About bounds for eigenvalues of the Laplacian with density

Let $M$ denote a compact, connected riemannian manifold of dimension $n\in{\mathbb{N}}$ with positive Ricci curvature. Let $\partial M$ be the boundary of $M$. We assume that $\partial M$ is connected and non-empty. Denote by $g$, ${\mathrm{d}v_g}$, $|M|$ respectively the Riemannian metric on $M$, the associated volume element, the volume of $M$. For simplicity, we also denote by ${\mathrm{d}v_g}$ the volume element for the induced metric on $\partial M$. If $\Delta$ is the Laplace-Beltrami operator on $M$ equipped with the weighted volume form ${\mathrm{d}m}: ={e^{-h}} {\mathrm{d}v_g}$, we are interested in the operator $L_h={e^{-h(\alpha-1)}}\left( {\Delta}\cdot +\alpha g(\nabla h,\nabla\cdot)\right)$, where $\alpha>1$ and $h\in C^2(M)$ are given. The main result in this paper states about the existence of upper bounds for the eigenvalues of the weighted Laplacian $L_h$.


Introduction
Let (M, g) be a compact connected n-dimensional Riemannian manifold. Let h ∈ C 2 (M ) and ρ the positive function define by ρ := e −h . Let dv g , ∆ and ∇ denote respectively, the Riemannian volume measure, the Laplace and the gradient operator on (M, g). The Witten Laplacian (also called drifting, weighted or Bakry-Emery Laplacian) with respect to the weighted volume measure ρdv g is given by ∆ ·+g(∇ h, ∇ ·). We designate by {λ k (ρ, ρ)} k 0 its spectrum under Neumann conditions. Let S k be the set of all k−dimensional vector subspaces of H 1 (M ), the spectrum consists of a non-decreasing sequence of eigenvalues variationally defined by In recent years, the Witten Laplacian received much attention from many mathematicians (see [7], [6], [14], [12], [11], [8], [13], [10] and the references therein), in particularly the classical research topic of estimating eigenvalues.
A. Ndiaye Institut de mathématiques Université de Neuchâtel · Switzerland · Tel.: +41327182800 E-mail: aissatou.ndiaye@unine.ch When h is a constant, the Witten Laplacian is exactly the Laplacian. Another spectrum has a similar characterisation with the one of the Witten laplacian: the spectrum of the Laplacian associated with the metric ρ 2 n g, which is conformal to g. It is natural to denote its spectrum by {λ k (ρ, ρ n−2 n )} k 0 , since the eigenvalues are variationally characterised by In the present work, we are interested in the expanded eigenvalue problem of the Dirichlet energy functional weighted by ρ α , with respect to the L 2 inner product weighted by ρ, where α 0 is a given constant. These eigenvalues are those of the operator L ρ = L h := −ρ −1 (ρ α ∇ u) = e h(α−1) (∆ · + αg(∇ h, ∇ ·)) on M endowed with the weighted volume form dm := ρdv g , under Neumann conditions on the boundary if it is non-empty. The spectrum consists of an unbounded increasing sequence of eigenvalues which are given by for all k 0. As already mentioned, S k is the set of all k−dimensional vector subspaces of H 1 (M ). The particular cases where α = 1 and α = n−2 n correspond to the problems mentioned above.
A main interest is to investigate the interplay between the geometry of (M, g) and the effect of the weights, looking at the behaviour of λ k (ρ, ρ α ), among densities ρ of fixed total mass. The more general problem where the Dirichlet energy functional is weighted by a positive function σ, not necessarily related to ρ is presented by Colbois and El-Soufi in [4].
In the aforementioned paper, Colbois and El-Soufi exhibit an upper bound for the singular case where α = 0 ([4, Cor 4.1]): where C n depends only on the dimension n. Whereas, in [5, Th 5.2], Colbois, El Soufi and Savo prove that, when α = 1, there is no upper bound among all manifolds. Indeed, on a compact revolution manifold, one has λ 1 (ρ, ρ) as large as desired (see [5]). In their work in [9], Kouzayha and Pétiard give an upper bound for λ k (ρ, ρ α ), when α ∈ 0, n−2 n and prove that there is none for λ 1 (ρ, ρ α ) when α runs over the interval n−2 n , 1 . In this work, we treat the remaining cases, that is when α > 1. We prove that there is no upper bound for λ 1 (ρ, ρ α ), in the class of manifolds M with convex boundary and positive Ricci curvature.
Theorem 1 Let α > 1 a given real constant. Let (M, g) be a compact connected n−dimensional Riemannian manifold whose Ricci curvature satisfies Ric > A, for some positive constant A. If M has non-empty and convex boundary ∂M , then there exists a sequence of densities {ρ j } j 2 and j 0 ∈ N, such that Aj, ∀j j 0 .
This inequality provides a lower bound that grows linearly to infinity in j as j → ∞, showing that with respect to these densities, λ 1 (ρ, ρ α ) becomes as large as desired. Unfortunately, I do not know any other way to prove it, than the following long and painful calculation.
Our aim is to show that, there exists a family of densities ρ j = e −h j , j ∈ N, such that their corresponding first non-zero eigenvalues become as large as desired. For this, we use the extended Reilly formula given in Theorem2, to provide a lower bound that grows linearly to infinity in j, as j → ∞.
where D 2 denotes the Hessian tensor, ∇ ∂ the tangential gradient and ∆ ∂ the Laplace-Beltrami operator on ∂M , ∂n is the derivative with respect to the outer unit normal vector to ∂M . The second fundamental form on ∂M is defined by I(X, Y ) := g(∇ X n, Y ), H := trI denotes the mean curvature and Ric is the Ricci curvature on M .
In the next section, we prove these two theorems.

Proof of Theorem1
Let (M, g) be a compact connected n-dimensional Riemannian manifold with a convex boundary. Let h ∈ C 2 (M ) and assume that λ is the first non-zero eigenvalue of L h . Let u = 0 be an eigenfunction with corresponding eigenvalue λ, i.e. u satisfies L h u = λu with the Neumann boundary condition.