A perturbation approach for Paneitz energy on standard three sphere

We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetrization for higher order variational problems.

Lectures on the fourth order Q curvature equation

We discuss some open problems and recent progress related to the 4th order Paneitz operator and Q curvature in dimensions other than 4.

Paneitz operator for metrics near $S^3$

We derive the first and second variation formula for the Greens function poles value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of conformal deformation. We also introduce a new invariant of the Paneitz operator and illustrate its close relation with the second eigenvalue and Sobolev inequality of Paneitz operator.