### Article

Kyungpook Mathematical Journal 2022; 62(4): 751-767

**Published online** December 31, 2022

Copyright © Kyungpook Mathematical Journal.

### On the Paneitz-Branson Operator in Manifolds with Negative Yamabe Constant

Ali Zouaoui

Department of Mathematics, Mustapha Stambouli University, Mascara 29000, Maâmounia-Algeria

e-mail : ali.zouaoui@univ-mascara.dz

**Received**: September 15, 2021; **Revised**: October 9, 2022; **Accepted**: October 10, 2022

### Abstract

This paper deals with the Paneitz-Branson operator in compact Riemannian manifolds with negative Yamabe invariant. We start off by providing a new criterion for the positivity of the Paneitz-Branson operator when the Yamabe invariant of the manifold is negative. Another result stated in this paper is about the existence of a metric on a manifold of dimension 5 such that the Paneitz-Branson operator has multiple negative eigenvalues. Finally, we provide new inequalities related to the upper bound of the mean value of the

**Keywords**: Paneitz-Branson operator, Yamabe invariant,

### 1. Introduction

Let

where _{g}

This operator is conformally covariant in the following sense: if

Moreover, this operator is intimately related to a conformally invariant

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

Given a smooth compact Riemannian manifold

where

The Paneitz-Branson operator is also conformally covariant in this sense: if

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 _{g}

then the Paneitz-Branson operator

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

**Theorem 1.2.** Given

where _{1}

Moreover, we also prove in this section, that it is possible to choose a metric

**Theorem 1.3.** Let

where

### 2. Positivity of the Paneitz-Branson operator

Let

The conditions under which the operator

Yang and Xu in [22] showed that when the dimension of

In this section, we are concerned with finding sufficient conditions on the curvature of

As a first result we have

**Theorem 2.1.** Let

then the Paneitz-Branson operator is positive.

First, we multiply both sides of (1.2) by

where _{n}

which implies the positivity of

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

**Lemma 2.2.**

The tensorial norm of

Since

Therefore, an integration by parts give us

Now, by the density of

one can assume that

We are now in position to prove Theorem 1.1.

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

### 3. Multiple Negative Eigenvalues for the Paneitz-Branson Operator

Our first observation is that the negativity of the quantity

is a sufficient condition for the negativity of the first eigenvalue _{1}

Indeed by the variational definition of _{1}

where

It follows, in the particular case of

So, if

then

**Example 3.1.** Let

with the product metric

Thus ^{7},g)

and consequently, the scalar curvatures are

Finally, the scalar curvature of ^{7}

Thus, as a conclusion, the manifold ^{7}_{1}

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 _{g}

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.

In the particular case when

So, if

is strictly negative, which implies immediately the negativity of the first eigenvalue _{1}_{1}

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 _{g}

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

**Theorem 3.3.** Let _{0}

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

_{0}_{0}

Since

Thus,

which implies

In particular, for

So, if we set

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

Let

Moreover,

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

of dimension

where _{1})

**Theorem 3.4.** For every _{2}

If we further assume that

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

Indeed, if

**Theorem 3.5.** Let

of dimension 5, equipped with a product metric

where _{2}

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

After scaling by a positive constant we may assume that the scalar curvature _{g}

In other words,

Let λ be an eigenvalue of

Combining this with (3.9), we then see that

Now, let _{2}

Therefore, we infer by (3.10) that, for each eigenvalue λ of

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

If the first eigenvalue _{1}

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

**Lemma 3.7.** Let

then

using the formula

with the Bochner formula

we obtain

So, if (3.11) holds, then

Now, if the first eigenvalue _{1}_{1}

which implies that

or, equivalently

Thus,

### 4. Upper Bound for the Mean Value of the Q -curvature

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

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

There is no analogue to this theorem for manifolds of greater dimension (

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

**Lemma 4.2.** Let

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

By the Hölder's inequality and since

which implies that the number

Hence

Now, we state our main result

**Theorem 4.3.** Let

where

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

If we integrate (4.1) over

So, it appears that (4.3) is an analogous inequality to (4.5), and with the same dimensional constant

Now, we prove Theorem 4.3.

_{g}

where

Now, by the definition of the Yamabe constant _{g}(M)

and combining this with (4.6) one gets

Or equivalently, since

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

we deduce that

Therefore, from (4.9) and the fact

which proves (4.3).

Now, it remains for us to prove Theorem 1.3.

_{g}

where _{1}

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

where _{n}

and with (4.10) we infer

where

Recall now from [1] that the scalar curvature satisfied

where _{g}(M)

If

Combining (4.14) with (4.12), we infer

Recall that

### Acknowledgements.

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

### Footnote

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

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