Ω), L²(Ω; ΛT^∗Ω), and H^s(Ω; ΛT^∗Ω). Let us also recall that the L² spaces are those associated with the unit volume form for the Riemannian structure on Ω.

Let d be the exterior differential on C^∞(Ω; ΛT^∗Ω),

d^(p):C^∞(Ω; Λ^pT^∗Ω)→ C^∞(Ω; Λ^p+1T^∗Ω),

andd^∗ its formal adjoint with respect to theL²-scalar product inherited from the Riemannian structure,

d^(p),∗ :C^∞(Ω; Λ^p+1T^∗Ω)→ C^∞(Ω; Λ^pT^∗Ω).

Set, for a function f ∈ C^∞(Ω;R) and h >0, the distorted operators defined on C^∞(Ω; ΛT^∗Ω):

df,h=e^−f(x)/h(hd)e^f(x)/h and d_f,h^∗ =e^f(x)/h(hd^∗)e^−f(x)/h. (2.1) The Witten Laplacian is the differential operator defined on C^∞(Ω; ΛT^∗Ω) by:

∆f,h=d^∗_f,hdf,h+df,hd^∗_f,h= (df,h+d^∗_f,h)². (2.2) The last equality follows from the property dd=d^∗d^∗ = 0 which implies:

df,hdf,h=d^∗_f,hd^∗_f,h = 0. (2.3) It means, by restriction to the p-forms inC^∞(Ω; Λ^pT^∗Ω):

∆^(p)_f,h =d^(p),∗_f,h d_f,h^(p) +d^(p−1)_f,h d^(p−1),∗_f,h . (2.4) The Hodge operator ⋆,

⋆:C^∞(Ω; Λ^pT^∗Ω)→ C^∞(Ω; Λ^n−pT^∗Ω),

is locally defined in a pointwise orthonormal frame (E1, . . . , En) by:

(⋆ωx)(Eσ(p+1), . . . , Eσ(n)) = ε(σ)ωx(Eσ(1), . . . , Eσ(p)),

forω_x ∈Λ^pT_x^∗Ω and with any permutationσ ∈Σ(n) of{1, . . . , n} preserving {1, . . . , p} (ε(σ) denotes the signature ofσ).

This operator is essentially mentionned here in order to recall the following formulas wich will be useful further:

⋆(⋆ωx) = (−1)^p(n−p)ωx , ∀ωx ∈Λ^pT_x^∗Ω, (2.5) hω1|ω2i_Λ^p_L² =R

Ωω1∧⋆ω2 , ∀ω1, ω2 ∈Λ^pL² , (2.6)

⋆d^∗,(p−1) = (−1)^pd^(n−p)⋆ , ⋆d^(p)= (−1)^p+1d^{∗,(n−p−1)}⋆ . (2.7) Let us now recall a first result about the localization of the small eigenval- ues of ∆^(p)_f,h used by E. Witten in his proof of the Morse inequalities (c.f.

[Wit][HeSj4]).

Theorem 2.1. For p∈ {0, . . . , n}, the Witten Laplacian ∆^(p)_f,h starting from C^∞(Ω; Λ^pT^∗Ω) is essentially self adjoint on L²(Ω; Λ^pT^∗Ω) and its self ad- joint realization, still denoted by ∆^(p)_f,h, has for domain the Sobolev space H²(Ω; Λ^pT^∗Ω).

Moreover, if f is a Morse function, there exists h0 > 0, such that, for h ∈ (0, h0], ∆^(p)_f,h has the following property: for p ∈ {0, . . . , n}, the spec- tral subspace Ran 1_[0,h^3/2₎(∆^(p)_f,h) has rank mp(f), the number of critical points of f with index p. The kernel Ker ∆^(p)_f,h has moreover rank bp, the p-th Betti number of the manifold Ω.

Let us also recall the Witten complex structure, which is the core of Witten’s approach to Morse inequalities.

Denote byF^(p)the spectral subspace Ran 1_[0,h^3/2₎(∆_f,h^(p)). Then,B_f,h^(p) =d^(p)_f,h F^(p)

and B_f,h^(p),∗ =d^(p−1),∗_f,h

F^(p) define two complexes of finite dimensional spaces:











0 −→ F^{(0) B}

(0)

−→f,h F^{(1) B}

(1)

−→f,h . . . ^B

(n−1)

−→f,h F⁽ⁿ⁾ −→ 0

0 ←− F^{(0) B}

(0),∗

←−f,h F^{(1) B}

(1),∗

←−f,h . . . ^B

(n−1),∗

f,h←− F⁽ⁿ⁾ ←− 0. Furthermore, this complex structure linked to Theorem 2.1 directly leads to the Morse inequalities:



 Pk

p=0 (−1)^k−pm_p(f) ≥ Pk

p=0(−1)^k−pb_p , 0≤k < n Pn

p=0 (−1)^pmp(f) = Pn

p=0(−1)^pbp.

3 Labelling of the critical points and defini- tion of the injective maps j

, j

and k

3.1 Labelling of the extrema, definition of j

and j

The ordering of the minima and the construction/definition of the one to one mapping j01 are done as follows (see also [Nie] or [HeKlNi] for a slightly weaker assumption than our generic one). Let us also recall that they corre- spond, in the probability framework, to the ordering of exit times, according to [BoEcGaKl][BoGaKl].

Ordering of the minima and definition of j₀₁

Remember first that for a real number λ, the number of connected compo- nents of the sublevel set {f < λ} is the dimension of its 0-th order homology group b0({f < λ}) = dimH0({f < λ}).

1) SetU₁⁽⁰⁾ = minx∈Ωf,z1 =∞,f(z1) = +∞and considerb0({f < λ}), for λ decreasing from f(z1) = +∞.

2) When U_k⁽⁰⁾₀ and zk0 are defined for k0 = 1, . . . , K0 −1, decrease λ from f(z_K0−1) until b₀({f < λ}) increases by 1. Denote by λ_K0 this value.

3) Due to the structure of the level sets of a Morse function, there exists a point in U⁽¹⁾, which has to be unique according to Assumption 1.1, that we denote by z_K0, satisfying f(z_K0) =λ_K0. Denote then byU_K⁽⁰⁾₀ the global minimum of the new connected component.

4) Iterate 2) and 3) until all the local minima have been considered.

5) Lastly, permute thek0’s to make the sequence

f(zk0)−f(U_k⁽⁰⁾₀ )

k0∈{1,...,m0}

strictly decreasing, which is possible by Assumption 1.1.

6) The one to one mapping j01 is defined from U⁽⁰⁾\n U₁⁽⁰⁾o

into U⁽¹⁾ by:

∀k0 ≥2, U_j⁽¹⁾_01(k0) =zk0 .

In order to compute the small eigenvalues of ∆⁽²⁾_f,h, we are going to apply the results of [HeKlNi] to the Witten Laplacian ∆⁽⁰⁾_−f,h, whose minima and saddle points (with index 1) are nothing but the maxima and saddle points (with index 1) of f.

The construction of j21 is thereby the same as the one of j01, after having replaced f by −f. An equivalent way to see it is to work withf again, but to consider the connected components of the upperlevel sets of f instead of its sublevel sets and to make λ increase:

Ordering of the maxima and definition of j21

1) Set U₁⁽²⁾ = maxx∈Ωf,z1 =∞,f(z1) =−∞and considerb0({f > λ}), for λ increasing fromf(z₁) = −∞.

2) When U_k⁽²⁾₂ and zk2 are defined for k2 = 1, . . . , K2 −1, increase λ from f(zK2−1) until b0({f > λ}) increases by 1. Denote by λK2 this value.

3) By Assumptions 1.1, there exists a unique point inU⁽¹⁾, that we denote by zK2, satisfying f(zK2) = λK2. Denote then by U_K⁽²⁾₂ the global maximum of the new connected component.

4) Iterate 2) and 3) until all the local maxima have been considered.

5) Lastly, permute thek2’s to make the sequence

f(U_k⁽²⁾₂ −f(zk2))

k2∈{1,...,m2}

strictly decreasing, which is possible by Assumption 1.1.

6) The one to one mapping j₂₁ is defined from U⁽²⁾\n U₁⁽²⁾o

into U⁽¹⁾ by:

∀k2 ≥2, U_j⁽¹⁾_21(k2) =zk2 .

3.2 Labelling of the saddle points and definition of k

Let us now label the saddle points U_j⁽¹⁾, j ∈ {1, . . . , m1}, in order to define the one to one mapping k fromU⁽¹⁾ into U⁽⁰⁾∪ U⁽²⁾.

Owing to Proposition 1.2 which ensures that the ranges of j01 and j02 are disjoint, the subset

n

U_j⁽¹⁾_p1(kp), p∈ {0,2}, kp ∈ {2, . . . , mp}o

of U⁽¹⁾

consists of (m0 −1) + (m2 − 1) saddle points, or equivalently, due to the Morse inequalities, of m1−b1 saddle points.

The b1 points of U⁽¹⁾ which are not reached by bothj01 and j21 are denoted byU₁⁽¹⁾, . . . , U_b⁽¹⁾₁ . The ordering of the remainding saddle points and the con- struction of the one to one mapping k are done as follows.

Once U_j⁽¹⁾ and k(j) are defined for j = b1 + 1, . . . , J −1, U_J⁽¹⁾ and k(J) are chosen in the following way:

• takeJ =jp1(Kp) such as|f(U_K^(p)_p)−f(U_j⁽¹⁾_p1(Kp))|is the unique minimizer of the setn

|f(U_k^(p)_p )−f(U_j⁽¹⁾_p1(kp))|o

, wherep∈ {0,2}, 2≤kp ≤mp, and (k0, k2)∈ {k(b/ 1+ 1), . . . , k(J −1)},

• set then k(J) = Kp.

The one to one mapping k is therefore defined from U⁽¹⁾\ {U₁⁽¹⁾, . . . , U_b⁽¹⁾₁ } into U⁽⁰⁾ ∪ U⁽²⁾. It is even a bijection from U⁽¹⁾ \ {U₁⁽¹⁾, . . . , U_b⁽¹⁾₁ } onto

U⁽⁰⁾\ {U₁⁽⁰⁾}

∪

U⁽²⁾\ {U₁⁽²⁾} .

Remark 3.1. Actually, the ordering of the saddle points U_j⁽¹⁾’s (j > b1) and the construction of the one to one mapping k are just made in such way as the two following conditions are fulfilled:

• k is the inverse of jp1 on the range of jp1, for p∈ {0,2},

• the sequence

|f(U_k(j)^(p))−f(U_j⁽¹⁾)|

j>b1

is strictly decreasing.

4 Proofs and remarks

Proof of Proposition 1.2.

Let us first define, for a∈R,

Ω^a₋ :={x∈Ω, f(x)≤a} and Ω^a₊ :={x∈Ω, f(x)≥a}.

Ifa is not a critical value of f, the sets Ω^a₋ and Ω^a₊are two oriented compact 2-manifolds with boundary Ω^a := {x ∈ Ω, f(x) = a}, which is an oriented compact 1-manifold without boundary. To be more precise, Ω^a is homotopic to a union of disjoint circles. The number of connected components of these sets are respectively denoted byb₀(Ω^a₋),b₀(Ω^a₊) andb₀(Ω^a). According to the usual Morse theory (see e.g. [Mil] or [Gra]), b0(Ω^a_∓) is constant whena stays between two critical values of f.

Take now U¹ a critical point of f with index 1 and a < b two real num- bers such thatU¹ is the only critical point off inf⁻¹([a, b]). This is possible by Assumption 1.1 which ensures that the critical values of f are distinct.

According again to the usual Morse theory (see again [Mil] or [Gra]), if we denote by b1(Ω^a₋) (resp. b1(Ω^a₊)) the dimension of the first order homology group H₁(Ω^a₋) (resp. H₁(Ω^a₊)), two possibilities can happen:

b0(Ω^a₋) = b0(Ω^b₋) + 1 b₁(Ω^a₋) = b₁(Ω^b₋) or

b0(Ω^a₋) =b0(Ω^b₋) b₁(Ω^a₋) =b₁(Ω^b₋)−1 ( resp.

b₀(Ω^a₊) + 1 =b₀(Ω^b₊) b1(Ω^a₊) =b1(Ω^b₊) or

b₀(Ω^a₊) = b₀(Ω^b₊)

b1(Ω^a₊)−1 =b1(Ω^b₊) ). Therefore, four cases are a priori possible and owing to the definitions of the one to one mappings j01 and j21, Proposition 1.2 amounts to show that one of these a priori possible cases cannot happen, or equivalently that the two following propositions cannot occur simultaneously (but they can a priori both not happen):

b₀(Ω^a₋) =b₀(Ω^b₋) + 1, (4.1)

b0(Ω^b₊) =b0(Ω^a₊) + 1. (4.2) Let us also notice that this incompatibility is implied by the two following implications:

If b0(Ω^a₋) =b0(Ω^b₋) + 1, then b0(Ω^a)> b0(Ω^b), (4.3) If b0(Ω^b₊) =b0(Ω^a₊) + 1, then b0(Ω^b)> b0(Ω^a). (4.4) These implications are obviously “symmetric” and we are only going to prove (4.3).

Let us then assume that Equation (4.1) is satisfied and let us denote by Ω₋^b,1, . . . ,Ω^b,N₋ (resp. Ω^a,1₋ , . . . ,Ω₋^a,N+1) the N := b0(Ω^b₋) (resp. N + 1 = b0(Ω^a₋)) connected components of Ω^b₋ (resp. Ω^a₋), with U⁽¹⁾ ∈Ω^b,N₋ .

SinceU¹ is the only critical point off inf⁻¹([a, b]), according to Theorem 3.1 of [Mil] p. 12, these connected components can be ordered such as, for k in {1, . . . , N − 1}, Ω^a,k₋ is a deformation retract of Ω^b,k₋ (nothing happens for these components since they do not contain any critical point in f⁻¹([a, b])).

In particular, the following relation is satisfied,

∀k ∈ {1, . . . , N −1}, b0(Ω^b,k) =b0(Ω^a,k), and the proof amounts to check that

b₀(Ω^a,N) +b₀(Ω^a,N+1)> b₀(Ω^b,N). (4.5) According to [Gra] pp. 60-61, one of the three following equalities occurs,

b0(Ω^a,N) +b0(Ω^a,N+1) =b0(Ω^b,N)±1 or b0(Ω^a,N) +b0(Ω^a,N+1) =b0(Ω^b,N), and the fact that Ω is oriented ensures that the third case cannot happen (see Proposition 5 of [Gra], p. 81).

It appears now clearly that, if the number of connected components of the boundary {f = λ} decreases when the critical value f(U¹) is crossed (for λ =b thenλ =a), i.e. if

b0(Ω^a,N) +b0(Ω^a,N+1) =b0(Ω^b,N)−1,

then the number of connected components of {f ≤ λ} cannot increase, i.e.

Ω^b,N does not give two connected components, Ω^a,N and Ω^a,N+1. Hence the only possibility is b₀(Ω^a,N) +b₀(Ω^a,N+1) =b₀(Ω^b,N) + 1, which implies (4.5), and Proposition 1.2 is proven.

The rest of the proof of Theorem 1.3 is quite easy if we keep in mind the Morse inequalities and the following results.

Lemma 4.1. (Stated for a compact manifold with dimension n∈N^∗) Let up be an eigenvector of ∆^(p)_f,h associated with the eigenvalue λp, for some p ∈ {0, . . . , n}. Then ⋆up is an eigenvector of ∆^(n−p)_−f,h associated with the eigenvalue λp. The multiplicity of λp is moreover the same for ∆^(p)_f,h and

∆^(n−p)_−f,h since the Hodge operator ⋆ is bijective.

Proof. Coming back to the definitions of df,h and d^∗_f,h (see (2.1)), and owing to Equation (2.7), let us write the following relations:

⋆d^∗,(p−1)_f,h = (−1)^pd^(n−p)_−f,h ⋆ and ⋆ d^(p)_f,h= (−1)^p+1d^{∗,(n−p−1)}_−f,h ⋆ .

This implies the following formal relations which are the main point of the proof:

⋆∆^(p)_f,h = ∆^(n−p)_−f,h ⋆ and ⋆∆^(n−p)_f,h = ∆^(p)_−f,h⋆ .

Hence the proof only comes down to check the action of the Hodge operator

⋆ on the operator domain, H²(Ω; ΛT^∗Ω), which is actually preserved under its action.

Lemma 4.2. (Stated for a compact manifold with dimension n∈N^∗) Let up be an eigenvector of ∆^(p)_f,h associated with the eigenvalue λp, for some p∈ {0, . . . , n}. Then, if λp 6= 0, one of the two following cases occurs:

• either df,hup is an eigenvector of ∆^(p+1)_f,h associated with the eigenvalue λp,

• or d^∗_f,hup is an eigenvector of ∆^(p−1)_f,h associated with the eigenvalue λp. Moreover, a basis of Ker

∆^(p)_f,h−λp

can be chosen such as, for each ele- ment of the basis, only one of the above cases occurs, the other vector being 0.

Proof. Let us first recall, due to the ellipticity of the Witten Laplacian, that an eigenvector up of ∆^(p)_f,h is in C^∞(Ω; ΛT^∗Ω). Hence, there is no problem of regularity. Assume now that both df,hup and d^∗_f,hup are zero and let us show that it leads to a contradiction.

From the relation

∆^(p)_f,hup =λpup,

we obtain, looking at the scalar product with up:

0 =||df,hup||²+||d^∗_f,hup||² =λp||up||². This is a contradiction since up 6= 0 and λp 6= 0.

The rest of the proof is a consequence of the complex structure and Hodge theory. Assume that λp is an eigenvalue of ∆^(p)_f,h and take (u¹_p+1, . . . , u^ℓ_p+1) (resp. (u¹_p−1, . . . , u^ℓ_p−1^′ )) an orthogonal basis of Rand^(p)_f,h

Ker

∆^(p)_f,h−λp

(resp. Rand^(p−1),∗_f,h

Ker

∆^(p)_f,h−λp

). The following relations are hence satisfied:

∀1≤k ≤ℓ, df,hu^k_p+1 = 0 and ∀1≤k ≤ℓ^′, d^∗_f,hu^k_p−1 = 0. Owing to the first part of the proposition, the set

(d^∗_f,hu¹_p+1, . . . , d^∗_f,hu^ℓ_p+1, d_f,hu¹_p−1, . . . , d_f,hu^ℓ_p−1^′ )

then consists of ℓ+ℓ^′ non zero eigenvectors of ∆^(p)_f,hassociated withλp. Since each of these vectors is obviously cancelled by df,h or d^∗_f,h, for proving the end of the proposition, it is sufficient to show that they form a basis of

Ker

∆^(p)_f,h−λp

.

This family of vectors is linearly independent since orthogonal:

∀1≤k6=k^′ ≤ℓ, ∀1≤j 6=j^′ ≤ℓ^′,

hd^∗_f,hu^k_p+1, d^∗_f,hu^k_p+1^′ i=h∆^(p+1)_f,h u^k_p+1, u^k_p+1^′ i=λphu^k_p+1, u^k_p+1^′ i= 0, hdf,hu^j_p−1, df,hu_p−1^j′ i=h∆^(p−1)_f,h u_p−1^j , u^j_p−1^′ i=λphu^j_p−1, u^j_p−1^′ i= 0, hd^∗_f,hu^k_p+1, df,hu^j_p−1i=hu^k_p+1, df,hdf,hu^j_p−1i= 0.

To conclude, it reminds only to check that this family generates Ker

∆^(p)_f,h−λp

. Take thenv_p in Ker

∆^(p)_f,h−λ_p

withv_porthogonal tod^∗_f,hu¹_p+1, . . . , d^∗_f,hu^ℓ_p+1 and todf,hu¹_p−1, . . . , df,hu_p−1^ℓ′ , and show thatvp is zero. For allkin{1, . . . , ℓ}, the relation

0 =hvp, d^∗_f,hu^k_p+1^′ i=hdf,hvp, u^k_p+1^′ i implies thatd^(p)_f,hvp belongs to Rand^(p)_f,h

Ker

∆^(p)_f,h−λp

⊥

. Henced^(p)_f,hvp is zero, since d^(p)_f,hvp obviously belongs to Rand^(p)_f,h

Ker

∆^(p)_f,h−λp

. By the same argument,d^(p−1),∗_f,h vp is also zero. This implies the nullity ofvp and ends up the proof of Lemma 4.2.

Let us now go back to the case of a manifold with dimension 2. The last lemma directly implies the following result:

Lemma 4.3. Let∆f,hdenotes the self adjoint realization of the Witten Lapla- cian. The set of the non zero eigenvalues of ∆⁽¹⁾_f,h (counted with multiplicity) is the union of the set of the non zero eigenvalues of ∆⁽⁰⁾_f,h with the set of the non zero eigenvalues of ∆⁽²⁾_f,h.

Let us now prove Theorem 1.3.

Proof of Theorem 1.3. The result given for ∆⁽⁰⁾_f,his the main result of [HeKlNi]

and the result given for ∆⁽²⁾_f,h is a direct consequence of [HeKlNi] (with the Morse function −f) and Lemma 4.1. Recall moreover that the number of small eigenvalues of ∆⁽⁰⁾_−f,h is actuallym2.

Owing to Lemma 4.3, we obtain all the non zero eigenvalues of ∆⁽¹⁾_f,h. The ordering of the saddle points as well as the definition of the one to one map- ping k then allow us to write the asymptotic formulas stated in Theorem 1.3 for ∆⁽¹⁾_f,h.

Remark 4.4. 1) The trick used here unfortunately does not work for higher dimensions, not even for n = 3. In this last case, we cannot, following only the previous analysis, determinate all the non zero smallest eigenvalues of

∆⁽¹⁾_f,h and ∆⁽²⁾_f,h, but we can nevertheless obtain m0−b0 small non zero eigen- values of ∆⁽¹⁾_f,h and m3−b3 small non zero eigenvalues of ∆⁽²⁾_f,h.

2) The main obstacle to the complete analysis, forn ≥3, is the construction of global quasimodes for general p-forms. At this time, we actually only know how to construct global quasimodes for functions, owing to the fact that e^−fh is in the kernel of ∆⁽⁰⁾_f,h. Such a construction means probably a deeper under- standing of the Morse geometry than the one needed in [HeKlNi][HeNi][Lep1].

3) Having in mind Lemma 4.2, one expects, in the general framework of p- forms in a n-dimensional space, to write the non zero eigenvalues of ∆^(p)_f,h on the form Ak(h)e⁻²

|f(U(p±1) i(k) )−f(U(p)

k )|

h (1 +hck(h)), where i is a one to one map- ping from U^(p)\ {U₁^(p), . . . , U_b^(p)_p }into U^(p−1)∪ U^(p+1); the pointsU₁^(p), . . . , U_b^(p)_p being those associated with the bp zero eigenvalues of ∆^(p)_f,h.

Acknowledgement: The author would like to thank T. Jecko, F. Nier and M. Klein for profitable discussions.

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