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
Department of Mathematics, Mustapha Stambouli University, Mascara 29000, Maâmounia-Algeria
e-mail : email@example.com
Received: September 15, 2021; Revised: October 9, 2022; Accepted: October 10, 2022
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,
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
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 , , , , and , 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
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
Moreover, we also prove in this section, that it is possible to choose a metric
Theorem 1.3. Let
2. Positivity of the Paneitz-Branson operator
The conditions under which the operator
Yang and Xu in  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
which implies the positivity of
Now, to prove Theorem 2.1, we need the following lemma.
The tensorial norm of
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
Indeed by the variational definition of
It follows, in the particular case of
Example 3.1. Let
with the product metric
and consequently, the scalar curvatures are
Finally, the scalar curvature of
Thus, as a conclusion, the manifold
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
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
is strictly negative, which implies immediately the negativity of the first eigenvalue
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  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
Below, we extend this result to the Paneitz-Branson operator on compact connected Einstein manifolds of dimension
Theorem 3.3. Let
The proof of this result is based on an idea of Canzani et al .
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 . Before stating this result, we need to introduce the notion of the GJMS operator
A lot of work has been devoted to the study of the GJMS operator, see, for example, ,,,, and , and the references therein.
Having defined the GJMS operator, we can now state the result of Canzani et al.
Theorem 3.4. For every
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.
Theorem 3.5. Let
of dimension 5, equipped with a product metric
It should be mentioned that the basic idea underlying the presented proof was borrowed from .
After scaling by a positive constant we may assume that the scalar curvature
In other words,
Let λ be an eigenvalue of
Combining this with (3.9), we then see that
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
To prove this result we need the following lemma of Tashiro . For the following version one can see  page
Lemma 3.7. Let
using the formula
with the Bochner formula
So, if (3.11) holds, then
Now, if the first eigenvalue
which implies that
4. Upper Bound for the Mean Value of the
Our purpose in this section is to get an upper bound for the quantity
in terms of curvatures such as scalar curvature and Ricci curvature. A famous result in this subject is the result of Gursky in  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.
denotes the norm of a function
By the Hölder's inequality and since
which implies that the number
Now, we state our main result
Theorem 4.3. Let
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.
Now, by the definition of the Yamabe constant
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.
Now, An integration by parts of the formula (4.1) over
and with (4.10) we infer
Recall now from  that the scalar curvature satisfied
Combining (4.14) with (4.12), we infer
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
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