Paneitz operator for metrics near $S^3$

TL;DR

This paper investigates the stability of the Paneitz operator on the standard three sphere by deriving variation formulas, revealing nonpositive second variation, and introducing a new invariant linked to eigenvalues and Sobolev inequalities.

Contribution

It provides the first and second variation formulas for the Green's function pole's value of the Paneitz operator on $S^3$, and introduces a new invariant related to eigenvalues and Sobolev inequalities.

Findings

First variation vanishes at the standard sphere.

Second variation is nonpositively definite.

New invariant relates to eigenvalues and Sobolev inequalities.

Abstract

We derive the first and second variation formula for the Green's function pole's 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.