Let be a compact Riemannian -manifold, and let and denote the Ricci tensor and the scalar curvature of , respectively.
A natural conformally invariant in dimension four is
This is commonly refered to the -curvature of metric . The term
is commonly denoted by where
the Schouten tensor of and
the second elementary symmetric polynomial in its eigenvalues.
Note that, under a conformal change of the metric
the quantity transforms according to
where denotes the Paneitz operator with respect to .Keep in mind that the Paneitz operator is conformally invariant in the sense that
for any conformal metric
For any , the operator acts on a smooth function on via
which plays a similar role as the Laplace operator in dimension two.
We know from the Gauss-Bonnet-Chern theorem that
where denotes the Weyl tensor. It now follows from the fact that is conformally invariant that
is also conformally invariant. Indeed, this can be seen from the rule developed in this note.
The prescribed -curvature problem can be formulated as follows:
Whether on a given four-manifold there exists a conformal metric for which the -curvature of equals the prescribed function ?
In terms of PDE, this is related to solving the following equation
This equation is the Euler-Lagrange equation of the functional
A lot of progresses have been done on this problem.