Sign of Green's function of Paneitz operators and the Q curvature

TL;DR

This paper establishes a conformally invariant condition for positive Q curvature existence in certain metrics and compares Green's functions of key conformal operators.

Contribution

It provides a necessary and sufficient condition for positive Q curvature and analyzes inequalities between Green's functions of the conformal Laplacian and Paneitz operators.

Findings

Derived a conformally invariant criterion for positive Q curvature.

Proved inequalities relating Green's functions of conformal Laplacian and Paneitz operators.

Established conditions in metrics with positive Yamabe invariant.

Abstract

In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive Q curvature. The condition is conformally invariant. We also prove some inequalities between the Green's functions of the conformal Laplacian operator and the Paneitz operator.