On a 2-dimensional compact Riemannian manifold without boundary, the prescribed scalar curvature problem in the flat case is equivalent to solving the following PDE
with is a given non-constant smooth function on and is the Laplace-Beltrami operator associated with the metric .
Simply by integrating both sides of the PDE, it is immediate to see that if solves the PDE, it would satisfy ; hence the candidate function must change sign in . In their elegant paper published in 1974, Kazdan and Warner showed that in addition to the sign-changing property of , it must also satisfy the following inequality
This is just a simple observation from integration by parts if we multiply both sides of the PDE by . Interestingly, Kazdan and Warner were able to show that the above two properties are also sufficient in the sense that it is enough to prove that the PDE is solvable.
In higher dimensions, perhaps, the most natural generalization of the operator is the well-known Paneitz operator and its corresponding Q-curvature, see this link.
Assume that is a compact Riemannian 4-manifold without boundary. We denote by the so-called Paneitz operator acting on any smooth function via the following rule
where by and we mean the Ricci tensor and the scalar curvature of , respectively.
Then the prescribed Q-curvature problem in the null case can be formulated as solving the following PDE
where again is the candidate, or the prescribed function. Notice that by flatness, we mean that the Q-curvature of the background metric is identically zero, i.e. in , see this link for details.
As far as we know, the prescribed Q-curvature problem in the null case was first studied by Ye and Xu in 2008. In their paper, they proved the same result as that obtained by Kazdan and Warner in 1974. Precisely, they showed that if
- The total integral and
then the PDE always admits a smooth solution.
Ye and Xu’s proof is variational. First, we try to minimize the associated energy functional to the problem given by
over the set
Then the condition helps us to conclude that the set is non-empty. Standard arguments plus the Lagrange multiplier then implies that there exist a constant and a function solving the following equation
Since is invariant up to an additive constant, we would expect that the new function will solve our PDE. However, to this end, it must be shown that . To this purpose, Ye and Xu showed that the minimum value of the above minimizing problem is in fact belong to a wider set
Then, if , they can construct a new test function belonging to the set which admits smaller energy. Hence, the proof follows.
However, as they already mentioned in the paper, the two conditions and are not necessary. Indeed, let be the standard 4-dimensional torus and be the standard flat metric. Then there exists a conformal change in such a way that the new Q-curvature is sing-changing with non-negative total integral .
As a consequence of a result due to Gursky, see this link, we expect that must hold. Notice that, only is invariant in the sense that for any . However, simply by multiplying by a large constant, we know that if the PDE is solvable for some with , then it is also solvable for any with arbitrarily large . Hence, any upper bound for should depend on as well.