Neumann eigenvalues of the biharmonic operator on domains: geometric bounds and related results

We study an eigenvalue problem for the biharmonic operator with Neumann boundary conditions on domains of Riemannian manifolds. We discuss the weak formulation and the classical boundary conditions, and we describe a few properties of the eigenvalues. Moreover, we establish upper bounds compatible with the Weyl's law under a given lower bound on the Ricci curvature.

We recall that in the case M = R n with the Euclidean metric, problem (1.1) is wellknown and has increasingly gained attention in recent years. We refer to [3,8,9,11,12,13,29,35,36,37] for the eigenvalue problem and to [39] for the corresponding boundary value problem. Problem (1.1) in dimension two represents Kirchhoff's solution to the problem of describing the transverse vibrations of a thin elastic plate with free edges. We refer to [7,22,31,32] for more details and for historical information.
We also note that the corresponding Dirichlet problem for the biharmonic operator, which for planar domains is related to the study of the transverse vibrations of a thin elastic plate with clamped edges [20], has been extensively studied not only in the Euclidean setting, but also for domains in Riemannian manifolds, see e.g., [16,40,41]. In particular, the Dirichlet problem on Euclidean domains and the analogous problem on Riemannian manifolds share many properties which can be derived by using similar arguments.
On the other hand, we have not been able to find the analogue of the biharmonic Neumann problem on Riemannian manifolds in the literature. The first aim of the present paper is to introduce problem (1.1) on domains of Riemannian manifolds in a suitable way, derive the boundary conditions as well as the variational formulation. The problem which we obtain turns out to be the genuine generalization of the biharmonic Neumann problem for Euclidean domains. We remark that the standard technique used to derive the boundary conditions and the variational formulation of problem (1.1) in the Euclidean case it to multiply the eigenvalue equation ∆ 2 u = µu by a test function φ ∈ C ∞ (Ω), integrate the resulting equality over Ω and perform suitable integrations by parts. Computations become easy since we can exchange the order of partial derivatives. This is no more possible in the case of Riemannian manifolds, hence we have to take a longer path, described in Subsection 3.1. An essential tool is Reilly's identity. It turns out that the strategy described in Subsection 3.1 allows to define alternatively problem (1.1) also in the Euclidean case.
If Ω is a bounded domain of R n it is known that the eigenvalues are non-negative and satisfy the Weyl's asymptotic law where ω n denotes the volume of the unit ball in R n and |Ω| denotes the Lebesgue measure of Ω, see e.g., [29]. An important question regarding the eigenvalues of Neumann-type problems is that of finding upper bounds which are compatible with the Weyl's law. One of the main purposes of the present paper is that of finding upper bounds for µ j which can be compared with (1.2) and which contain the correct geometric information.
In the case of Euclidean domains, Weyl-type upper bounds for µ j are well-known and are of the form (1.3) µ j ≤ A n j |Ω| 4 n with A n = 4+n 4 4/n 16π 4 ω 4/n n , see e.g., [29]. The proof in [29] is in the spirit of the analogous result of Kröger for the Neumann eigenvalues m j of the Laplacian on Euclidean domains, namely where B n = 2+n 2 2/n 4π 2 ω 2/n n , see [28]. , as j → +∞, The proofs of (1.3) and (1.4) rely on harmonic analysis techniques and are hardly adaptable to the case of manifolds.
In the case of the eigenvalues of the Laplacian on manifolds, one of the first result in this direction is presented in [10], where it is proved that Here m j denote, with abuse of notation, the eigenvalues of the Laplacian on a compact manifold (without boundary) M with Ric ≥ −(n − 1)κ 2 , κ ≥ 0. Results on domains have been obtained more recently. In [18] the authors prove the following upper bound for the Neumann eigenvalues m j of the Laplacian on a domain Ω of a complete Riemannian manifold with Ric ≥ −(n − 1)κ 2 , κ ≥ 0. The authors adopt a metric approach for the proof of (1.7). We will use this approach also in the present paper in order to obtain upper bounds for µ j . In view of (1.2), (1.5) and (1.7), it is natural to conjecture that the inequality holds for any bounded domain of a complete Riemannian manifold with Ric ≥ −(n−1)κ 2 , κ ≥ 0. We are able to prove (1.8) for certain classes of domains and manifolds. In particular we prove (1.8) for domains of manifolds with non-negative Ricci curvature and n = 2, 3, 4 (see Theorem 5.11), for domains of the standard sphere (see Theorem 5.15) and of the standard hyperbolic space (see Theorem 5.17), for boundaryless manifolds (see Corollary 5.21), and for convex domains (see Corollary 5.23).
In the general case, we are able to prove an estimate of the form (1.9) µ j ≤ A n j |Ω| 4 n + C(g) (see Theorem 5.7), where C(g) has an explicit form and depends on κ, r inj,Ω , |∂Ω|, where r inj,Ω is the injectivity radius relative to Ω (see (5.3) for the definition) and |∂Ω| is the n − 1-dimensional Hausdorff measure of the boundary. Estimate (1.9) is improved if we put additional hypothesis on Ω and M . In particular we provide more refined estimates in the case of domains with sufficiently small diameter in manifolds with non-negative Ricci curvature (see Theorem 5.14) and in the case of Cartan-Hadamard manifolds (see Theorem 5.18).
It is important to remark that a bound of the form (1.9) is good in the sense that the geometry of the domain and of the manifold appears as an additive constant in front of the term encoding the asymptotic behavior, which has the correct form compared with the asymptotic law (1.2).
As already mentioned, in order to prove the upper bounds, we adopt a metric approach. In particular, we exploit a result of decomposition of a metric measure space by disjoint capacitors, see [23,25] for more details, see also [17,18]. Namely, given a domain Ω, we find, for each j ∈ N, a family of j disjointly supported sets A i , i = 1, ..., j, in Ω with sufficient volume. Associated to each set A i we build a function u i to test in the min-max formula for the eigenvalues (see formula (3.33)). Since the u i are disjointly supported, from (3.33) we deduce that it is sufficient to bound the Rayleigh quotient of each u i in order to upper bound µ j . Hence the functions u i have to be constructed in a proper way.
We remark that test functions for the biharmonic operator need to belong to the standard Sobolev space H 2 (Ω). Usually, test functions are built in terms of distance-type functions, which are Lipschitz, but are not in general H 2 (Ω) functions. In particular, test functions for the Laplacian eigenvalues are cut-off functions which are just Lipschitz regular. The application of the technique used in [18] for the Laplacian is no straightforward in our situation, in fact it is notoriously a difficult task to build cut-off functions enjoying precise estimates for first and second derivatives, see e.g., [5,15,24]. We pay the price of the fact that we need cut-off functions in H 2 (Ω) with well-behaved gradient and Laplacian by introducing into the estimates the quantities r inj,Ω and |∂Ω|. Getting rid of these quantities in the general case seems a very difficult issue.
Looking at (1.2) and (1.5), one may wonder if there is some sort of relationship between µ j and m 2 j and if it is possible, in general, to recover upper estimates for µ j from upper estimates on m j . The answer is negative in general, in fact we provide examples showing that the ratio µj m 2 j may be made arbitrarily large or arbitrarily close to zero.
Another interesting feature of problem (1.1) is that it is possible to produce negative eigenvalues. This does not happen with the eigenvalues of the biharmonic Dirichlet problem on domains of manifolds. In particular, in subsection 4.3 we prove that any domain of the standard hyperbolic space H n admits at least n negative eigenvalues. Moreover, we prove that there exist domains with an arbitrarily large number of negative eigenvalues, the absolute value of which can be made arbitrarily large. On the other hand, for domains in manifolds with positive Ricci curvature we prove a lower bound for the eigenvalues µ j in terms of m j , η j and a lower bound on the Ricci curvature (see Theorem 4.3), where η j denote the eigenvalues of the rough Laplacian on Ω.
The present paper is organized as follows. In Section 2 we recall some preliminaries and introduce the notation. In Section 3 we describe the classical Neumann boundary conditions in (1.1) and derive the weak formulation of the problem, proving that it is wellposed and characterizing its spectrum. In Section 4 we discuss a few properties of the eigenvalues. In particular we provide examples where the ratio µj m 2 j can be made arbitrarily large or close to zero. We prove that any domain of the hyperbolic space admits at least n negative eigenvalues, and that there exists domains with an arbitrarily large number of negative eigenvalues with arbitrarily large absolute value. We also prove a lower bound for µ j for domains on manifolds with positive Ricci curvature. In Section 5 we recall the main technical results of decomposition of a metric measure space by capacitors, which allow to prove the upper estimates for the eigenvalues µ j presented in the same section.

Preliminaries and notation
Let (M, g) be a complete n-dimensional smooth Riemannian manifold. For a bounded domain Ω in M , by L 2 (Ω) we denote the space of measurable functions f on Ω such that The Sobolev space H 2 (Ω) is the completion of C ∞ (Ω) with respect to the norm The space L 2 (Ω) is a Hilbert space when endowed with the standard scalar product The space H 2 (Ω) is a Hilbert space when endowed with the standard scalar product which induces the norm (2.1).
The space H 2 0 (Ω) is the completion of C ∞ c (Ω) with respect to (2.1), there C ∞ c (Ω) is the space of functions in C ∞ (Ω) compactly supported in Ω. We refer to [26] for an introduction to Sobolev spaces on Riemannian manifolds.
We denote here by dv the Riemannian volume element of M and by dσ the induced n − 1-dimensional volume element of ∂Ω.
Through all the paper, we denote by ·, · the inner product on the tangent spaces of M associated with the metric g, and, with abuse of notation, we shall denote by ·, · also the induced metric on ∂Ω. Let ∇, D 2 and ∆ denote the gradient, the Hessian and the Laplacian on M , respectively, and let ∇ ∂Ω , div ∂Ω and ∆ ∂Ω denote the gradient, the divergence and the Laplacian on ∂Ω with respect to the induced metric, respectively. We denote by ν the outer unit normal to ∂Ω. The shape operator of ∂Ω, denoted by S, is defined for any X ∈ T ∂Ω as S(X) := ∇ X ν, where ∇ X ν is the covariant derivative of ν along a vector field X. The second fundamental form of ∂Ω, denoted by II(X, Y ), is defined as II(X, Y ) := S(X), Y for all X, Y ∈ T ∂Ω. We recall that the eigenvalues of S are the principal curvatures of ∂Ω. We will denote by H := 1 n−1 trS = 1 n−1 divν | ∂Ω the mean curvature of ∂Ω. Let Ric(·, ·) denote the Ricci tensor of M . Finally, for an open set E ∈ M we denote by |E| the standard Lebesgue measure of E. For a closed set G ∈ M of finite n − 1-dimensional Hausdorff measure, we still denote by |G| the n − 1-dimensional Hausdorff measure of G.
We recall Bochner's formula: holding pointwise for smooth functions f on Ω. It is also useful to recall Reilly's formula, see [38]: holding for smooth functions f on Ω.
We also recall Green's identity for smooth functions f, g : We recall that for any smooth vector field F on T ∂Ω and any function f defined on ∂Ω, that is, the divergence operator is the adjoint of the gradient. In particular, (2.7) holds with ∂Ω replaced by any complete smooth (boundaryless) Riemannian manifold (M, g), and div ∂Ω , ∇ ∂Ω replaced by the divergence and gradient on M , respectively.
Finally, by N we denote the set of positive natural numbers.
3. The eigenvalue problem for the biharmonic operator with Neumann boundary conditions In this section we describe the classical Neumann boundary conditions in (1.1), as well as the weak formulation of the problem. This is done in Subsection 3.1.
Then we prove that problem (3.1) is well-posed under suitable assumptions on the domain, and admits an increasing sequence of eigenvalues of finite multiplicity bounded from below and diverging to +∞. This is done in Subsection 3.2.
3.1. Classical Neumann boundary conditions and weak formulation. We consider the following variational problem: in the unknowns u ∈ H 2 (Ω) and µ ∈ R. Problem (3.1) is the variational (weak) formulation of problem (1.1), as stated in the following theorem. We remark that it is not straightforward to recognize the left-hand side of (3.1) as the right quadratic form for an eigenvalue problem for the biharmonic operator with Neumann boundary conditions. One would like to take the simpler quadratic form Ω ∆u∆φdv, which however provides an ill-defined problem, see Remark 3.3. Actually, we will prove that (3.1) is the weak formulation of the following eigenvalue problem: in Ω, (n − 1)H ∂u ∂ν + ∆ ∂Ω u − ∆u = 0, on ∂Ω, ∆ ∂Ω ∂u ∂ν − div ∂Ω S(∇ ∂Ω u) + ∂∆u ∂ν = 0, on ∂Ω, in the unknowns u (the eigenfunction) and µ (the eigenvalue). Then, we will show that the two boundary conditions in (3.2) coincide with those of (1.1), namely we will prove the following lemma.
Proof of Theorem 3.1. Assume that a function u ∈ C 4 (Ω) ∩ C 3 (Ω) and a real number µ are solution of the eigenvalue equation We multiply both sides of (3.5) by a function φ ∈ C ∞ and integrate over Ω, obtaining thanks to (2.6) We set, for a function f ∈ C 2 (Ω) We note that Reilly's formula (2.5) implies that , thus from (3.9) and (3.10) we deduce that Using (3.12) in (3.6), we can re-write (3.6) as (3.13) We note now that (3.14) where the second equality follows from (2.7), and that Thanks to (3.14) and (3.15), (3.13) can be rewritten as follows (3.16) Assume now that the function u satisfies the boundary conditions in (3.2). Then From the definition of H 2 (Ω) we deduce the validity of (3.1).
We prove now Lemma 3.2 Proof of Lemma 3.2. We start by proving (3.3).
is a orthonormal frame of ∂Ω and E n = ν is the outward unit normal to ∂Ω. For a Lipschitz vector field F in a neighborhood of ∂Ω, we denote by F ∂Ω := Note that where we have used the fact that ∇ ν F ∂Ω , ν = 0. Moreover, and that, by definition we immediately obtain from (3.18) the following identity This proves (3.3). We prove now (3.4). Let us consider the second boundary condition in (3.2). We need to show that which can be re-written as Actually we will prove that We note that for any vector fiel X ∈ T M ∇ ( ∇u, ν ) , X = ∇ X ∇u, ν + ∇ X ν, ∇u = ∇ ν ∇u, X + ∇ ∇u ν, X , since D 2 u and ∇ν are symmetric. Thus ∇ ( ∇u, ν ) = ∇ ν ∇u + ∇ ∇u ν. We have then In fact ∇ ν ν = 0 and ∇ ∇ ∂Ω u ν, ν = 0. This proves (3.4). The proof is now concluded.
Since we will be interested in the variational problem (3.1), we can relax the hypothesis on the smoothness of Ω. A sufficient condition for the solvability of (3.1) is, e.g., that Ω is of class C 1 , see Subsection 3.2.  in the unknowns u ∈ H 2 (Ω), µ ∈ R. We note that this problem is not well-posed: it is immediate to see that all harmonic functions in H 2 (Ω) are eigenfunctions corresponding to the eigenvalue µ = 0 . This is due to the fact that the quadratic form (3.19) is not coercive in H 2 (Ω), indeed we can add to the quadratic form (3.19) a term γ Ω uφdv with γ > 0 arbitrarily large and obtain a scalar product whose induced norm is not equivalent to the standard one of H 2 (Ω), see also Lemma 3.4. In [37] it is proved that (3.20) has an infinite kernel consisting of all harmonic functions in H 2 (Ω). Moreover, if we rule out the kernel, problem (3.20) admits an increasing sequence of positive eigenvalues of finite multiplicity which coincide with the Dirichlet eigenvalues of the biharmonic operator. It is not difficult to adapt the results of [37] to the case of domains in a Riemannian manifold. The classical formulation of problem (3.20) reads We remark that Neumann boundary conditions are usually called "natural boundary conditions" and in a certain sense arises from "solving a variational problem on the largest possible energy space", which in this case is H 2 (Ω). In this space, problem (3.20) is evidently not well posed.
We also remark that the situation is completely different if we impose Dirichlet boundary conditions, namely if we consider problem in the unknowns u (the eigenfunction) and Λ (the eigenvalue). In this case, the corresponding weak formulation is in the unknowns u ∈ H 2 0 (Ω), Λ ∈ R. In this case boundary conditions are no more "natural" but are "imposed" with the choice of the energy space H 2 0 (Ω). Actually, problem (3.23) can be written in the form (3.1) with the space H 2 (Ω) replaced by H 2 0 (Ω). In fact, it is easy to see that (3.24) for all u, φ ∈ H 2 0 (Ω), see (3.6) and (3.16). It turns out that (3.1) and (3.20) are equivalent in H 2 0 (Ω). The situation is similar if Ω = M is a compact complete (boundaryless) smooth Riemannian manifold. In this case H 2 (M ) = H 2 0 (M ) (see [26]), hence (3.24) holds for all u, φ ∈ H 2 (M ). Thus, the weak formulation of the biharmonic closed problem on M is fairly simple, and actually it turns out that the eigenvalues of the biharmonic operator on M coincide with the squares of the eigenvalues of the Laplacian on M , the eigenfunctions being the same. We refer to Subsection 5.8 for more details.
Finally, we remark that one can also consider the variational problem in the unknowns u ∈ H 2 (Ω), µ ∈ R. As it is done in Subsection 3.2 it is possible to prove that problem (3.25) is well-posed and admits an increasing sequence of nonnegative eigenvalues of finite multiplicity. However, it is not always possible to recover an eigenvalue problem of the form (1.1) starting from a smooth solution of (3.25) as in the proof of Theorem 3.1, except for few particular cases. In fact, by following the proof of Theorem 3.1, we are left to deal with the term Ω Ric(∇u, ∇φ)dv, and we would like to have an identity of the form for suitable differential operators L, B 1 , B 2 . It is not in general possible to have explicit form for L, B 1 , B 2 (they exist by Riesz Theorem). If (M, g) is an Einstein manifold, that is, Ric = Kg, then L(u) = −K∆u, B 1 (u) = K ∂u ∂ν and B 2 (u) = 0. Thus, any smooth solution of (3.25) solves in Ω, Problem (3.26) contains lower order terms in the eigenvalue equation and in the second boundary condition.

3.2.
Neumann eigenvalues of the biharmonic operator. We prove here that, under suitable hypothesis on Ω, problem (3.17) admits an increasing sequence of eigenvalues of finite multiplicity bounded from below and diverging to +∞. To do so we recast problem (3.1) into an eigenvalue problem for a compact self-adjoint operator acting on a Hilbert space. First we note that (3.1) can be re-written as where γ ∈ R is fixed, in the unknowns u ∈ H 2 (Ω) and Γ ∈ R. Clearly a pair (u, µ) ∈ H 2 (Ω) × R is a solution of (3.1) if and only if the pair (u, µ + γ) ∈ H 2 (Ω) × R is a solution of (3.27). We will study the eigenvalue problem in the equivalent formulation (3.27) for suitable choices of γ.
We consider on H 2 (Ω) the bilinear form with γ > 0. We denote by H 2 (Ω) the space H 2 (Ω) endowed with the form (3.28). We also set We state the following lemma, whose proof we postpone at the end of this section. Through all this subsection, we fix once and for all a positive number γ > γ 0 , where γ 0 is as in Lemma 3.4.
Then we define the operator P as an operator from H 2 (Ω) to its dual H 2 (Ω) ′ by setting By the Riesz Theorem it follows that P is surjective isometry. Then we consider the If the embedding H 2 (Ω) ⊂ L 2 (Ω) is compact, then the operator J is compact. Finally, we set If J is compact, since P is bounded, then also T is compact. Moreover for all u, φ ∈ H 2 (Ω). Hence T is self-adjoint. Note that Ker T = Ker J = {0} and the non-zero eigenvalues of T coincide with the reciprocals of the eigenvalues of (3.27), the eigenfunctions being the same. If Ω is of class C 1 , then the embeddings H 2 (Ω) ⊂ H 1 (Ω) ⊂ L 2 (Ω) are compact (see e.g., [4, § 2]).
We are now ready to prove the following theorem.
Theorem 3.5. Let (M, g) be a smooth n-dimensional Riemannian manifold and let Ω be a bounded domain in M with C 1 boundary. Then the eigenvalues of (3.1) have finite multiplicity and are given by a non-decreasing sequence of real numbers {µ j } ∞ j=1 bounded from below defined by where each eigenvalue is repeated according to its multiplicity. Moreover, there exists a Hilbert basis of {u j } ∞ j=1 of H 2 (Ω) of eigenfunctions u j associated with the eigenvalues µ j . By normalizing the eigenfunctions with respect to (3.29), then uj √ µj +γ ∞ j=1 define a Hilbert basis of L 2 (Ω) with respect to its standard scalar product.
Proof. By the Hilbert-Schmidt Theorem applied to the compact self-adjoint operator T it follows that T admits an increasing sequence of positive eigenvalues {q j } ∞ j=1 , bounded from above, converging to zero and a corresponding Hilbert basis {u j } ∞ j=1 of eigenfunctions of H 2 (Ω). Since q = 0 is an eigenvalue of T if and only if µ = 1 q − γ is an eigenvalue of (3.1) with the same eigenfunction, we deduce the validity of the first part of the statement. In particular, formula (3.33) follows from the standard min-max formula for the eigenvalues of compact self-adjoint operators.
To prove the final part of the theorem, we recast problem (3.27) into an eigenvalue problem for the compact self-adjoint operator T ′ = i • P (−1) • J ′ , where J ′ denotes the map from L 2 (Ω) to the dual of H 2 (Ω) defined by and i denotes the embedding of H 2 (Ω) into L 2 (Ω). We apply again the Hilbert-Schmidt Theorem and observe that T and T ′ admit the same non-zero eigenvalues, and that the eigenfunctions of T ′ can be chosen in H 2 (Ω) and coincide with the eigenfunctions of T . From (3.27) we deduce that the normalized eigenfunction u j of T with respect to (3.29), divided by √ µ j + γ, form a orthonormal basis of L 2 (Ω). This concludes the proof.
We prove now Lemma 3.4.

Proof of Lemma 3.4.
It is easy to see that there exists C > 0 (possibly depending on Ω, M and γ) such that for any u ∈ H 2 (Ω) in fact we can trivially take C = max 1, Ric L ∞ (Ω) , γ .
We prove now the opposite inequality possibly re-defining the constant C. We note that for ε ∈ (0, 1) Hence, in order to prove (3.35) is is sufficient to prove that for any fixed B > 0 there exists a constant A > 0 such that We argue by contradiction and assume that such constant does not exists. We find a sequence We normalize the functions u k by setting Ω |∇u k | 2 dv = 1. Hence Passing to a subsequence, we have that u k ⇀ū in H 2 (Ω) as k → +∞ (we have re-labeled the elements of the subsequence as u k ) and u k →ū in H 1 (Ω) by the compactness of the embedding H 2 (Ω) ⊂ H 1 (Ω). Hence Ω |∇ū| 2 dv = lim k→+∞ Ω |∇u k | 2 dv = 1 and Ωū 2 dv = lim k→+∞ Ω u 2 k dv = 0. Then we have found a functionū ∈ H 2 (Ω) such that Ω |∇ū| 2 dv = 1 and Ωū 2 dv = 0, a contradiction. This concludes the proof of (3.35) and of the lemma.

A few properties of Neumann eigenvalues
In this section we investigate a few properties of the eigenvalues µ j of problem (3.1). In particular we study the behavior of the ratio µj m 2 j , where m j are the Neumann eigenvalues of the Laplacian on Ω. In fact, in view of the asymptotic laws (1.2) and (1.5), it is natural to compare µ j with m 2 j . In particular we show that this ratio can be arbitrarily large or arbitrarily close to zero. We denote by 0 = m 1 < m 2 ≤ · · · ≤ m j ≤ · · · ր +∞ the Neumann eigenvalues of the Laplacian on Ω, which are given by Here H 1 (Ω) denotes the closure of C ∞ (Ω) with respect to the norm Ω |∇u| 2 + u 2 dv. We also consider the sign of the eigenvalues, proving that in some situations negative eigenvalues may appear. As a consequence we also provide examples where the ratio µj m 2 j can be made negative and with arbitrarily large absolute value. In order to produce suitable examples, we restrict our analysis to the Euclidean space, to manifolds with Ric ≥ (n − 1)K > 0 and to the standard hyperbolic space H n . 4.1. Domains of the Euclidean space. Through this subsection (M, g) is the standard Euclidean space R n . It is well-known that if Ω is a bounded Lipschitz domain, then 0 = µ 1 = µ 2 = · · · = µ n+1 < µ n+2 ≤ · · · ≤ µ j ≤ · · · ր +∞, and the eigenspace corresponding to the eigenvalue µ = 0 is generated by {1, x 1 , ..., x n }, see e.g., [37].
We have the following theorem.
Proof. The domains providing the result are obtained by connecting with thin junctions a fixed domain Ω to N domains Ω 1 , ..., Ω N , N ∈ N, of fixed volume and disjoint from Ω, and by letting the size of the junctions go to zero. We will prove the theorem for n = 2 and N = 1.
, where in the union the eigenvalues have been ordered increasingly, see e.g., [2]. In particular, m ε 1 = 0, m ε 2 → 0 as ε → 0 + , and m ε j are uniformly bounded from below by some positive constant independent on ε for j ≥ 3.
Let now µ ε j denote the eigenvalues of (3.1) in Ω ε . We prove that µ ε j ≤ Cε for j ≤ 6, where C > 0 does not depend on ε.
To do so, let φ L (x 1 , These functions are linearly independent and belong to H 2 (Ω ε ). Moreover, any u in the space generated by u 1 with C independent of ε. This implies from (3.33) that µ ε j ≤ Cε for j ≤ 6. The proof is now complete in the case n = 2 and N = 1.
The proof for n > 2 and N > 1 is a standard adaptation of the arguments above.
Remark 4.2. Let us consider a domain Ω ε,N as in the proof of Theorem 4.1. Such a domain is usually called a N + 1-dumbbell. We observe that if N > n, we have that m j , µ j → 0 as ε → 0 + for n + 2 ≤ j < N + 2. It is well-known that m 1 = 0 and m j = O(ε n+1 ) as ε → 0 + for 2 ≤ j ≤ N + 1, see e.g., [1,27]. Moreover, it is possible to show that in the case of a sufficiently regular N + 1-dumbbell domain Ω ε,N , µ j → 0 as ε → 0 + for n + 2 ≤ j ≤ (N + 1)(n + 1), and µ (N +1)(n+1)+1 is bounded away from zero, uniformly in ε. We refer to [3] for the proof in the case N = 1. Thanks to this fact, with the same arguments of [1] (see also [27]) it is possible to prove that µ j = O(ε n−1 ) as ε → 0 + for n + 2 ≤ j ≤ (N + 1)(n + 1). We omit the details of the computations which are standard but quite technical and go beyond the scopes of the present article. Anyway, we have that

Domains in manifolds with
Ric ≥ (n − 1)K > 0. Through all this subsection (M, g) will be a complete n-dimensional smooth Riemannian manifold with Ric ≥ (n − 1)K > 0. We note that for any domain Ω of class C 1 of M we have µ 1 = 0 and µ 2 > 0. In fact, from (3.33) we immediately deduce that µ j ≥ 0 for all j ∈ N and that µ 1 = 0 is an eigenvalue with corresponding eigenfunctions the constant functions. Constant functions are the only eigenfunctions associated with µ 1 = 0. In fact, any eigenfunction u corresponding to the eigenvalue µ = 0 satisfies the eigenvalues of the rough Laplacian on Ω with Neumann boundary conditions. They are characterized by where H 1 (Ω) is the space of 1-forms of class H 1 (Ω), see e.g., [19] for more information on the eigenvalues of the rough Laplacian. We have the following.
Proof. The inequality is trivially true for j = 1. Hence we assume j ≥ 2. We observe that for any non-constant u ∈ H 2 (Ω) Hence, for any subspace This implies where in the last inequality we have used the fact that H 2 (Ω) ⊂ H 1 (Ω), hence the minimum decreases. The proof is now complete.
with equality if and only if Ω is isometric to an n-dimensional Euclidean hemisphere of curvature K, see e.g., [21]. This result is in the spirit of the well-known Obata-Lichnerowicz inequality, see [14,30,33]. Hence, for any bounded domain Ω with II ≥ 0 we have from Theorem 4.3 that It is natural to conjecture that Open problem. Prove (4.5).
Thanks to Theorem 4.3 we have the following inequality for all j ∈ N, j ≥ 2 We recall now that for any N ∈ N there exists a sequence {Ω ε,N } ε∈(0,ε0) such that m j ≤ Cε for all j ≤ N . These domains are obtained by connecting to a fixed domain Ω, N domains of fixed volume and disjoint from Ω with thin junctions, and by letting the width of the channels, represented by the parameter ε > 0, go to zero. This is a standard construction (see [1,2], see also Subsection 4.1). This implies the validity of the following theorem. On the other hand, if Ω is such that the second fundamental form of its boundary is non-negative, that is, II ≥ 0, we have that We refer to Subsection 5.9 for the proof of (4.8).

4.3.
Domains of the hyperbolic space. Given an eigenvalue µ of (3.1) and a corresponding eigenfunction u µ ∈ H 2 (Ω), we have It is well-known, from Bochner's formula (2.4) and integration by parts, that, for any u ∈ H 2 0 (Ω) Note that for a complete, compact (boundaryless) smooth Riemannian manifold M In view of this, the natural question arises whether the biharmonic Neumann eigenvalues µ j can be negative or not. Clearly, a necessary condition for the appearance of negative eigenvalues is that Ric ≥ 0. In this section we consider domains of the standard hyperbolic space H n . We have the following theorem. Then Ω admits at least n strictly negative eigenvalues. Proof. We start by proving the result for n = 2. To do so, we will use Fermi coordinates for H 2 . In this case the metric is given by g(x, y) = dx 2 + cosh 2 (x)dy 2 .
The proof for n ≥ 3 is similar, however we shall highlight only the main differences, omitting the standard but quite long analogous computations.
The main tool in order to prove the result for n ≥ 3 is the Borsuk-Ulam Theorem which states that if g : S n → R n is an odd function (that is, g(p) = −g(−p) where −p is the antipodal point to p in S n ), then there exists p ∈ S n such that g(p) = 0.
Let q ∈ Ω and let H 1 be an hyperplane containing q. Let f 1 be the signed distance from H 1 . Then |D 2 f 1 | 2 + Ric(∇f 1 , ∇f 1 )dv < 0. The proof is analogous to that of the case n = 2. It follows by explicit computations in Fermi coordinates (x 1 , ..., x n ) where x 1 represents the signed distance from H 1 and (x 2 , ..., x n ) are normal coordinates on H 1 = H n−1 (in this system q = (0, ..., 0)). Therefore µ 1 < 0 with u 1 a corresponding eigenfunction.
Let π 2 be a fixed plane in T q H n and let v 1 ∈ π 2 be a non-zero vector. For θ ∈ [0, 2π), let H θ be the hyperplane whose tangent space at q is normal to the vector v θ ∈ π 2 , where v θ is a unit vector which forms with v 1 an angle of width θ in π 1 . Let f θ be the signed distance from H θ . The Rayleigh quotient of this function is again strictly negative. Moreover, we have that f 0 = −f π and if we define ρ 1 (θ) := Ω f θ u 1 dv, we find out that there exists θ ∈ [0, 2π) such that ρ 1 (θ) = 0. Thus we deduce the existence of a function with strictly negative Rayleigh quotient orthogonal to u 1 . As in the case n = 2, we deduce that µ 2 < 0. Assume now that we have µ 1 , ..., µ k < 0, with k < n, and with associated eigenfunctions u 1 , ..., u k . We prove that µ k+1 < 0.
This concludes the proof.
Let N ∈ N be fixed. Consider the domain D δ,N L . Let the points p 1 , ..., p N ∈ γ be given by p i = L i − 1 2 , 0, ..., 0 for i = 1, ..., N and let B i := B p i , L 4 and 2B i := B p i , L 2 . The balls 2B i are disjoint. Associated with each 2B i we define cut-off functions φ i by The functions u i := x k φ i , i = 1, ...N (for some k = 2, ..., n) are N disjointly supported functions in H 2 (D δ,L ), hence from (3.33) we deduce that The same computations above show that the Rayleigh quotient of each u i is bounded above by − C δ 2 , hence we have N negative eigenvalues with arbitrary large absolute value. By choosing N L = 1 δ n−1 we have also that |D δ,N L | = O(1) as δ → 0 + . This concludes the proof. Remark 4.9. A consequence of Theorem 4.8 is that we can always locally perturb a fixed domain of the hyperbolic space H n in order to obtain an arbitrary large number of negative eigenvalues with arbitrary large absolute value. This is done exactly as for the Neumann Laplacian, in which case we can deform locally a domain in order to have an arbitrary large number of eigenvalues arbitrarily close to zero. Indeed, it is sufficient to join to the domain a sufficient number of small balls by sufficiently thin junctions.
Remark 4.10. A natural problem is to find classes of domains of the hyperbolic space which have exactly n negative eigenvalues. A first immediate conjecture is that hyperbolic balls admit exactly n negative eigenvalues. However, a simple proof of this fact is currently unavailable and we leave this as an open question. Nevertheless, it is possible to prove the result in the case of sufficiently small balls. In fact, from Remark 4.7 we deduce that if η n+1 > 1, where η n+1 is the n + 1-th eigenvalue of the rough Laplacian on Ω, then Ω admits exactly n negative eigenvalues. Using normal coordinates with origin in the center of a ball B ε of radius ε it is possible to prove by means of explicit computations and max-min formula for the eigenvalues that η n+1 → +∞ as ε → 0 + . Thus we deduce the existence of ε 0 > 0 such that every ball of radius smaller than ε 0 admits exactly n negative eigenvalues.
Open problem. Prove that hyperbolic balls admit exactly n negative eigenvalues.

Upper estimates for eigenvalues
In this section we provide upper bounds for the eigenvalues µ j of (3.1) which are compatible with Weyl's law (1.2). As we have highlighted in Section 4, there is in general no monotonicity between the eigenvalues µ j and the squares of the Neumann eigenvalues of the Laplacian m j , hence in general there is no hope to recover upper bounds for µ j from the known upper bounds on m j . We remark that the situation is very different if we think of Dirichlet eigenvalues of the Laplacian and the biharmonic operator. Indeed, if we denote by λ j and by Λ j the eigenvalues of the Laplace and biharmonic operator respectively, with Dirichlet boundary conditions on a domain Ω of a complete n-dimensional smooth Riemannian manifold, then λ 2 j ≤ Λ j , for all j ∈ N. This is an immediate consequence of the min-max principle for λ j and Λ j . Hence, lower bounds for Λ j can be obtained from lower bounds on λ j . 5.1. Decomposition of a metric measure space by capacitors. In this subsection we present the main technical tools which will be used to prove upper bounds for eigenvalues. We start with some definitions.
We denote by (X, dist, ς) a metric measure space with a metric dist and a Borel measure ς. We will call capacitor every couple (A, D) of Borel sets of X such that A ⊂ D. By an annulus in X we mean any set A ⊂ X of the form where a ∈ X and 0 ≤ r < R < +∞. By 2A we denote Moreover, for any F ⊂ X and r > 0 we denote the r-neighborhood of F by F r , namely We recall the following metric construction of disjoint capacitors from [23]. As we shall see, Theorem 5.1 is not easy to use in many concrete cases, e.g., when X is a domain in the standard hyperbolic space H n and ς is the restriction of the Lebesgue measure on X. In fact, hypothesis i) fails to hold with Γ depending only on the dimension because of the exponential growth of the volume of balls. We state now the following lemma, which improves [17, Lemma 4.1]. We postpone its proof at the end of this subsection.
Lemma 5.2. Let (X, dist, ς) be a compact metric measure space with a finite measure ς. Assume that for all s > 0 there exists an integer N (s) such that each ball of radius 5s can be covered by N (s) balls of radius s. Let β > 0 satisfying β ≤ ς(X) 2 and let r > 0 be such that for all x ∈ X ς(B(x, r)) ≤ β 2N (r) .  A clever merging of Theorem 5.1 and Lemma 5.3 allows to obtain the following Theorem, which provides a further construction of disjoint families of capacitors. This is a construction which we will widely use in the next subsections. Its proof follows exactly the same lines as those of [ Theorem 5.4. Let (X, dist, ς) be a compact metric-measure space with ς a non-atomic finite Borel measure and let a > 0. Assume that there exists a constant Γ such that any metric ball of radius 0 < r ≤ a can be covered by at most Γ balls of radius r 2 . Then, for every j ∈ N there exists two families

Then there exist two open sets
We remark that for a sufficiently large integer j it is always possible to apply the construction of Theorem 5.1 and obtain a decomposition of the metric measure space by annuli (Theorem 5.4 i),ii) and iii)-a)). In particular we have the following. Corollary 5.6. Let the assumptions of Theorem 5.1 hold. Then each annulus A i has either internal radius r i such that where V (r) := sup x∈X ς(B(x, r)) and v j = c ς(X) j , or is a ball of radius r i satisfying (5.1).
It turns out that Corollary 5.6 applies to the case iii)-a) of Theorem 5.4, see also [25].
We conclude this subsection with the proof of Lemma 5.2 Proof of Lemma 5.2.
Step 1. We construct the points x i by induction. Let Ω 1 = X.
Step 2. We write −1 , 5r)). Let j be the smallest integer such that x ∈ B(x j , 5r). Then x ∈ Ω j . In fact, if it not the case, x ∈ B(x 1 , 5r) ∪ ... ∪ B(x j−1 , 5r)) and this would contradict the fact that j was minimal.
As this is a disjoint union, we get Step 3. For each i, let us show that In fact by definition, and B(x i , 5r) is covered by N (r) balls of radius r. Let B(y, r) one of these balls. We have Step 4. We deduce . We can argue as before and deduce that This concludes the proof.

5.2.
A first general estimate. In this subsection we prove an upper bound which holds for any domain with C 1 boundary of a complete n-dimensional smooth Riemannian manifold (M, g) with a given lower bound on the Ricci curvature of the form Ric ≥ −(n − 1)κ 2 , κ ≥ 0.
We need a few preliminary definitions. We denote by r inj (p) the injectivity radius of the manifold (M, g) at p. We denote by r inj the injectivity radius of the manifold (M, g), which is defined as the infimum of r inj (p) for p ∈ M . We will use also the injectivity radius relative to Ω ⊂ M , defined by: If Ω is bounded, then the infimum in (5.3) is actually a minimum and it is strictly positive.
We are ready to state the main result of this subsection. . Then for all j ∈ N, where A n , B n , C n are positive constants which depend only on the dimension.
The strategy of the proof of Theorem 5.7 is to build, for each j ∈ N, j disjointly supported functions u 1 , ..., u j ∈ H 2 (Ω). Hence, the linear space U j spanned by u 1 , ..., u j is j-dimensional and we can use U j in the min-max formula (3.33). The fact that the functions u 1 , ..., u j have disjoint support makes easy to estimate the Rayleigh quotient of any function in U j : it is in fact sufficient to estimate the Rayleigh quotient of each of the u i .
Suitable test functions for the Rayleigh quotient in (3.33) are built, in this subsection, in terms of the Riemannian distance function from a point p ∈ M . For any x, p ∈ M we denote by δ p (x) the function δ p (x) := dist(x, p). We also denote by cut(p) the cut-locus of a point p ∈ M . We will make use of the Laplacian Comparison Theorem, see e.g., [34, § 9] for details.
We prove now the following lemma.
In particular, for any κ ≥ 0 Proof. We prove point ii). Let p ∈ M and let x ∈ B p, and x belongs to the geodesic joining p and p ′ . From Theorem 5.8 it follows that Moreover, since x belongs to the geodesic connecting p with p ′ we see that Point ii) is now proved. Point i) is proved exactly in the same way. The last statement follows by observing that cosh(t) ≤ 1 t − 1 for all t > 0, which implies (5.5).
We are now ready to prove Theorem 5.7.
Proof of Theorem 5.7. We first apply Theorem 5.4 with a = min 1 κ , rinj,Ω 2 (if κ = 0, then we take a = rinj,Ω 2 ). We take X = Ω endowed with the induced Riemannian distance, and with the measure ς defined as the restriction to Ω of the Lebesgue measure of M , namely ς(E) = |E ∩ Ω| for all measurable set E.
Step 1 (large j). From Lemma 5.5 we deduce that there exists j Ω ∈ N such that for all j ≥ j Ω there exists a sequence {A i } 4j i=1 of 4j annuli such that 2A i are pairwise disjoint and The constant c depends only on Γ of Theorem 5.4, hence it depends only on the dimension and can be determined explicitly (see [25]). Since we have 4j annuli, we can pick at least 2j of them such that Among these last 2j annuli, we can pick at least j of them such that We take this family of j annuli, and denote it by {A i } j i=1 . Subordinated to this decomposition we construct a family of j disjointly supported functions u 1 , ..., u j . If u 1 , ..., u j belong to H 2 (Ω), then from (3.33) Thus, in order to estimate µ j it is sufficient to estimate the Rayleigh quotient of each of the test functions. Let f : [0, ∞) → [0, 1] be defined as follows: By construction f ∈ C 1,1 [0, +∞). Moreover f ∈ C 2 ( 1 2 , 1 ). We consider test functions of the form f (ηδ p (x)) for some η ∈ R and p ∈ M . We note that (5.11) ∇f (ηδ p (x)) = ηf ′ (ηδ p (x))∇δ p (x) and (5.12) ∆f (ηδ p (x)) = η 2 f ′′ (ηδ p (x)) + ηf ′ (ηδ p (x))∆δ p (x).
In ( and center p i . Associated to A i we define a function u i as follows By construction, u i| Ω ∈ H 2 (Ω). Standard computations (see (5.11)-(5.14)) and Lemma 5.9 show that When estimating the Rayleigh quotient of u i we will also need to estimate |∇|∇u i | 2 |. We have that Case b (annulus). Assume that A i is a proper annulus of radii 0 < r i < R i ≤ rinj,Ω 4 and center p i . Associated to A i we define a function u i as follows otherwise.
In any case then Moreover, from Lemma 5.9 we have otherwise.
In any case then We will also need an estimate on |∇|∇u i | 2 |. As for (5.18) we find that otherwise.
In any case then We also need an upper bound for the volume of 2A i . Since the outer radius of 2A i is by construction smaller than 1 κ , we have, from the volume comparison and standard calculus that From Bochner's formula (2.4) we deduce that We note that the boundary integrals are taken on ∂Ω ∩ 2A i since by construction u i ∈ H 2 0 (2A i where the last inequality follows from the fact that r i ≤ R i ≤ 1 2κ . Corollary 5.6 gives us information on the size of the radius r i , in fact We observe that each r ∈ B is such that by volume comparison, where |B(p ′ , r)| κ denotes the volume of the ball of radius r in the space form of constant curvature −κ 2 . If κ = 0 then each r ∈ B is such that c|Ω| j ≤ ω n r n .
Hence any r ∈ B is such that If κ > 0, thenr ≤ 2r i ≤ 2 κ by construction, and sincer = inf B, from volume comparison and standard calculus (see also (5.26)) c|Ω| j ≤ sinh(2) n−1 ω nr n . Therefore We note that (5.31) implies (5.32) which holds true for any κ ≥ 0. We conclude that In order to complete the estimates, it remains to bound the term Ω∩2Ai (∆u i ) 2 dv. We need to distinguish the case n = 2, 3, 4 and n > 4.
Case a' (lower dimensions). Let n ≤ 4. We note that in this case it irrelevant to know that |2A i ∩ Ω| ≤ |Ω| j . This fact is crucial only for higher dimensions. If A i is a ball of radius r i ≤ 1 2κ , and hence |2A i | ≤ 2 n sinh(1) n−1 ω n r n i , we have In the very same way it is possible to prove that (5.36) holds if A i is a proper annulus, possibly with a different β ′ n , but still dependent only on n. It is sufficient to split the integral 2Ai (∆u i ) 2 dv as the sum of the integrals of (∆u i ) 2 on the annulus ri 2 ≤ δ pi (x) ≤ r i and on the annulus R i ≤ δ pi (x) ≤ 2R i and use (5.22) and (5.26) in each case.
Case b' (higher dimensions). Let n > 4. Let A i be a ball of radius r i . Then, by Hölder's inequality, where (5.39) β ′′ n = 72(4(n + 1) 2 + (n − 1) 2 )(ω n sinh(1) n−1 ) 4 n . In the same way it is possible to prove that (5.38) holds if A i is a proper annulus, possibly with a different β ′′ n , but still dependent only on n . It is sufficient to split the integral 2Ai (∆u i ) 2 dv as the sum of the integrals of (∆u i ) 2 on the annulus ri 2 ≤ δ pi (x) ≤ r i and on the annulus R i ≤ δ pi (x) ≤ 2R i and use (5.22) and (5.26) on each annulus. We have then proved that, for all dimensions n ≥ 2, where β n is a constant which depends only on n and is explicitly computable. This concludes the estimate of the numerator of the Rayleigh quotient for u i . As for the denominator we have Then, we have for all i = 1, ..., j where we have used Young's inequality in the last passage. We have proved then that for all j ≥ j Ω , where (5.43) A n := 4β n + 3α n c and (5.44) B n := 2 α n c .
Step 2 (small j). Let now j < j Ω be fixed. By using Theorem 5.4 as in Step 1, we find that there exists a sequence of 4j sets If the sets A i are annuli, we can proceed as in Step 1 and deduce the validity of (5.42). Assume now that j is such that the sets A i of the decomposition are of the form 160 . Since we have 4j disjoint sets D i , we can pick j of them such that |Ω ∩ D i | ≤ |Ω| j and |∂Ω ∩ D i | ≤ |∂Ω| j . We take from now on this family of j capacitors. Note that D i is a disjoint union of l i balls B(x i 1 , 5r 0 ), · · · , B(x i li , 5r 0 ) of radius 5r 0 . Associated to each B(x i k , 5r 0 ), k = 1, ..., l we construct test functions u i k as in (5.15). Then we define the function u i associated with the capacitor (A i , D i ) by setting u i = u i k on B(x i k , 5r 0 ). We have j disjointly supported test functions in H 2 (Ω). We estimate the Rayleigh quotient of each of the u i as in Step 1. As in (5.16), (5.17) and (5.18) we estimate |∇u i |, |∆u i | and |∇|∇u i | 2 |. In particular, we find a universal constant c 0 such that By using (5.27), as we have done for (5.33), (5.36) and (5.38), we find that for all i = 1, ..., j, which implies, by using Young's inequality as in (5.41), that The proof of (5.4) follows by combining (5.42) and (5.46), possibly re-defining the constants A n , B n .
Remark 5.10. We point out that in the proof of Theorem 5.7 the inequality (5.46) appears. Apparently this may look like a nonsense, in fact the right-hand side of the inequality does not depend on j. However, note that this situation may occur only for a finite number of eigenvalues µ j , since, starting from a certain j Ω (of which it is possible in principle to give a lower bound), the capacitors of the decomposition given by (5.4) are of the form iii) − a), hence the estimate (5.42) holds starting from a certain j Ω .
In the next subsections we will present more refined estimates under additional assumptions.

Manifolds with
Ric ≥ 0 and n = 2, 3, 4. In this subsection we establish upper bounds for bounded domains in Riemannian manifolds satisfying Ric ≥ 0. In this case we will use Theorem 5.1, noting that the constant Γ depends only on n. Moreover, test functions in this case are not given in terms of the distance from a point, thus avoiding the obstruction of the lack of regularity in correspondence of the cut-locus.
We state the main theorem of this subsection.
Let Ω be a bounded domain of M with C 1 boundary. Then for all j ∈ N, the constant A n depending only on the dimension.
Proof of Theorem 5.11. We use Theorem 5.1, which provides, for all indexes j ∈ N, a family {A i } j i=1 of annuli such that |A i ∩ Ω| ≥ c |Ω| j and the annuli 2A i are pairwise disjoint. Moreover, from Corollary 5.6 we deduce that each annulus A i has either internal radius r i satisfying (5.1) or is a ball of radius r i satisfying (5.1). In this case B = r ∈ R : V (r) ≥ c |Ω| j and V (r) := sup x∈Ω |B(x, r) ∩ Ω|.
Since Ric ≥ 0, from volume comparison we know that every ball B(p, r) ⊂ M satisfies |B(x, r)| ≤ ω n r n . Hence any r ∈ B is such that we build a family of test functions {u i } j i=1 in H 2 (Ω) in the following way. If A i is a ball of radius r i , we take u i = φ ri , where φ ri is defined in Lemma 5.12. Hence u i is supported in 2A i and u i ≡ 1 on A i . If A i is a proper annulus of radii 0 < r i < R i , we take u i = φ Ri − φ ri . Again, u i is supported in 2A i and u i ≡ 1 on A i . The functions u i| Ω j i=1 are disjointly supported and belong to H 2 (Ω). Hence, from (3.33) we deduce that We estimate now the Rayleigh quotient of each of the u i . Assume that A i is a ball of radius r i . We have, for the numerator In the first inequality we have estimated Ω∩2Ai |D 2 u i | 2 + Ric(∇u i , ∇u i )dv with the integral on the whole ball 2A i , being the integrand a non-negative function. Moreover, from Bochner's formula, since u i ∈ H 2 0 (2A i ), we have that 2Ai |D 2 u i | 2 + Ric(∇u i , ∇u i )dv = 2Ai (∆u i ) 2 dv ≤ |2Ai|c(n) 2 r 4 i . Finally we have used (5.48) since n ≤ 4. For the denominator we have From (5.49) and (5.50) we deduce In the very same way it is possible to prove that (5.49) holds if A i is a proper annulus, possibly with a different dimensional constant in front of the term j |Ω| 4 n . It is sufficient to split the integral 2Ai (∆u i ) 2 dv as the sum of the integrals of (∆u i ) 2 on the annulus ri 2 ≤ δ pi (x) ≤ r i and on the annulus R i ≤ δ pi (x) ≤ 2R i . The proof is now complete.
Remark 5.13. We remark that, differently from the proof of Theorem 5.7, we did not choose the annuli of the decomposition in such a way that |Ω ∩ 2A i | ≤ |Ω| j . In fact, in the case n = 2, 3, 4, an upper bound on the size of the supports of test functions seems to be irrelevant in the estimates. Moreover, being Ric ≥ 0, the quadratic form |D 2 u| 2 + Ric(∇u, ∇u) is always non-negative, hence we can estimate its integral over Ω ∩ 2A i with the whole integral over 2A i and use Bochner's formula. Of course we can do this passage also for n > 4. However, in this case we would obtain Assume that A i is a ball of radius r i . The term ∂Ω∩2Ai |∆u i ||∇u i |dσ is estimated by For the term Ω∩2Ai (∆u i ) 2 dv we have Hence, as for (5.41), we find that for some constants A n , B n , C n which depend only on the dimension. In the case that A i is a proper annulus, inequality (5.54) still holds, with possibly different dimensional constants. Unfortunately an estimate of the form |∇|∇φ r | 2 | ≤ c(n) r 3 is not available for a function φ r as in Lemma 5.12. If such an inequality would hold, then we would immediately have (5.55) and therefore Open problem. Prove (5.56) for domains in complete smooth manifolds with Ric ≥ 0 and n > 4. Prove inequality (5.47) for domains in complete smooth manifolds with Ric ≥ 0 and n ≥ 2.

Manifolds with
Ric ≥ 0 and small diameter. If Ric ≥ 0 and the diameter is sufficiently small compared to r inj,Ω , it is possible to build test functions for the Rayleigh quotient in terms of the distance function. In particular, we have the following. Proof. In order to prove Theorem 5.14 we exploit Theorem 5.1 and Corollary 5.6. We find, as in the proof of Theorem 5.7, for all j ∈ N a family of j annuli where the constants A n , B n depend only on the dimension. This implies (5.57) for all j ∈ N. The last statement follows immediately from Theorem 5.11. This concludes the proof.
5.5. Domains on the sphere. In this section we obtain bounds for domains of the standard sphere S n R , namely we have the following theorem. Theorem 5.15. Let (M, g) = S n R be the sphere of radius R with standard round metric and let Ω be a domain in S n with C 1 boundary. Then Proof. Through all the proof we assume n > 4. The validity of the theorem for n ≤ 4 follows from Theorem 5.11. Given two points p 1 , p 2 ∈ S n R , the maximal distance among them is attained when they are antipodal points. In this case δ p1 (p 2 ) = πR. Moreover, µ 1 = 0 and the corresponding eigenfunctions are the constant functions on Ω, and µ 2 > 0 (see Section 4.2).
We apply Theorem 5.1 with X = Ω with the Riemannian distance, ς(A) = |A ∩ Ω|. Since we have positive Ricci curvature, point i) of Theorem 5.1 is satisfied for some Γ which depends only on n. Points ii) and iii) are easily seen to hold. Hence we deduce that there exists c which depends only on the dimension such that for any j ≥ 2, there exists A 1 , ..., A j annuli with j ; iv) each annulus 2A i has outer radius less than πR 2 . Points iii) and iv) follow from the fact that we shall actually apply Theorem 5.1 with 2j + 1 and by observing that we can choose first 2j among the 2j + 1 annuli given by the construction so that iv) holds. Indeed, if an annulus A i with center p i has outer radius strictly greater than πR 2 , the set S n R \ 2A i is B(p 1 , r) ∪ B(p ′ i , r ′ ), where p ′ i is the antipodal point of p i , r, r ′ < πR 2 (B(p ′ , r ′ ) = ∅ if A i is a ball), and all other annuli 2A k of the decomposition need belong to B(p 1 , r) ∪ B(p ′ i , r ′ ). Hence, no more than one annulus can satisfy iv). Moreover, among the remaining 2j annuli, we can chose j annuli such that iii) holds (see also the proof of Theorem 5.7). We shall denote by r i and R i the inner and outer radius of A i , if A i is an actual annulus, while we shall denote by r i the radius, if A i is a ball. By p i we denote the center of the annuli A i .
Associated to each of the j annuli A 1 , ..., A j satisfying i)-iv) we construct test functions u i as in (5.15) (if A i is a ball) or in (5.19) if A i is a proper annulus. The functions u i are of the form u i (x) = f (δ p (x)).
In fact, one easily checks that all the eigenvalues are non-negative, and that there is only one zero eigenvalue with associated eigenfunctions the constant functions on M .
We prove now that the eigenvalues µ j of (3.1) are exactly the squares of the eigenvalues of the Laplacian on M . Recall that the weak formulation of the closed eigenvalue problem for the Laplacian is  and the corresponding eigenfunctions can be chosen to be the same.
Proof. Let v i denote the eigenfunctions associated with m i normalized by Ω v i v k dv = δ ik . Since the metric is smooth, we have that v i ∈ H 2 (M ) and −∆v i = m i v i in L 2 (M ). Hence, by setting V :=< v 1 , ..., v j >, we have that V is a j-dimensional subspace of H 2 (M ) and a function v ∈ V is of the form v = j i=1 α i v i for some α 1 , ....α j ∈ R. Hence, from (3.33) we have On the other hand, the well-known min-max principle for the eigenvalues of the Laplacian on M states that m j = min We choose U :=< u 1 , ..., u j > where u 1 , ..., u j are the eigenfunctions associated with the eigenvalues µ 1 , ..., µ j of the biharmonic operator on M normalized by M u i u k dv = δ ik . Then M ∆u i ∆u k dv = µ i δ ik . Any u ∈ U is of the form u = j i=1 α i u i for some α 1 , ..., α j ∈ R. We note that The rest of the proof is straightforward.
From Theorem 5.20 and from (1.6) we deduce the following corollary. 5.9. Domains with convex boundary. In this last subsection we shall present a case in which upper bounds for biharmonic Neumann eigenvalues µ j can be deduced directly by comparison with Neumann eigenvalues of the Laplacian and by (1.7). Let (M, g) be a complete n-dimensional smooth Riemannian manifold with Ric ≥ −(n − 1)κ 2 , κ ≥ 0, and let Ω be a bounded domain in M with C 2 boundary. If II ≥ 0 then we can compare the eigenvalues of (3.1) with the squares of the eigenvalues of the Neumann Laplacian on Ω. We recall that the weak formulation of the Neumann problem for the Laplace operator on Ω is given by (5.68) with M replaced by Ω. Neumann eigenvalues of the Laplacian have finite multiplicity, are non-negative and form an increasing sequence 0 = m 1 < m 2 ≤ · · · ≤ m j ≤ · · · ր +∞. The associated eigenfunctions are denoted by {v i } ∞ i=1 can be chosen to form a orthonormal basis of L 2 (Ω). We have the following Since the domain is of class C 2 , by standard elliptic regularity we have that the eigenfunctions v i of the Neumann Laplacian belong to H 2 (Ω). Therefore −∆v i = m i v i in L 2 (Ω) and ∂vi ∂ν = 0 in L 2 (∂Ω). We deduce that for any linear combination Consider then V :=< v 1 , ..., v j > the j-dimensional space spanned by the first j eigenfunctions of the Neumann Laplacian. This is a subspace of H 2 (Ω) of dimension j. Each v ∈ V is of the form v = j i=1 α i v i for some α 1 , ..., α j ∈ R. Moreover From (3.33) we have that This concludes the proof.
Theorem 5.22 and inequality (1.7) imply the following corollary. We remark that this bound holds independently of the size of Ω, its diameter and the injectivity radius of M . Hence it is natural to pose the following question, whose answer seems quite complicated at this stage.