Eigenvalue Estimates on Weighted Manifolds

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for f-minimal hypersurfaces and the spectrum of the Hodge Laplacian.


Introduction and results
The aim of spectral geometry is to obtain deep insights into both topology and geometry of Riemannian manifolds from the spectrum of differential operators.However, in the case of an arbitrary Riemannian manifold, it is in general not possible to explicitly calculate the spectrum of a given differential operator.For this reason, it is very important to obtain characterizations of the eigenvalues of such operators in terms of geometric data as for example curvature.The most prominent spectral problem one can study is the case of the Laplace operator acting on functions for which we recall several important results.Let Ω ⊂ M be a bounded domain of a Riemannian manifold M with smooth boundary ∂Ω and let λ be an eigenvalue of the Dirichlet problem ∆u = λu on Ω u = 0 on ∂Ω (1.1) where u ∈ C ∞ (M ) and ∆ is the Laplace-Beltrami operator.From general spectral theory, we know that the spectrum of the Dirichlet problem (1.1) is discrete and that the eigenvalues satisfy 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ . . .≤ λ j ≤ . . .→ ∞.
In the case that the manifold M is the Euclidean space with the flat metric (R n , can), Payne, Pólya and Weinberger [PGW1,PGW2] derived the following so-called universal inequalities for the Dirichlet problem of the Laplace operator (1.1).Namely, they show that for any integer k ≥ 1 An immediate consequence of (1.2) is the following inequality, which allows to get an estimate of all the eigenvalues if an upper bound is known on the lowest one λ 1 .Another kind of such inequality has been further developed by Hile and Protter (see [HP]) and by Yang in [Y] who established the following estimate This inequality improves all the previous estimates.In [CY], Cheng and Yang were able to turn the recursion formula (1.3) into an estimate for the (k + 1)-th eigenvalue of (1.1) in terms of some power of k as follows where C 0 (n, k) is a constant only depending on n and k.For more details on the spectral properties of the Dirichlet problem (1.1) we refer to the lecture notes [Ash].
Many of these formulas have been generalized to the case of a Riemannian manifold.In this setup, Chen and Cheng [CC] obtained an extrinsic estimate for eigenvalues of the Dirichlet problem (1.1) of the Laplacian on a complete Riemannian manifold (M n , g) that is isometrically immersed in an (n + m)-dimensional Euclidean space R n+m (see also [EHI]).The estimate is k i=1 (1.5) Here, H := 1 n trace(II) denotes the mean curvature of the immersion and II is the second fundamental form.As a consequence, they deduce an upper bound for the (k + 1)-th eigenvalue in terms of a power of k that involves the mean curvature.In this article, we focus on the study of geometric differential operators on an interesting class of Riemannian manifolds, the so-called Bakry-Emery manifolds.A Bakry-Emery manifold is a triple (M, g, dµ f ), where instead of the Riemannian measure dv g one considers the weighted measure dµ f := e −f dv g with f ∈ C ∞ (M ).One often refers to this kind of manifolds as weighted manifolds.While the initial motivation to study such kind of manifolds was to model diffusion processes [BE] they have by now become famous in the study of self-similar solutions of the Ricci flow, the so-called Ricci solitons.Also they appear in the analysis of shrinkers, which represent a special class of solutions of the mean curvature flow.We refer to [CM,Section 2] and [IRS] for more details.Due to the presence of the weight, differential operators on weighted manifolds contain additional contributions and also their spectrum is different compared to the case of standard Riemannian manifolds.For example, the Laplace operator acting on functions on a weighted manifold is defined as follows where ∆ = δd is the standard Laplace-Beltrami operator.A direct computation shows that ∆ f , usually called drift Laplacian, is elliptic and self-adjoint (if M is compact) with respect to the weighted measure e −f dv g .Therefore, its spectrum is discrete and consists of an increasing sequence of eigenvalues As in the standard case, the eigenvalue 0 corresponds to constant functions.Several results have been obtained on the spectrum of this operator (see for example [MD,ML,ML1]) of which we give a non-exhaustive list below.In particular, when a compact Bakry-Emery manifold (M n , g) is isometrically immersed into the Euclidean space, Xia and Xu [XX] established the following recursion formula which clearly reduces to Inequality (1.5) when f is zero.Also several explicit calculations for the drift Laplacian on self shrinkers were carried out in [CP, Z, BK].This article is devoted to the study of the Hodge Laplacian acting on differential forms on weighted manifolds, which we will often refer to as drifting Hodge Laplacian, and also denote it by ∆ f (see Subsection 2.2 for the definition).As for functions, the drifting Hodge Laplacian on differential p-forms has a discrete spectrum that entirely consists of nonnegative eigenvalues denoted by (λ i,p,f ) i .In particular, we will derive various eigenvalue estimates as well as different recursion formulas which will characterize the corresponding spectrum.
In the following, we present the main results of this article.First, we extend the results of [Asa, S] to the case of weighted manifolds: Theorem 1.1.Let (M n , g, dµ f = e −f dv g ) be a compact Bakry-Emery manifold that is isometrically immersed into the Euclidean space R n+m .For p = 1, . . ., n, we let λ 1,p,f be the first nonnegative eigenvalue of ∆ f on p-forms and λ ′ 1,p,f the first positive eigenvalue of ∆ f on exact p-forms.Then, we have the following estimate (1.7) Also, we have the inequality where Scal M denotes the scalar curvature of M .
The tensor f appearing in the first statement of Theorem 1.1 is the so-called p-Ricci tensor, see (2.6) for the precise definition.Also, II [p] νs is some canonical extension of the second fundamental form II of the immersion to differential p-forms in the direction of the normal vector field ν s , see (3.3) for more details.
(1) It is well-known that λ 1,f = λ ′ 1,1,f where λ 1,f is the first positive eigenvalue of ∆ f on functions.Hence we get the estimate for λ 1,f which is the same estimate as in [BCP].(2) It was shown in [JMZ] that when M is an embedded shrinker of revolution in R 3 such that the intersection of M with some sphere has only two connected components and such that M is symmetric with respect to reflection across the axis of revolution, then equality is attained in (1.9).Therefore, the inequality in Theorem 1.1 is attained for p = 1.
In addition to the eigenvalue estimates presented in Theorem 1.1, we establish the following recursion formulas for the eigenvalues of the drifting Hodge Laplacian on weighted manifolds: Here, ω i represents the i-th eigenform of ∆ f .
Remark 1.4.In Theorem 1.1 and Theorem 1.3 the tensor B [p] appears in the estimates.It is possible to express this tensor in terms of extrinsic geometric data via equation (4.5) but as this would further complicate the presentation of the eigenvalue estimates we do not make use of this option.
In order to eliminate the dependence on ω i in the statements of Theorem 1.3, we take the infimum over Ω of the smallest eigenvalue of the endomorphsim f which we denote by δ 1 to deduce that Corollary 1.5.Let X : (M n , g) → (R n+m , can) be an isometric immersion.For any p ∈ {0, . . ., n}, the eigenvalues of the drifting Hodge Laplacian ∆ f acting on p-forms on a domain Ω of M with Dirichlet boundary conditions satisfy for any k ≥ 1 for α ≥ 2. Here, we set Remark 1.6.
(1) Choosing α = 2 in the estimates of Corollary 1.5 generalizes the results obtained by Xia and Xu [XX] for functions to the case of p-forms.
(2) In the case of α = 2 and f = |X| 2 2 recursion formulas similar to the ones of Corollary 1.5 were established in [CZ].
Finally, we finish by establishing an estimate for the index of a compact f -minimal hypersurface of R n+1 as in [S2,IRS].Recall that a f -minimal hypersurface M of the weighted manifold (R n+1 , can, e −f dv g ) is a hypersurface such that the f -mean curvature vanishes, where H is the mean curvature.The stability operator of M , also called Jacobi operator, is then defined for any smooth function u on M by where Ric R n+1 f denotes the Bakry-Emery Ricci tensor.On an arbitrary Riemannian manifold (N, h), the Bakry-Emery Ricci tensor (or the ∞-Bakry-Emery Ricci tensor) is defined as follows Ric N f = Ric N + Hess N f, where Hess N f is the Hessian of f on N .In the case of N = (R n , can) we simply have that Ric R n+1 f = Hess R n+1 f and the stability operator acquires the form The f -index of M is the number of negative eigenvalues of the Jacobi operator L f .In order to estimate this number, we will establish an upper bound for the eigenvalues λ k (L f ) k of the Jacobi operator that depends on the eigenvalues of the drifting Hodge Laplacian ∆ f on p-forms.This will generalize the result obtained in [IRS, Theorem A] when p = 1.
Theorem 1.7.Let (M n , g) be a compact hypersurface of the weighted manifold (R n+1 , can, dµ f = e −f dv g ).Assume that M is f -minimal and that Ric R n+1 f = Hess R n+1 f ≥ a > 0. We denote by k 1 , . . ., k n the principal curvatures of the hypersurface.Then, for all l ≥ 1 and 1 ≤ p ≤ n, we have that Then, for any 1 ≤ p ≤ n, we have that A particular case of Corollary 1.8 is when M is self-shrinker.Recall that a self-shrinker manifold is a compact and connected submanifold of the Euclidean space R n+m satisfying the equation X ⊥ = −nH.Here X = (x 1 , x 2 , . . ., x n+m ) are the components of the immersion into R n+m .By taking the weight f = |X| 2 2 , one can easily check that any self-shrinker hypersurface is automatically f -minimal and that Hess R n+1 f = can.Therefore, Corollary 1.9.Let (M n , g) → R n+1 be a compact self-shrinker such that p(p − 1)γ M ≤ p + 1 for some 1 ≤ p ≤ n.Then, we have Remark 1.10.We would like to point out that Theorem 1.7 allows to obtain a statement on the stability of f -minimal hypersurfaces in terms of its p-th Betti number which was only known for p = 1 so far.However, the drawback is the fact that the stability estimate also depends on the curvature of the hypersurface which is not the case for p = 1.
This article is organized as follows: In Section 2 we present a number of general features on Dirac and Laplace type operators on weighted manifolds.Section 3 provides the proofs of the main results while Section 4 is devoted to a number of geometric applications.Throughout this article, we will use the geometers sign convention for the Laplacian.
When finalizing this manuscript, the article "Spinors and mass on weighted manifolds" [BO] appeared, where most of the results presented in Subsection 2.1 have been obtained independently from our calculations.
2. Dirac and Laplace type operators on weighted manifolds 2.1.The Dirac operator on weighted manifolds.In this section, we study the question of defining the fundamental Dirac operator on a weighted Riemannian spin manifold.In the famous article of Perelman [Per,Remark 1.3] some results in this direction were already presented.
In particular, this will allow to get a new vanishing result on the kernel of the Dirac operator based on the scalar curvature of weighted manifolds, see also [Per,Remark 1.3].
In order to approach this question, we first recall some basic facts on spin manifolds [LM].Let (M n , g) be a Riemannian spin manifold of dimension n.The spinor bundle ΣM is a vector bundle over the manifold M that is equipped with a metric connection ∇ and a Hermitian scalar product •, • .On this bundle we have the Clifford multiplication of spinors with tangent vectors which we denote by the symbol " • ".Remember that Clifford multiplication is compatible with the connection on the spinor bundle.Moreover, it is skew-symmetric in the sense for all Y ∈ T M and ψ, ϕ ∈ Γ(ΣM ).The fundamental Dirac operator D : Γ(ΣM ) → Γ(ΣM ) acting on spinors is defined by where {e i } i=1,...,n is an orthonormal basis of T M .The Dirac operator is an elliptic, first order differential operator which is self-adjoint with respect to the L 2 -norm (when M is compact).Therefore, it admits a sequence of eigenvalues of finite multiplicities.Also, its square is related to the Laplacian through the so-called Schrödinger-Lichnerowicz formula which is given by [LM] where ∇ * ∇ = − n i=1 ∇ e i ∇ e i + n i=1 ∇ ∇e i e i and Scal M is the scalar curvature of the manifold M .Let us now consider a smooth real valued function f on M and the weighted measure dµ f := e −f dv g .In order to define the Dirac operator on a compact weighted manifold, we use the following variational principle.The standard Dirac action with a weighted measure is where ψ ∈ Γ(ΣM ).
Proposition 2.1.The critical points of (2.2) are characterized by the equation Proof.We consider a variation of the spinor ψ defined as Therefore, the critical points of the energy E are exactly solutions of Equation (2.2).This completes the proof.

Motivated by the above calculation, we define
Definition 2.2.The fundamental Dirac operator on a Bakry-Emery manifold (M n , g, dµ f ) is given by Proof.It is straightforward to check that D and D f have the same principal symbol, hence D f is clearly elliptic.To prove the second part, we choose two spinors ψ, ϕ and calculate The proof is complete.
Remark 2.4.It is straightforward to check that (2.2) is invariant under conformal transformations of the metric on M .Hence, we cannot expect that we can relate the properties of D and D f by conformally transforming the metric on M .
However, the spectral properties of D f are essentially the same as the ones of the standard Dirac operator D as is shown by the following proposition.
Proposition 2.5.On a Bakry-Emery manifold (M n , g, dµ f ), we have In particular, if ψ is an eigenspinor of D associated with the eigenvalue λ, then e f /2 ψ is an eigenspinor of D f associated with the same eigenvalue λ.
Proof.By a straightforward calculation, we have for any spinor field ψ In order to obtain a corresponding Schrödinger-Lichnerowicz formula for D f we make the following observation: The L 2 -adjoint of ∇ with respect to the weighted measure e −f dv g will be denoted by ∇ * f and is given by the following expression

Hence, we have
Proposition 2.6.The square of D 2 f satisfies the following formula: Combining with (2.1) and using the definition of ∇ * f completes the proof.As in the case of the fundamental Dirac operator on Riemannian spin manifolds [Li], we deduce the following vanishing result Proof.This follows directly from Proposition 2.6 and the invariance of the spectrum of the Dirac operator in Proposition 2.5.
Remark 2.8.Suppose that (M, g) is a closed Riemannian surface.If we integrate the condition Scal M > 2∆f + |df | 2 with respect to the standard Riemannian measure dv g , we get where χ(M ) represents the Euler characteristic of the surface.Hence, this inequality can only be satisfied on a surface of positive Euler characteristic.
2.2.The Bochner-Laplacian on weighted manifolds.In this section, we will recall the Hodge Laplacian on weighted manifolds and define a twisted Laplacian motivated from the expression of the Dirac operator in the previous section.This will also allow to get a new vanishing result on the cohomology groups of the manifold.
For this, let (M n , g, dµ f = e −f dv g ) be a Riemannian Bakry-Emery manifold.We denote by d the exterior differential and by δ the codifferential.The weighted codifferential is defined by δ f := δ + df where we identify here (and in all the paper) vectors with one-forms through the musical isomorphism.The weighted codifferential is the L 2 -formal adjoint of d with respect to the measure dµ f , when M is compact.By a straightforward computation, one shows that The drifting Hodge Laplacian on differential forms is then defined as Clearly, the drifting Hodge Laplacian commutes with d and δ f and, therefore, it preserves the spaces of exact and weighted coexact forms.Moreover, this operator is elliptic and, when M is compact, it is self-adjoint with respect to the weighted measure dµ f .Therefore, as for the ordinary Hodge Laplacian on compact manifolds, the drifting Hodge Laplacian restricted to differential p-forms (1 ≤ p ≤ n) has a spectrum that consists of a nondecreasing, unbounded sequence of eigenvalues with finite multiplicities, that is The eigenvalue 0 corresponds to the space of f -harmonic forms, that is differential forms ω satisfying dω = 0 and δ f ω = 0. Notice that, by standard elliptic theory, we have an isomorphism [L,Formula 2.13], H meaning that the f -Betti numbers do not depend on f .Now it is shown in [PW], that the drifting Hodge Laplacian ∆ f has a corresponding Bochner-Weitzenböck formula which is where B [p] = n i,j=1 e * j ∧ e i R M (e i , e j ) is the Bochner operator that appears in the Bochner-Weitzenböck formula for ∆ = dδ + δd.Here, R M (X, for all X 1 , . . ., X p ∈ T M and {e 1 , . . ., e n } is a local orthonormal frame of T M .The tensor f is called the p-Ricci tensor and is denoted by Ric (p) f (see [P]).For p = 1, the p-Ricci tensor is just Ric (1) which corresponds to the so-called N -Bakry-Emery Ricci tensor for p = 1.It is now a wellknown fact that the Bochner-Weitzenböck formula gives rise to the Gallot-Meyer estimate on manifolds having a lower bound on the Bochner operator B [p] [GM].In the following, we will adapt this technique to give an estimate for the eigenvalues of the drifting Hodge Laplacian on a Bakry-Emery manifold having a lower bound on Ric (p) N,f .For this, we denote by λ ′ 1,p,f the first positive eigenvalue of ∆ f restricted to exact p-forms.Then, we have Proposition 2.9.Let (M n , g, dµ f ) be a compact Bakry-Emery manifold.If Ric (p) N,f ≥ p(n − p)γ for some γ > 0, then the first eigenvalue λ ′ 1,p,f satisfies the estimate Proof.Let ω be any exact p-eigenform.Applying the Bochner-Weitzenböck formula (2.6) yields Here, we use the inequality (p+q) 2 s ≥ p 2 N − q 2 N −s for all N such that N (N − s) > 0. Therefore, by integrating over M and using the fact λ ′ 1,p,f M |ω| 2 dµ f = M |δ f ω| 2 dµ f , we get the estimate.In the last part of this section, we will define a new drifting Hodge Laplacian acting on differential p-forms that will allow us to get a new vanishing result on the cohomology groups.For this, recall that the weighted Dirac operator defined in the previous section is When restricted to differential forms, the operator D f can be written as Here, we use the fact that X • ω = X ∧ ω − X ω for any vector field X and a differential form ω. It is not difficult to check that d 2 f = δ 2 f = 0 and that δ f = δ f − 1 2 df is the L 2 -adjoint of d f with respect to the measure dµ f = e −f dv g .Also the square of the Dirac operator gives rise to the twisted Laplacian has the same spectrum as ∆ by Proposition 2.5.Now, we establish a Bochner-Weitzenböck formula for ∆ f in order to get a new vanishing result for the cohomology groups.
Proposition 2.10.Let (M, g, dµ f ) be a Bakry-Emery manifold.Then, we have the Bochner-Weitzenböck formula for the twisted drifting Hodge Laplacian (2.8) In particular, if M is compact and In the last equality, we used the identity [S1, Lem.2.1] f .Now, an easy computation shows that Using the fact that T i=1 ∇ e i df ∧ (e i ) which can be proven by a straightforward computation, we get after using Equation (2.6) and replacing the last equality, Note that (2.8) has the same structure as the corresponding Schrödinger-Lichnerowicz formula for the Dirac operator on weighted manifolds (2.5).The vanishing result on the cohomology is obtained by just applying (2.8) to a ∆ f -harmonic form and integrating over M .Finally, the fact that the set of ∆ f -harmonic form is isomorphic to H p (M ) allows to finish the proof.

Proofs of the main results
In this section, we provide the proofs of the main results.
Proof of Theorem 1.1.We start by proving the first part of the theorem.We assume that (M n , g, dµ f = e −f dv g ) is a compact Bakry-Emery manifold that is isometrically immersed into R n+m .For p = 1, . . ., n, we let λ 1,p,f the first nonegative eigenvalue of ∆ f on p-forms and λ ′ 1,p,f the first positive eigenvalue of ∆ f on exact p-forms.For every i = 1, . . ., n + m, the parallel unit vector field ∂x i decomposes into where ι is the isometric immersion.Now, take any p-eigenform ω of ∆ f and consider the (p − 1)-form φ i := (∂x i ) T ω for each i = 1, . . ., n + m.By the Rayleigh min-max principle, we have that In the following, we will adapt some computations done in [GS] to the context of the drifting Hodge Laplacian (see also [Asa], [CGH,Thm. 5.8] for a similar computation).Denoting by {e 1 , . . ., e n } a local orthonormal frame of T M we find that n+m i=1 Here, we use the fact that α = 1 p n s=1 e * s ∧ e s α, for any p-form α.Moreover, by using the Cartan formula and [GS,Eq. (4.3)], we write where Z is the canonical extension of the second fundamental form II of the immersion to differential p-forms in the normal direction Z.More precisely, if we write II Z (X), Y = II(X, Y ), Z for any tangent vector fields X, Y ∈ T M and Z ∈ T ⊥ M , we define (II for any differential p-form ω on M and X 1 , . . ., X p ∈ T M .Now, as we did in (3.1), we decompose (∂x i ) T and (∂x i ) ⊥ in the frames {e 1 , . . ., e n } and {ν 1 , . . ., ν m } to compute the norm square of dφ i , we deduce after summing over i that n+m i=1 Here, {ν 1 , . . ., ν m } is a local orthonormal frame of T ⊥ M .In the above computation, we use the fact that all cross terms involving (∂x i ) T and (∂x i ) ⊥ are zero.By writing Replacing Equalities (3.2), (3.4) and (3.5) into Inequality (3.1), we deduce after using the Bochner-Weitzenböck formula (2.6) that This finishes the proof of the first part.To prove the second part of Theorem 1.1, we use the same technique as in [S] and adapt it to the case of the drifting Hodge Laplacian.We consider the exact p-form for i k = 1, . . ., n + m, k = 1, . . ., p.By applying the min-max principle and summing over i 1 , . . ., i p , we get (3.6) Now, we manipulate both sums the same way as in [S].The sum on the left hand side of ( 3 In the last line, we use the fact that, for any p-form ω, the equality n s=1 |e s ∧ ω| 2 = (n − p)|ω| 2 holds true.Hence, by induction, we deduce that i 1 ,...,ip Now, we aim to compute the sum on the right hand side of (3.6).To this end we recall some computations done in [S].First, by [S,Eq. 3.3], where, on differential q-forms, the operator T
It was shown in [S, p. 592] that and the latter can be computed explicitly in terms of |II| 2 and the scalar curvature of M through the formula (see [S,Lemma 2.5]) Therefore, we compute First, we show that the mixed term in (3.9) vanishes.For this, we write Hence, summing over i 1 , . . ., i p , we deduce that Now, it remains to compute the sum over i 1 , . . ., i p of the second term in (3.9).We establish the following lemma: Lemma 3.1.For any vector field X ∈ T M , we have Proof.Using the formula X (α ∧ ω) = g(X, α ♯ )ω − α ∧ X ω, which is valid for any one-form α, we write Taking the norm and summing over i 1 , . . ., i p , we get that i 1 ,...,ip Using the fact that n+m i=1 g(X, (∂x i ) T ) 2 = |X| 2 and Equation (3.7), the first sum in the above equation is equal to (p − 1)! n p − 1 |X| 2 .Concerning the second sum, we make use of the , and deduce that the second sum is equal to After using X = n+m i=1 g(X, (∂x i ) T )(∂x i ) T , the last sum reduces to Here, we conclude that Thus, by induction, we deduce that i 1 ,...,ip After some algebraic manipulations, Equality (3.10) reduces to the following This finishes the proof of the Lemma.
Using Lemma 3.1 with X = df and Equation (3.8), we deduce that the sum over i 1 , . . ., i p of Equation (3.9) gives the following Hence, by plugging Equation (3.7) into Inequality (3.6) we get the estimate completing the proof of Theorem 1.1.
Before we turn to the proof of Theorem 1.3, we recall the following powerful results from [AH] in which the authors show the following: Theorem 3.2.[AH, Thm.2] Let H be a complex Hilbert space with a given inner product •, • .Let A : D ⊂ H → H be a self-adjoint operator defined on a dense domain D which is semi-bounded below and has a discrete spectrum λ 1 ≤ λ 2 ≤ λ 3 . ... Let {B k : A(D) → H} N k=1 be a collection of symmetric operators which leave D invariant and let {u i } ∞ i=1 be the normalized eigenvectors of A, u i corresponding to λ i .This family is assumed to be an orthonormal basis of H. Let g be a nonnegative and nondecreasing function of the eigenvalues {λ i } m i=1 .Then we have the inequality Here [A, B] := AB − BA is the commutator of the two operators A and B.
As a corollary of this result and by taking g(λ) = (λ m+1 − λ) α−2 for α ≤ 2, we get the following inequality [AH,Cor. 3] (3.11) In the same paper, the authors deal with another type of inequalities treated by Harell and Stubbe [HaSt].For this, we say that a real function f satisfies condition (H1) if there exists a function r(x) such that f (x) − f (y) x − y ≥ r(x) + r(y) 2 (H1).
An example of such a function f is whenever f ′ is concave, in this case r = f ′ .In [AH], Ashbaugh and Hermi prove the following: Theorem 3.3.[AH,Thm. 7] Under the same assumptions as in the previous theorem, and if f is a function satisfying the condition (H1), we have For the particular case, when f (λ) = (λ m+1 − λ) α with α ≥ 2, they deduce the following inequality [AH,Cor. 8] (3.12) Proof of Theorem 1.3.In the following we will use Inequalities (3.11) and (3.12) in the case of the drifting Hodge Laplacian defined on a manifold with boundary.Recall that on a compact Riemannian manifold (Ω, g) with boundary ∂Ω the Dirichlet problem on differential p-forms is given by ∆ f ω = λ p,f ω on Ω ω = 0 on ∂Ω. (3.13) We now take A = ∆ f and B k a function on Ω, which we will denote by G, in Inequalities (3.11) and (3.12).We follow closely the computations done in [IM].Throughout the proof we choose an orthonormal frame {e i } i=1,...,n of T M such that ∇e i = 0 at a fixed point.First, using Equation (2.6), we compute Here, we recall that ∆ f G = ∆G + g(df, dG).Therefore, we get that In the last equality, we use the fact that ∆ f G 2 = 2G∆ f G − 2|dG| 2 which can be shown by a straightforward computation.Also, we have that In the following, we will assume that the manifold Ω is a domain in a complete Riemannian manifold (M n , g) that is isometrically immersed into the Euclidean space R n+m endowed with its canonical metric.We choose in the above formulas the function G to be G = x l , l = 1, . . ., n + m the components of the immersion X = (x 1 , . . ., x n+m ).Recall that (∂x l ) T = d(x l • ι) where ι is the isometric immersion.In addition, ω = ω i are eigenforms of the problem (3.13) associated to the eigenvalues λ i,p,f that are chosen to be of L 2 -norm equal to 1.The computation done in (3.14) gives after summing over l that Recall here that n+m l=1 |dx l | 2 = n.By taking G = x l in Equation (3.15) and summing over l yields Here, we used that the mean curvature of the immersion is given by H = 1 n (∆x 1 , . . ., ∆x n+m ).Applying the Bochner-Weitzenböck formula (2.6) to ω i and taking the scalar product with ω i itself, the above equality acquires the form: Inserting the above equations into (3.11) and (3.12) completes the proof of Theorem 1.3.
Proof of Theorem 1.7.First of all, let u be the function given by where ω is a p-form on M and ν is the inward unit normal vector field.Throughout the proof we will denote by ∇ R n+1 the connection on R n+1 and by ∇ the connection on M , as well as for d R n+1 and d.To simplify the notations, we will denote u i 1 ,...,i p+1 by u in the following technical lemma (see [IRS,Lem. 3.1] for p = 1).
Lemma 3.4.Suppose that M is a f -minimal hypersurface of the weighted manifold (R n+1 , g = can, e −f dv g ).Then, we have

Here T [p]
f is defined in the same way as in Section 2.2 on R n+1 .Proof.For any X ∈ T M , we have Here, we used the fact that (∇ R n+1 X ω) T = ∇ X ω, since ω is a differential form on M .Differentiating again with respect to X yields In the last part, we used the Gauss formula and the identity ν ∇ R n+1 X ω = II(X) ω.Now tracing over an orthonormal frame {e i } of T M gives that In the last equality, we used the fact that δII = −ndH and the expression of ∇ * f ∇.Now, combining Equation (4.5) with the Bochner-Weitzenböck formula (2.6) combined with (4.5) that is shown in the next section, the above equality reduces to Now, for a f -minimal hypersurface, we write Hence, by differentiating along a vector X ∈ T M , we get that Therefore, we find f ω + nHII [p] ω.
Also, we have that Hence, plugging these computations into the expression of ∆ f u, we arrive at f ω .The proof of the lemma then follows from the expression of L f .In order to complete the proof of Theorem 1.7, we proceed as in [IRS].Let {ϕ j } be an L 2 (e −f dv g )-orthonormal basis of eigenfunctions of L f associated to λ j (L f ).Let E d be the sum of the first d-eigenspaces of ∆ f : For all l ≥ 2, we consider the following system of equations M u i 1 ,...,i p+1 ϕ 1 dµ f = . . .= M u i 1 ,...,i p+1 ϕ l−1 dµ f = 0, where we recall that u i 1 ,...,i p+1 = ∂x i 1 ∧. . .∂x i p+1 , ν ∧ω .This system consists of n + 1 p + 1 (l−1) homogeneous linear equations in the variable ω ∈ E d .Hence, if we take we can find a non-trivial ω ∈ E d which is orthogonal to the first (l − 1)-eigenfunctions of L f .Therefore, we deduce that Summing over i 1 < . . .< i p+1 , we first have for the l.h.s. that For the r.h.s. of the previous inequality, we use the previous lemma to get that Hence, we deduce that In the last equality, we used the fact that any eigenvalue of the operator T

[p]
f is the sum of p-distinct eigenvalues of Hess R n+1 f .Now, we have that l) .Also, we have that Finally, we deduce that λ l (L f ) ≤ λ d(l),p,f − (p + 1)a + γ M p(p − 1), which finishes the proof of the theorem.

Now as we have that
This proves the first corollary.To prove the second one, we take as before then, we clearly have that d(l 0 ) ≤ b p (M ).Therefore, λ d(l 0 ),p,f = 0.In the case of a self-shrinker, we recall that for f = |X| 2 2 the hypersurface M is f -minimal and that Hess R n+1 f = g = can.Hence with the assumption p(p − 1)γ M ≤ p + 1, we deduce that λ l 0 (L f ) ≤ 0. Finally, we get that the index is at least n + 1 + l 0 by the fact that L f has at least n + 1 eigenvalues equal to −1.This finishes the proof.

Geometric applications of the main results
In this section we will provide several geometric applications of Theorem 1.3 by making explicit choices of the function f .First, we will consider the case when f is the Riemannian distance function in order to compute explicitly the different terms in the statement of Theorem 1.3.To this end, let M n → R n+m be an isometric immersion and Ω a domain in M .Consider the function f : Ω → R given by f 2 , where a is a real positive number.This particular choice of f has important applications in mean curvature flow as described in the introduction of the article.Here, | • | denotes the Euclidean norm in R n+m and X = (x 1 , . . ., x n+m ) are the components of the immersion.In other words, the function f is the square of the distance function from the origin point 0 ∈ R n+m to a point X ∈ Ω.Using the decomposition X = X T + X ⊥ , we have that d R n+m f = aX and therefore, df = aX T .Hence, we deduce that For any Y ∈ T M , we compute where II X ⊥ is defined as in the proof of Theorem 1.1.By tracing (4.2), we deduce that ∆f = a(−n − n(X, H)).
(4.3) Also, plugging (4.2) into the expression of T where II [p] Z is the canonical extension of II Z to p-forms as defined in (3.3).Hence, we deduce In the following, we will bound the term B [p] ω i , ω i in Theorem 1.3 by using the results of [S3].For this, recall that A. Savo shows in [S3, Thm.1] that for any isometric immersion (M n , g) → (N n+m , g), the Bochner operator B [p] on p-forms of M splits as where B [p] ext is the operator defined by Here, {ν 1 , . . ., ν m } is a local orthonormal frame of T M ⊥ , and the operator B [p] res is the operator that satisfies p(n − p)γ N ≤ B [p]  res ≤ p(n − p)Γ N , where γ N and Γ N are respectively a lower bound and an upper bound for the curvature operator of N .Hence for an isometric immersion M → (R n+m , can), we deduce that The result in Corollary 4.1 generalizes the one in [Z1, Thm.1.1] when taking a = 1 2 , α = 2 and Ω = M being a domain in R m .In this case, II = 0 and we get that Moreover, Corollary 4.1 generalizes the result in [Z1, Thm 1.2] when taking for some integer l > 1, the number a = l − 1, α = 2 and (M n , g) = (R n−l × S l (1), , R n−l ⊕ , S l ) where , S l is the standard metric on the unit round sphere S l (1) of curvature 1.Indeed, we take the immersion M ֒→ R n−l × R l+1 where the second fundamental form is given by the matrix 0 0 0 Id .Thus, we have X ⊥ = −ν and nH = trace(II) = lν and, therefore, aX ⊥ + nH = ν.Also, using that II [p] ω = n i=1 e i ∧ II(e i ) ω for any p-form ω, we deduce by a straightforward computation that Hence, we get that Corollary 4.1 also generalizes the one in [CP] on compact self-shrinkers.In this case, for a = 1, we get that In the last part of this section, we will consider the case when f is the distance function from some fixed point in M with respect to the Riemannian metric g on M .For this, fix a point x 0 ∈ M and consider the distance function and ρ x 0 (x) := 1 2 d x 0 (x) 2 .We recall from [Pe] that the distance function d x 0 is smooth in the complement of the cut locus of x 0 and that its gradient is of norm 1 almost everywhere.Thus we will choose a domain Ω ⊂ M such that it is contained in this complement.Let us recall the comparison theorem [Pe] (see also [HaSi]).For this, we consider for any l ∈ R, the function Theorem 4.2.[Pe] Let (M n , g) be a complete Riemannian manifold.
(1) If Ric M ≥ (n − 1)l for some l ∈ R, then for every x 0 ∈ M , the inequalities ∆d x 0 (x) ≥ −H l (d x 0 (x)), and ∆ρ x 0 (x) ≥ −(1 + d x 0 (x)H l (d x 0 (x)), hold at smooth points of d x 0 .Moreover, these inequalities hold on the whole manifold in the sense of distributions.
We will now use Theorem 4.2 to compute the different terms in Theorem 1.3.We begin with the case when Ric M ≥ (n − 1)l for some l ∈ R on the manifold M .By taking f = aρ x 0 where a > 0, we deduce that the inequalities |df | 2 = a 2 d 2 x 0 , and ∆f (x) ≥ −a (1 + d x 0 (x)H l (d x 0 (x)) hold in the sense of distributions.Therefore, we deduce from Theorem 1.3 the following Corollary 4.3.Let X : (M n , g) → (R n+m , can) be an isometric immersion.Assume that Ric M ≥ (n − 1)l for some l ∈ R. Let x 0 be fixed point in M and Ω a domain in the complement of the cut locus of x 0 .Let f = aρ x 0 for some positive real number a > 0. The eigenvalues of the drifting Hodge Laplacian ∆ f acting on p-forms on Ω with Dirichlet boundary conditions satisfy for any k ≥ 1 In the following, we will consider the case when the sectional curvature of M is bounded from below by l 1 and from above by l 2 .We let δ 2 be the number given by Proof.We proceed as in [S1].At any point x ∈ Ω \ {x 0 }, there is an orthonormal frame {e 1 (x), . . ., e n−1 (x), e n = ∇d x 0 (x)} such that ∇ 2 ρ x 0 has the eigenvalues η 1 (x), . . ., η n−1 (x), 1.By the comparison theorem 4.2, we get that for any j = 1, . . ., n − 1 Therefore, we find that |df | 2 = a 2 d 2 x 0 and ∆f ≥ −a(1 + d x 0 H l 1 (d x 0 )).
Recall that the endomorphism T

[p]
f is by definition the sum of p distinct eigenvalues of ∇ 2 f .Hence, for any p-form ω, we get that T ≤ 1 if l 2 > 0. Replacing all these inequalities in Theorem 1.3, we get the result after combining with Equation (4.5) for the curvature term.