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
where and .
Hence, for 3-manifolds, the Q-curvature of is now given as follows
while the Paneitz operator acts on a smooth function on via
In a joint paper published in Calculus of Variations and Partial Differential Equations in 2004, see this link, Hang and Yang defined some new conditions called (NN), (P), (NN+) and (P+) associated to the Paneitz operator for 3-manifolds. To be precise, associated to the Paneitz operator, we define
Definition. For any , we say:
- satisfies condition (NN) if whenver for some then ;
- satisfies condition (P) if whenver for some then ;
- satisfies condition (NN+) if whenver for some and then ;
- satisfies condition (P+) if whenver for some and then .
The main use of the (NN) condition is that if satisfies (NN) and there exists some such that for some and then
Therefore, if then is simply a constant multiple of the Green function of the Paneitz operator with the pole .
In future posts, I will explain these conditions in details.