Conformal upper bounds for the eigenvalues of the $p$-Laplacian

In this note we present upper bounds for the variational eigenvalues of the $p$-Laplacian on smooth domains of complete $n$-dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given manifold $(M,g)$ for $11$ when we fix a metric $g$. To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio.

Here ∂u ∂ν denotes the derivative of u in the direction of the exterior unit normal to the boundary, ν. If M is compact (boundaryless) and Ω = M , then ∂Ω = ∅ and we consider the closed problem −div(|∇u| p−2 ∇u) = µ|u| p−2 u in M .
Eigenvalue problems for the p-Laplacian with Dirichlet boundary conditions on Euclidean domains have been largely investigated in the last decades. We mention [19] and especially [28] for extensive references on the subject. As for Neumann boundary conditions, less information is available. We refer to [27] for an exhaustive presentation of eigenvalues problems for the p-Laplacian on Euclidean domains subject to various boundary conditions. It is well-known that problem (1.1) admits an increasing sequence of non-negative eigenvalues obtained through the Ljusternik-Schnirelman principle 0 = µ 1,p < µ 2,p ≤ · · · ≤ µ k,p ≤ · · · ր +∞, which are also called the variational eigenvalues. It is still an open problem whether other eigenvalues exist (except for the case p = 2 or n = 1). Results concerning the first positive eigenvalue and Neumann boundary conditions are available in [16,22] for Euclidean domains. Concerning higher eigenvalues, results in the case of Dirichlet conditions for Euclidean domains can be found in [1,4,36] (continuity and limits with respect to p, multiplicity, etc.). As for Riemannian manifolds, we mention [12,30,32,35,41] for sharp estimates for the first positive eigenvalue in case of compact manifolds or domains and Neumann boundary conditions, under a given lower bound on the Ricci curvature. We also mention [31,33] for estimates in a conformal class of a given metric for the first positive eigenvalue of a compact manifold. Results for the first eigenvalue of the corresponding Dirichlet problem on domains are available in [15,29].
In this paper we investigate upper bounds for all variational eigenvalues that agree with the Weyl's law under suitable geometrical assumptions on the manifold M . In particular, we will investigate upper bounds in the conformal class of a given metric and upper bounds for a fixed metric.
Results in this spirit are classical and well-known in the case of the Laplacian (p = 2). For Euclidean domains we recall in particular the classical result of Kröger [25] which is stated as follows: (1.2) µ k,2 ≤ C n k |Ω| We note that the constant of the upper bounds (1.2) does not coincide with the constant in the Weyl's law. The validity of (1.2) with C n = 4π 2 ω −2/n n is a famous open problem: the Polya's conjecture for the Neumann Laplacian (see [38,39]). We also note that for the eigenvalues of the Neumann Laplacian, lower bounds for the eigenvalues of the form (1.2) do not hold in general, at least for the bottom of the spectrum. In fact it is well-known that one can produce domains with fixed volume and an arbitrary number of eigenvalues close to zero. Thus, for Neumann-type problems, lower bounds necessarily depend on the geometry of the domain in a more involved form (and not merely on the volume), see [5,6] for a discussion on Cheeger lower bounds on domains and on compact manifolds, respectively. On the other hand, uniform upper bounds (depending on the dimension and the volume only) are relevant for Neumann-type problems. Upper bounds (1.2) are obtained in [25] by classical techniques of harmonic analysis and are easily extended to the Neumann eigenvalues of higher order linear elliptic operators on Euclidean domains, see [26].
As for the eigenvalues of the Laplacian on a compact n-dimensional Riemannian manifold (M, g) with Ric g ≥ −(n − 1)κ 2 , κ ≥ 0, uniform upper bounds were proved by Buser [3]. Namely where C ′ n depends only on the dimension. We note that an additive term appears in the estimate. When Ric g is supposed to be non-negative, then we have an analogous estimate as (1.2), however the presence of an additive term of the form (n−1) 2 4 κ 2 is necessary, see [2]. Again, these upper bounds agree with the Weyl's law which is given for a compact manifold by (1.3) with Ω = M . We also note that the geometry of the manifold enters the bounds as an additive constant, while the term encoding the asymptotic behavior, as expected, contains only information on the dimension and the volume of the manifold.
Another relevant problem is to provide bounds within the conformal class [g] of a given metric g on a compact manifold M . The first results in this sense for the eigenvalues of the Laplacian are due to [42] for n = 2 and [24] for n ≥ 2 (see also [17] for results concerning the first positive eigenvalue). The result of [24] can be summarized as follows: for any compact n-dimensional Riemannian manifold (M, g) ) depends only on n and on the conformal class of g. Inequality (1.6) has been then improved in [21]. In particular, in [21] the following inequality is proved for any compact n-dimensional Riemannian manifold (M, g) ) denotes the min-conformal volume (see (1.2) below for the definition). Inequalities (1.4) and (1.5) can be deduced by (1.7) when we fix a metric. In particular, in [21] an analogous inequality as (1.7) is proved for the Neumann eigenvalues on domains, which implies the bound (1.5).
Analogous estimates, all in the same spirit of [3,11,21] have been proved for the Steklov problem [8], for hypersurfaces [9], and for the Neumann eigenvalues of the biharmonic operator on domains of Riemannian manifolds with Ricci curvature bounded from below [14].
The classical approach to prove upper bounds for the eigenvalues of linear elliptic operators on Euclidean domains, as mentioned, relies on harmonic analysis. However this approach is no more suitable for compact manifolds or, in general, for domains of complete manifolds. The approach which has been used in [11] (which is in the same spirit of [3], see also [20]) makes use of a metric construction. Namely, in order to bound µ k,2 one considers A 1 , ..., A k disjoints subsets of Ω of measure of the order of |Ω| k , and introduce test functions u 1 , ..., u k subordinated to these sets. A clever estimate of the Rayleigh quotient of these functions provides the upper bounds (1.5) or (1.7) (depending on the particular construction).
As for the variational eigenvalues of the Neumann p-Laplacian with p = 2 much less is known. Asymptotic estimates are available, which are consequence, as in the case of the Laplacian, of the Weyl's law. Actually, the validity of a Weyl's law for the Neumann (and Dirichlet) variational eigenvalues of the p-Laplacian has been conjectured by Friedlander in [18], who proved an asymptotic estimate of the form C 1 λ n/p ≤ N (λ) ≤ C 2 λ n/p with C 1 , C 2 depending only on n, p, as λ → +∞, for the counting function N (λ) := ♯{k : µ k,p ≤ λ}. The conjecture C 1 = C 2 has been recently proved in [34] (see also [23] for asymptotic estimates for the eigenvalues of the fractional p-Laplacian).
In order to state our main results we need to fix some notation and introduce some definitions. Through the rest of the paper, we will denote by µ g k,p either the Neumann eigenvalues of the p-Laplacian on a domain Ω of a complete Riemannian manifold (M, g), or the eigenvalues of the p-Laplacian on a compact Riemannian manifold (M, g), unless otherwise specified. For a complete Riemannian manifold (M, g), we shall denote by | · | g the Riemannian measure associated with the metric g. By [g] we denote the conformal class of a given metric g.
The first main result of the paper is the following.
with Ricci curvature bounded below Ric g0 ≥ −(n − 1)κ 2 , κ ≥ 0 and let 1 < p ≤ n. Then, for all bounded domains Ω ⊂ M with smooth boundary and all metric g ∈ [g 0 ] there exist constants A p,n , B p,n > 0 only depending on p and n such that for all k ∈ N.
Theorem 1.1 can be stated also for compact manifolds. To do so, we need the following definition (see also [21]) Definition 1.2. Let (M, g) be a compact n-dimensional Riemannian manifold. We define the min-conformal volume V ([g]) by It is a standard to verify that, if we denote by κ(g) the smallest non-negative number such that Ric g ≥ −(n − 1)κ(g) 2 , then We have the following theorem. The proof of Theorem 1.3 follows easily from that of Theorem 1.1 (see Remark 4.2). We note that inequalities of the type (1.8)-(1.9) are not possible for p > n. In fact, it is proved in [33] that if p > n, for any n-dimensional Riemannian manifold (M, g 0 ) there exists a metric in g ∈ [g 0 ] and of volume one with µ g 2,p arbitrarily large. However, if we fix the metric, we have upper bounds for any p > 1. The second main result of this paper is the following. Theorem 1.4. Let (M, g) be a complete n-dimensional Riemannian manifold with Ricci curvature bounded below Ric g ≥ −(n − 1)κ 2 , κ ≥ 0, and let p > 1. Then, for all bounded domains Ω ⊂ M with smooth boundary there exist constants A p,n , B p,n > 0 only depending on p and n such that for all k ∈ N.
As in the case of Theorem 1.1, Theorem 1.4 can be stated also for compact manifolds (see Remark 4.3).
Theorem 1.5. For all p > 1, n ≥ 2, there exist constants A p,n , B p,n > 0 only depending on the p and n, such that for all compact n-dimensional Riemannian manifolds with Ricci curvature bounded below Ric g ≥ −(n − 1)κ 2 , κ ≥ 0 we have for all k ∈ N. Theorem 1.3 carries a number of corollaries. In particular, as in [7] it is possible to introduce the p-conformal variational eigenvalues which are defined by . Theorem 1.3 implies the following bounds on the p-conformal variational eigenvalues. Corollary 1.6. For all 1 < p ≤ n, n ≥ 2, there exist constants A p,n , B p,n > 0 only depending on p and n, such that for all compact n-dimensional Riemannian manifolds and all k ∈ N we have In the case of surfaces (i.e., n = 2) we know that any orientable Riemannian 2-manifold of genus γ is conformally equivalent to the standard sphere of constant curvature 1 if γ = 0, to the flat torus if γ = 1, and to a manifold with constant curvature −1 if γ ≥ 2 (Uniformization Theorem). In the first two cases V ([g]) = 0, while in the third case V ([g]) ≤ 4π(γ − 1). From this and Theorem 1.3 we deduce the following corollary. Corollary 1.7. For all 1 < p ≤ 2, there exist constants A p , B p > 0 only depending on p, such that for all compact orientable Riemannian surfaces (Σ γ , g) of genus γ and all k ∈ N we have Finally, we want to remark that Theorem 1.4 holds in particular for all Euclidean domains. Corollary 1.8. For all bounded domains Ω in R n , n ≥ 2, with smooth boundary and all p > 1 there exist constants A p,n , B p,n > 0 only depending on p and n such that for all k ∈ N, where µ k,p denotes the k-th variational eigenvalue of the Neumann p-Laplacian on Ω and |Ω| denotes the Lebesgue measure of Ω.
We note that in the Euclidean case it is not possible to prove (1.15) as in [25,26] (except for the case p = 2), where the Hilbert structure of L 2 (Ω) is deeply involved in the proof.
The proofs of Theorems 1.1 and 1.4 are given in the same spirit of [11,14,21]. In fact, in this paper we formalize a general metric approach suitable to prove upper bounds for variational eigenvalues which has been refined more and more starting from [20], passing through [11,21] and ending with [14] and with the present note.
Not only we formalize this metric approach stating a general decomposition result (Theorem 3.4): we also show how it can be exploited in two different ways, whose differences may seem slight but are substantial. A first way allows to provide conformal upper bounds for 1 < p ≤ n (see the proof of Theorem 1.1), while an alternative way allows to prove upper bounds for all p > 1 when we fix a metric g (see the proof of Theorem 1.4). We note that the main decomposition result, Theorem 3.4, is a clever merging of a classical decomposition result of a metric measure space by capacitors (Theorem 3.1, see also [20, Theorem 1.1]) and of a refinement of a method introduced in [11] (Lemma 3.3). Actually, one can see that the proof of Theorem 1.4 can be obtained just by exploiting Lemma 3.3, i.e., morally it can be obtained with (a refinement of) the technique introduced in [11]. Therefore, we deduce a sort of "metatheorem" stating that, if we are able to prove a result for the eigenvalues of the Laplacian with a method in the spirit of [11], the result is likely to hold also for the variational eigenvalues of the p-Laplacian. On the other hand, this is no more true if the classical decomposition of [20] is used.
We mention that a similar behavior has been observed for upper bounds on the Neumann eigenvalues of linear elliptic operators of order 2m, m ∈ N and density on Euclidean domains, see [13]. In particular, if 2 ≤ 2m ≤ n uniform upper bounds hold, which morally correspond to the conformal upper bounds discussed in this paper. For n < 2m these bounds do not hold (counter-examples are provided in [13]).
We also recall that Lemma 3.3 has been exploited to prove upper bounds for the eigenvalues on Σ, where Σ is an hypersurface in a complete n-dimensional Riemannian manifold (M, g) bounding some smooth domain Ω. In fact, upper bounds in term of the isoperimetric ratio for the Laplacian eigenvalues on Σ are obtained in [9]. As already mentioned, when we have a result obtained by means of Lemma 3.3, that result is likely to hold also for the variational eigenvalues of the p-Laplacian. This is in fact the case.
We include the precise statements of the analogous results for the variational eigenvalues of the p-Laplacian, along with their proofs, in the Appendix A.
The paper is organized as follows. In Section 2 we set the notation and recall some preliminary results. In Section 3 we collect all the main technical results of decomposition of a metric measure space by capacitors. In Section 4 we prove Theorems 1.1 and 1.4. In Appendix A we discuss upper bounds on hypersurfaces in terms of the isoperimetric ratio.

Preliminaries and notation
By W 1,p (Ω) we denote the Sobolev space of functions u ∈ L p (Ω) with weak first derivatives in L p (Ω). The space W 1,p (Ω) is endowed with the norm where ∇ g denotes the gradient associated with the metric g and dv g denotes the Riemannian volume element associated with g. For u ∈ L p (Ω) we denote by u L p (Ω) its standard norm given by A sequence of eigenvalues for (2.3) can be obtained through the Ljusternik-Schnirelman (see [27] for a detailed discussion). These eigenvalues, which form an increasing sequence of non-negative numbers diverging to +∞, are called the variational eigenvalues as they can be characterized variationally as follows: In order to prove upper bounds for µ g k,p we need suitable sets F k ∈ Γ k to test in (2.4). The following lemma provides us a useful way to build such F k . Lemma 2.1. Let k ∈ N, k ≥ 1, and let u 1 , ...u k ∈ W 1,p (Ω), with u i = 0 and with pairwise disjoint supports U 1 , ..., U k . Let where e k i , i = 1, ..., k, denotes the standard basis of R k . The function f k is an odd homeomorphism between F k and S k−1 p := x ∈ R k : k i=1 |x i | p = 1 , which is the unit sphere of R k with respect to the ℓ p norm. This implies that γ(F k ) = γ(S k−1 p ) (see also [40,Proposition 2.3]). Finally, by the Borsuk-Ulam Theorem we deduce that γ(S k−1 p ) = k.

Decomposition of a metric measure space by capacitors
In this section 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 The prototype of the construction of a decomposition of a metric measure space by disjoint sets is given in [20]. We recall it here for the reader's convenience. Theorem 3.1 provides a decomposition of a metric measure space by annuli of the size at least c ς(X) k . The common idea of the proofs of Theorems 1.1 and 1.4 is to build for each k ∈ N, suitable test functions u i supported on 2A i and such that u i = 1 on A i , and then to compute their Rayleigh quotients. In principle, in order to obtain the estimate of Theorems 1.1 and 1.4 with this method, we need that the constant c which controls the volume of the A i 's depends only on n. This is true, by the Bishop-Gromov volume comparison Theorem, in the case Ric g ≥ 0, but this is no longer true in the case of a negative lower bound on the Ricci curvature. The direct application of Theorem 3.1 would lead to an estimate of the form (1.8)-(1.10), but with B p,n depending also on the diameter of Ω (see also [20] for more details), and such an estimate is bad: the geometry of the domain enters as a multiplicative constant in front of the Weyl-term. Moreover, the diameter should not affect the upper bounds.
We state now the following lemma, which improves [9, Lemma 4.1]. This Lemma will be useful to construct a decomposition of a metric measure space with capacitors alternative to Theorem 3.1. The proof of this Lemma can be found in [14]. 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) .

Then there exist two open sets
. A consequence of Lemma 3.2 is the following result providing a decomposition of a metric measure space by capacitors which is alternative to that of Theorem 3.1.  A clever merging of Theorem 3.1 and Lemma 3.3 allows to obtain the following Theorem, which provides a further construction of disjoint families of capacitors. This is the construction that we will use in the proofs of Theorems 1.1 and 1.4. Its proof follows exactly the same lines as those of [ Theorem 3.4. 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 k ∈ N there exists two families We remark that for a sufficiently large integer k it is always possible to apply the construction of Theorem 3.1 and obtain a decomposition of the metric measure space by annuli (Theorem 3.4 i),ii) and iii)-a)). In particular we have the following. We refer to [21,Proposition 2.1] for the proof of Lemma 3.5. We state now a useful corollary of Theorem 3.1 which gives a lower bound of the inner radius of the annuli of the decomposition, see [20,Remark 3.13].
Corollary 3.6. Let the assumptions of Theorem 3.1 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 k = c ς(X) k , or is a ball of radius r i satisfying (3.1).
It turns out that Corollary 3.6 applies to the case iii)-a) of Theorem 3.4, see also [21].

Proof of Theorems 1.1 and 1.4
In this section we present the proofs of Theorems 1.1 and 1.4.
Proof of Theorem 1.1. We apply Theorem 3.4 with a = 1 κ (if κ = 0 we take a = +∞). We take X = Ω endowed with the Riemannian distance induced by g 0 , and with the measure ς g defined as the restriction to Ω of the Lebesgue measure of M associated with the metric g, namely ς g (E) = |E ∩ Ω| g for all measurable sets E.
Step 1 (large k). From Lemma 3.5 we deduce that there exists k Ω ∈ N such that for all k ≥ k Ω there exists a sequence {A i } 3k i=1 of 3k annuli such that 2A i are pairwise disjoint and The constant c depends only on Γ of Theorem 3.4, hence it depends only on the dimension n and can be determined explicitly (see [21]). Since we have 3k annuli, we can pick at least k of them such that We take this family of k annuli, and denote it by {A i } k i=1 . Subordinated to this decomposition we construct a family of k disjointly supported functions u 1 , ..., u k ∈ W 1,p (Ω), with supp(u i ) = 2A i . Let us take From (2.4) and from Lemma 2.1 we deduce that which in particular implies, since u i are disjointly supported, that Thus, in order to estimate µ g k,p it is sufficient to estimate the Rayleigh quotient of each of the test functions. Now we describe explicitly the test functions u i which we will use in (4.4). Let f : [0, ∞) → [0, 1] be defined as follows: By construction f ∈ C 0,1 [0, +∞). We consider test functions of the form f (ηδ g0 p (·)) for some η ∈ R and p ∈ M , where δ g0 p (·) denotes the Riemannian distance from a point p ∈ M associated with the metric g 0 . We note that Standard computations, and the fact that |∇ g0 δ p | ≤ 1 a.e. in M , show that Let now A i be an annulus of the family {A i } k i=1 . We have two possibilities. Either A i is a proper annulus with 0 < r i < R i , or is a ball of radius r i > 0.
Case a (ball). Assume that A i is a ball of radius r i > 0 and center p i . Associated to A i we define a function u i as follows otherwise.

Case b (annulus).
Assume that A i is a proper annulus of radii 0 < r i < R i and center p i . Associated to A i we define a function u i as follows otherwise.
As for the denominator we have Then, combining (4.16) and (4.18) we deduce that for all i = 1, ..., k where B p,n := 3c −1 β p,n . We have proved that for any k ≥ k Ω (4.20) µ g k,p ≤ B p,n k |Ω| g p n .
Step 2 (small k). Let now k < k Ω be fixed. By using Theorem 3.4 as in Step 1, we find that there exists a sequence of 3k sets 3k . If the sets A i are annuli, we can proceed as in Step 1 and deduce the validity of (4.20). Assume now that k is such that the sets A i of the decomposition are of the form i are pairwise disjoint, and δ g0 k . We take from now on this family of k 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 j , 5r 0 ), j = 1, ..., l we construct test functions u i j as in (4.8). Then we define the function u i associated with the capacitor (A i , D i ) by setting u i = u i j on B(x i j , 5r 0 ). We have k disjointly supported test functions in W 1,p (Ω). We estimate the Rayleigh quotient of each of the u i as in Step 1. As in (4.9) we estimate |∇ g0 u i |. In particular, we find a universal constant c 0 > 0 such that Moreover, as for (4.15), it is easy to prove that where γ n > 0 depends only on n. Then, since 1 < p ≤ n, thanks to Hölder's inequality, inequality (4.22), and analogous computations of those in Step 1, we deduce that (4.23) As for the denominator From (4.23) and (4.24) we deduce that for all i = 1, ..., k, where A p,n > 0 depends only on p and n. This immediately implies The proof of (1.8) follows by combining (4.20) and (4.25), possibly re-defining the constants A p,n , B p,n .
Remark 4.1. We point out that in the proof of Theorem 1.1 the inequality (4.25) appears. Apparently this may look like a nonsense, in fact the right-hand side of the inequality does not depend on k. However, note that this situation may occur only for a finite number of eigenvalues µ p,k , since, starting from a certain k Ω (of which it is possible in principle to give a lower bound), the capacitors of the decomposition given by (3.4) are of the form iii) − a), hence the estimate (4.20) holds starting from a certain k Ω .
Remark 4.2. We note that if M is a compact manifold, we can choose Ω = M . In this case, the proof of Theorem 1.1 remains exactly the same. Thus inequality (1.8) holds also for compact manifolds. The left hand side of (1.8) does not depend on g 0 . Hence, we can take the infimum with respect to g 0 ∈ [g] such that Ric g0 ≥ −(n − 1)κ 2 , κ ≥ 0. From this fact and from Definition 1.2 we deduce the validity of (1.9) We prove now Theorem 1.4. The proof is similar to that of Theorem 1.1, however, though if the differences may seem slight and technical, they are substantial. In fact, as shown in [33], Theorem 1.1 does not hold for p > n.
In the statement and the proof of Theorem 1.4 the metric g is fixed (so everything will be expressed with respect to g). In particular, Theorem 3.4 will be exploited with X = Ω and the Riemannian distance, and the measure induced by the fixed metric g. The test functions u i for the variational principle (4.4) will be written in terms of the Riemannian distance associated to g, and not in terms of the Riemannian distance associated with some different g 0 ∈ [g] (as in the proof of Theorem 1.1).
Proof of Theorem 1.4. We apply Theorem 3.4 with a = 1 κ (if κ = 0 we take a = +∞). We take X = Ω endowed with the Riemannian distance induced by g, and with the measure ς g defined as the restriction to Ω of the Lebesgue measure of M associated with the metric g, namely ς g (E) = |E ∩ Ω| g for all measurable sets E.
Step 1 (large k). From Lemma 3.5 we deduce that there exists k ′ Ω ∈ N such that for all k ≥ k ′ Ω there exists a sequence {A i } 2k i=1 of 2k annuli such that 2A i are pairwise disjoint and The numbers k ′ Ω ∈ N and c ′ > 0 are in principle different from k Ω and c of the proof of Theorem 1.1. The constant c ′ depends only on Γ of Theorem 3.4, hence it depends only on the dimension n and can be determined explicitly (see [21]). Since we have 2k annuli, we can pick at least k of them such that We take this family of k annuli, and denote it by . As in the proof of Theorem 1.1, subordinated to this decomposition we construct a family of k disjointly supported functions u 1 , ..., u k ∈ W 1,p (Ω), with supp(u i ) = 2A i .
We also set F k := k i=1 α i u i : α i ∈ R, k i=1 |α i | p = 1 . Inequality (4.4) holds, thus, in order to estimate µ g k,p it is sufficient to estimate the Rayleigh quotient of each of the test functions. The test functions u i which we shall use are identical to those in the proof of Theorem 1.1, except for the fact that they are given in terms of the distance function associated with the fixed metric g.
In particular, if A i is a ball, the function u i associated with A i is given by (4.8), where we replace δ g0 p (x) by δ g p (x), while if A i is a proper annulus, the associated function u i is given by (4.11), where again we replace δ g0 p (x) by δ g p (x). It is straightforward, as in (4.9) and (4.13), to prove that Corollary 3.6 gives us information on the size of the radius r i , in fact The constant c ′ depends only on the dimension and is the same constant as in (4.26). 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. If κ = 0 then each r ∈ B is such that c ′ |Ω| g k ≤ ω n r n .
Hence any r ∈ B is such that r ≥ c ′ |Ω| g ω n k If κ > 0, thenr ≤ 2r i ≤ 1 κ by construction, and sincer = inf B, from volume comparison and standard calculus c|Ω| g k ≤ e n−1 ω nr n .
Therefore (4.32) r i ≥r 2 ≥ 1 2 c ′ |Ω| g e n−1 ω n k 1 n . We note that (4.31) implies (4.32) which holds true for any κ ≥ 0. We conclude, by (4.27) and ( Step 2 (small k). Let now k < k ′ Ω be fixed. By using Theorem 3.4 as in Step 1, we find that there exists a sequence of 2k sets 2k . If the sets A i are annuli, we can proceed as in Step 1 and deduce the validity of (4.37). Assume now that k is such that the sets A i of the decomposition are of the form where r 0 = 4a 1600 , D i = A 4r0 i are pairwise disjoint, and δ g x i l (x i j ) ≥ 4r 0 if l = j. By definition D i = B(x i 1 , 5r 0 ) ∪ · · · ∪ B(x i li , 5r 0 ). Since we have 2k disjoint sets D i , we can pick k of them such that |Ω ∩ D i | g ≤ |Ω|g k . We take from now on this family of k 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 j , 5r 0 ), j = 1, ..., l we construct test functions u i j as in (4.8) with δ g0 replaced by δ g . Then we define the function u i associated with the capacitor (A i , D i ) by setting u i = u i j on B(x i j , 5r 0 ). We have k disjointly supported test functions in W 1,p (Ω). We estimate the Rayleigh quotient of each of the u i as in Step 1. As in (4.9) we estimate |∇ g u i |. In particular, we find a universal constant c 0 > 0 such that (4.38) |∇ g u i | ≤ c ′ 0 r 0 .
Analogous computations of those in Step 1 (see (4.33) and (4.35)) allow us to conclude that p,n a p = A p,n κ p , for all i = 1, ..., k, where A ′ p,n > 0 depends only on p and n. This immediately implies (4.39) µ k,p ≤ A ′ p,n κ p . The proof of (1.10) follows by combining (4.37) and (4.39), possibly re-defining the constants A ′ p,n , B ′ p,n .
Remark 4.3. We note that if M is a compact manifold, we can choose Ω = M . In this case, the proof of Theorem 1.4 remains exactly the same. Thus inequality (1.10) holds also for compact manifolds. This is exactly the statement of Theorem 1.5.
eigenvalues of the p-Laplacian on Σ with the induced Riemannian metric. Then for any