Of recent interest is the prescribed Q-curvature on closed Riemannian manifolds since it involves high-order differential operators.

In a previous post, I have talked about prescribed Q-curvature on 4-manifolds. Recall that for 4-manifolds, this question is equivalent to finding a conformal metric for which the Q-curvature of equals the prescribed function ? That is to solving

where for any , the so-called Paneitz operator acts on a smooth function on via

which plays a similar role as the Laplace operator in dimension two and the Q-curvature of is given as follows

Sometimes, if we denote by the negative divergence, i.e. , we obtain the following formula

Generically, for -manifolds, we obtain

and

where and .