The Paneitz operator in any dimension

Let us recall from this topic the following fact: Let be a compact Riemannian -manifold, and let and denote the Ricci tensor and the scalar curvature of , respectively. The so-called Paneitz operator acts on a smooth function on via

which plays a similar role as the Laplace operator in dimension two where is the de Rham differential. Associated to this operator is the notion of -curvature given by

Under the following conformal change

passing from to is easy through the following formula

Prescribed Q-curvature on 4-manifolds

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.