Let (M,g) be a smooth 4-dimensional Riemannian manifold. The Paneitz operator discovered in [18] is the fourth order operator defined for all smooth functions u by

where Δg(u)=−divg(∇u) is the Laplacian of u with respect to the metric g, and Sc_g and Ricg denote the scalar and the Ricci curvatures of g respectively (we will use this notation throughout the paper).

This operator is conformally covariant in the following sense: if g˜ is e2φg where φ is a smooth function, then the following holds

Moreover, this operator is intimately related to a conformally invariant Qg4−curvature

Originally the Paneitz operator was introduced for physical motivations and has many applications in mathematical physics. This operator was generalized to manifolds of greater dimension (n≥5) by Branson.

Given a smooth compact Riemannian manifold (M,g) of dimension n≥5, the Paneitz-Branson operator is defined as

where

The Paneitz-Branson operator is also conformally covariant in this sense: if g˜=φ4n−4g is a metric conformal to the metric g where φ is a smooth positive function, then

Of course, as an object in conformal geometry, a lot of research has been devoted to this operator; see [9], [4], [14], [8], and [21], and the references therein.

Our aim in this paper is to investigate the influence of the geometry on the sign of the eigenvalues of this operator. As a first result we give conditions sufficient to ensure the positivity of the Paneitz-Branson operator even when the scalar curvature is negative. In particular we prove the following.

Theorem 1.1. Let (M,g) be a compact Riemannian manifold of dimension n≥ 5 with negative scalar curvature Sc_g not necessarily constant. If the following three conditions hold

then the Paneitz-Branson operator Pgn is positive.

The proof of this theorem is based on a new inequality which we give in the next section. Observe that in this theorem we allow the Yamabe constant of M to be negative (for the definition of the Yamabe constant one can see Section 4). Now, we give an outline of the rest of the paper. In Section 2 we prove Theorem 1.1 and give another result for the positivity of the operator Pgn when the scalar curvature is positive. In Section 3 we investigate the negativity of the eigenvalues of Pgn in an Einstein manifolds, in particular we prove the following.

Theorem 1.2. Given (M,g) an Einstein manifold of dimension n≥ 5 with negative scalar curvature. If

where λ1 is the first eigenvalue of the Laplacian operator Δg, then the first eigenvalue µ₁ of the Paneitz-Branson operator is strictly negative.

Moreover, we also prove in this section, that it is possible to choose a metric g on the manifold M such that Pgn has multiple negative eigenvalues. Finally, in the last section we give several inequalities related to the mean value of the Q-curvature, as an example we prove the following result

Theorem 1.3. Let M be a compact Riemannian manifold endowed with a Yamabe metric g. If the scalar curvature Scg≥n(n−1), then

where λ1 is the first eigenvalue of the Laplacian operator Δg and Sn,h denotes the round sphere endowed with its canonical metric.

Let (M,g) be a compact Riemannian manifold of dimension n≥ 5, we say that the Paneitz-Branson operator Pgn is positive [8] if

The conditions under which the operator Pgn is positive have been intensively studied. For example it was considered by Gursky [11] for dimension n=4, Yang and Xu [22] for dimension n≥ 5, Hebey and Robert [14], and recently by Gursky and Malchiodi [12].

Yang and Xu in [22] showed that when the dimension of (M,g) is n ≥ 6, if the Yamabe invariant of g is non negative and the Q-curvature is positive, then with respect to any conformal metric of positive scalar curvature, the Paneitz-Branson operator is positive.

In this section, we are concerned with finding sufficient conditions on the curvature of g, to ensure the positivity of the Paneitz-Branson operator Pgn.

As a first result we have

Theorem 2.1. Let (M,g) be a compact Riemannian manifold of dimension n≥ 5 with positive non constant scalar curvature and positive Q-curvature. If

then the Paneitz-Branson operator is positive.

Proof. To prove this theorem, we use an idea from [22].

First, we multiply both sides of (1.2) by u, and integrate by parts,

where a_n as in (1.3). An application of the Bochner formula together with condition (2.1) gives

which implies the positivity of Pgn.

Now, to prove Theorem 2.1, we need the following lemma.

Lemma 2.2. ∀u∈H22(M) we have

Proof. Let u be a smooth function, and consider the following tensor with local coordinates

The tensorial norm of T with respect to the metric g is

Since |T|g2≥0, it follows that

Therefore, an integration by parts give us

Now, by the density of C∞(M) in H22(M) with respect to the norm

one can assume that u∈H22(M) (instead of C∞(M)) and inequality (2.5) remains valid.

We are now in position to prove Theorem 1.1.

Proof. We begin with (2.2)

An application of the Bochner formula, together with condition (1.7) gives

Now, we apply inequality (2.4) with condition (1.8), to obtain

which implies by assumption (1.6), that Pgn is positive.

Our first observation is that the negativity of the quantity

is a sufficient condition for the negativity of the first eigenvalue µ₁ of the operator Pgn.

Indeed by the variational definition of µ₁, we have

where H22(M) is the Sobolev space defined as the completion of C∞(M) with respect to the norm

It follows, in the particular case of u≡1, that

So, if

then μ1<0. Thus it is clear that though the scalar curvature of the manifold is positive; the Paneitz-Branson operator can have negative eigenvalues. The following is a typical example.

Example 3.1. Let (S4,h) be the standard round sphere of dimension 4 and (Σ3,g0) a hyperbolic manifold. We equip the product manifold

with the product metric

Thus (M⁷,g) is a conformally flat manifold, since it is the product of two manifolds with sectional curvatures of opposite sign Kp(S4)=+1, Kp(Σ3)=−1. Moreover, the Ricci curvatures of S4 and Σ3 are respectively

and consequently, the scalar curvatures are

Finally, the scalar curvature of M⁷ is Scg=6 and |Ricg|g2=48.

Thus, as a conclusion, the manifold M⁷ is conformally flat manifold with positive scalar curvature (Scg=6) but according to formula (1.4) with negative Q-curvature (Qg=−218), which implies that the first eigenvalue µ₁ of the Paneitz-Branson operator is strictly negative.

However there is an important class of manifolds for which condition (3.3) is not satisfied, for example the Einstein manifolds (note that the previous example is not an Einstein manifold). Indeed, if the scalar curvature Sc_g of an Einstein manifold is a non zero constant, then the Q-curvature is strictly positive constant and satisfied

But still in this situation, the Paneitz-Branson operator has negative eigenvalues; Theorem 1.2 gives us an example. In what follow we prove this theorem.

Proof. Let (M,g) be an Einstein manifold of dimension n≥5 with negative scalar curvature, and u∈H22(M).

In the particular case when u≡φ1, where φ1 is an eigenfunction corresponding to the first eigenvalue λ1 of Δg; (3.5) becomes

So, if λ1 is satisfied (1.9), then it is obvious that the quantity

is strictly negative, which implies immediately the negativity of the first eigenvalue µ₁ of the operator Pgn by the variational definition (3.2) of µ₁.

It turns out that Riemannian manifolds with negative scalar curvature are the most favoured example for the Paneitz-Branson operator to have negative eigenvalues. In fact, in [7] Canzani et al provided several examples of manifolds of negative curvature for which the Yamabe operator

has multiple negative eigenvalues. In particular, they proved the following

Theorem 3.2. Let (M,g) be a compact connected Riemannian manifold. Then, for every m there is a metric g on M such that the Yamabe operator L_g has at least m negative eigenvalues counted with multiplicity.

Below, we extend this result to the Paneitz-Branson operator on compact connected Einstein manifolds of dimension n=5.

Theorem 3.3. Let (M,g) be a compact connected Einstein manifold of dimension 5. For every m∈ℕ⋆ there is a metric g₀ on M such that the Paneitz-Branson operator Pg0n has at least m negative eigenvalues counted with multiplicity.

The proof of this result is based on an idea of Canzani et al [7].

Proof. Let (M,g) be a compact connected Einstein manifolds of dimension n=5. By a result of Lohkamp [16], for any λ∈ℝ, there exist a metric g₀ such that λ is the first eigenvalue of Δg0 of multiplicity m, the volume Vg0(M) of M with respect to g₀ satisfies Vg0(M)=1, and Ricg0≤−m2g.

Since (M,g) is an Einstein manifold, so we may assume

Thus, ∀u∈H22(M), the Paneitz-Branson operator Pg0n take the form

which implies

In particular, for u≡φ1 the eigenfunction associated to the first eigenvalue λ of Δg0 with multiplicity m, (3.7) yields

So, if we set λ=12m2, then it follows from (3.8), that μ1=−7256m4 is a negative eigenvalue for the Paneitz-Branson operator with multiplicity m.

An other example of manifolds such that the Paneitz-Branson operator admits multiple negative eigenvalues is also presented by Canzani et al in [7]. Before stating this result, we need to introduce the notion of the GJMS operator Pgk.

Let (M,g) be a compact Riemannian manifold of dimension n≥ 3. Let k be a positive integer such that n>2k. In [10], Graham-Jenne-Mason-Sparling defined a differential operator denoted by Pgk. From the conformal geometric point of view, this operator can be considered as a generalization of both the Yamabe operator given in (3.6) and the Paneitz-Branson operator given in (1.2). More precisely, this operator is conformally covariant in the sense that if g˜:=φ4n−2kg where φ∈C∞(M),φ>0, then

Moreover, Pgk is self-adjoint with respect to the L2-scalar product. This operator is intimately related to the geometric quantity of Q-curvature, denoted as Qgk and satisfying

A lot of work has been devoted to the study of the GJMS operator, see, for example, [17],[3],[19],[15], and [21], and the references therein.

Having defined the GJMS operator, we can now state the result of Canzani et al.

First, Let (M,g) be a compact hyperbolic product manifold

of dimension n, equipped with a product metric

where (N,g₁) is an hyperbolic manifold of dimension n-2, and (Σ,g2) is an hyperbolic surface with genus s≥2. Notice that (M,g) is an Einstein manifold with Ricg=−g.

Theorem 3.4. For every m∈ℕ, we can choose the hyperbolic metric g₂ on Σ so that the GJMS operator Pgk has at least m negative eigenvalues for all odd integers k≤n−12.

If we further assume that n=4l or n=4l+1 for some l∈ℕ, then the same conclusion holds for all integers k≥n2.

It seems that this result does not cover the case of the Paneitz-Branson operator in manifolds of five dimension.

Indeed, if n=5, then either k is an odd integer less then 2, i.e, k=1 and in this case we have only the Yamabe operator Pg1, or l=1, i.e., n=4l+1, and in this case k≥52. This means all GJMS operators are of order greater than 3, and as is obvious, the Paneitz-Branson operator does not appears in this sequence of operators. So, in this section and by the same technique, we investigate the case of the Paneitz-Branson operator on manifolds of dimension 5, and we prove the following.

Theorem 3.5. Let (M,g) be a compact hyperbolic product manifold

of dimension 5, equipped with a product metric

where (N,g1) is an hyperbolic manifold of dimension 3, and (Σ,g2) is an hyperbolic surface with genus s≥ 2. For any m∈ℕ, there exists a metric g₂ on Σ, such that the Paneitz-Branson operator has at least m negative eigenvalues.

It should be mentioned that the basic idea underlying the presented proof was borrowed from [7].

Proof. Let (N,g1) be a hyperbolic manifold of dimension 3, and (Σ,g2) be a hyperbolic surface with genus s≥2. Consider the product manifold M=N×Σ equipped with the product metric g:=g1⊗1+1⊗g2.

After scaling by a positive constant we may assume that the scalar curvature Sc_g of the manifold M is

In other words, (M,g) is an Einstein manifold; therefore, the operator Pgn has the following formula

Let λ be an eigenvalue of Δg2 and let φ be an eigenfunction associated to λ. If we regard φ as a function on M, then

Combining this with (3.9), we then see that φ is an eigenfunction of Pgn with eigenvalue

Now, let m∈ℕ; Since 14∈740,38 and Σ has genus s≥ 2, so it follows by a result of Buser [6] that one can find an hyperbolic metric g₂ on Σ, such that the Laplacian Δg2 has at least m eigenvalues which belong to 740,38.

Therefore, we infer by (3.10) that, for each eigenvalue λ of Δg2 in the interval 740,38; there exists a negative eigenvalue μ(λ) for the Paneitz-Branson operator Pgn. As a conclusion, the operator Pgn has m negative eigenvalues. This completes the proof of Theorem 3.5.

We finish this section with a classification result which comes as a consequence of the negativity of the first eigenvalue of the Paneitz-Branson operator.

Proposition 3.6. Let (M,g) be a compact Riemannian manifold of dimension 5 with non negative Q-curvature, and negative scalar curvature. Assume that

If the first eigenvalue µ₁ of the Paneitz-Branson operator is non positive, then (M,g) is conformally diffeomorphic to the round sphere (S5,h).

To prove this result we need the following lemma of Tashiro [20]. For the following version one can see [13] page 291.

Lemma 3.7. Let (M,g) be a compact Riemannian manifold of dimension n≥2. If there exist a non constant function f∈C∞(M); such that

then (M,g) is conformally diffeomorphic to the round sphere Sn,h.

Proof. Let (M,g) be a compact Riemannian manifold of dimension 5; and let u∈H22(M). By formula (2.2) and since Qg≥0, we have

using the formula

with the Bochner formula

we obtain

So, if (3.11) holds, then

Now, if the first eigenvalue µ₁, of the Paneitz-Branson operator Pgn is non positive, then it follows from (3.15) in the particular case when u≡φ, where φ is the eigenfunction corresponding to the first eigenvalue µ₁ of Pgn, that

which implies that

or, equivalently

Thus, (M,g) is conformally diffoemorphic to the round sphere (S5,h) by Lemma 3.7.

Our purpose in this section is to get an upper bound for the quantity κg, where

with

in terms of curvatures such as scalar curvature and Ricci curvature. A famous result in this subject is the result of Gursky in [11] for four-dimensional Riemannian manifolds.

Theorem 4.1. Let (M4,g) be a smooth compact four-dimensional Riemannian manifold. If the Yg(M)≥0 then κg≤8π2. Moreover, Yg(M)≥0 and κg=8π2 if and only if (M4,g) is conformally equivalent to the round sphere.

There is no analogue to this theorem for manifolds of greater dimension (n≥5) or for manifolds with negative Yamabe constant. Almost all that is known in this direction is Theorem 1.3 in the case of manifolds with Yamabe metric.

Now, we begin by the following lemma which is useful for the sequel.

Lemma 4.2. Let (M,g) be a compact Riemannian manifold of dimension n≥5 with negative scalar curvature, not necessarily constant. Its Yamabe constant Yg(M) satisfies

It is worth noticing that the previous inequality give us a lower bound for the Yamabe constant, which is negative since the scalar curvature is negative. From now on, we use the following notations.

The quantity

denotes the norm of a function u in the Lebesgue space Lp(M), and 2⋆=2nn−2 the critical Sobolev exponent. Now we prove the lemma.

Proof. First let us recall the definition of the Yamabe constant:

By the Hölder's inequality and since Scg(x)<0, we have

which implies that the number −n−24(n−1)‖Scg‖n2 is a lower bound for

Hence

Now, we state our main result

Theorem 4.3. Let (M,g) be a compact Riemannian manifold of dimension n≥5 with non constant negative scalar curvature (Scg(x)<0, ∀x∈M). If the Q-curvature satisfies Qg≥0, then

where

Before proving this theorem, let us make the following remark.

If we integrate (4.1) over M, then it follows by the divergence theorem and the inequality |Ricg|g2≥1nScg2 that

So, it appears that (4.3) is an analogous inequality to (4.5), and with the same dimensional constant n2−48n(n−1)2.

Now, we prove Theorem 4.3.

Proof. Multiply (4.1) by Sc_g, we have, after integration by parts,

where an=n3−4n2+16n−168(n−1)2(n−2)2. Since |Ricg|2≥1nScg2, we have

Now, by the definition of the Yamabe constant Y_g(M) we get

and combining this with (4.6) one gets

Or equivalently, since Scg<0

Now using Lemma 4.3 together with the following Hölder's inequality

we deduce that

Therefore, from (4.9) and the fact Scg<0, it follows that

which proves (4.3).

Now, it remains for us to prove Theorem 1.3.

Proof. Let (M,g) be a compact Riemannian manifold endowed with a Yamabe metric. As is well known, the scalar curvature Sc_g of M is constant and satisfied by the Aubin estimate [2]

where λ₁ is the first eigenvalue of the Laplacian operator Δg.

Now, An integration by parts of the formula (4.1) over M gives us

where a_n as in (1.3). Since |Ricg|2≥1nScg2, we have

and with (4.10) we infer

where Vg(M) stands for the volume of M with respect to g.

Recall now from [1] that the scalar curvature satisfied

where Y_g(M) is the Yamabe constant of M, and Vh(Sn) is the volume of § with respect to h.

If Scg≥n(n−1), then it follows from (4.13) that

Combining (4.14) with (4.12), we infer

Recall that Qh=n(n2−4)8, the inequality (1.10) follows.

The author is very grateful to the reviewer and editor for providing valuable suggestions to improve the quality of this paper.

This work was supported by P.R.F.U. Research Grant number C00L03EP130120190001.