Remarks on GJMS operator of order six

TL;DR

This paper analyzes the sixth order GJMS operator, providing Green's function expansions and establishing existence results for prescribed Q-curvature on Einstein manifolds with positive scalar curvature.

Contribution

It offers new insights into the analysis of the sixth order GJMS operator and proves existence results for prescribed Q-curvature under specific geometric conditions.

Findings

Green's function expansions for $P_g^6$ in conformal normal coordinates

Existence of conformal metrics with prescribed Q-curvature on Einstein manifolds

Conditions involving Weyl tensor and function vanishing order for existence results

Abstract

We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Green's function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we manage to give some existence results of prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n \geq 10$, let $(M^n,g)$ be a closed Einstein manifold of positive scalar curvature and $f$ a smooth positive function in $M$. If the Weyl tensor is nonzero at a maximum point of $f$ and $f$ satisfies a vanishing order condition at this maximum point, then there exists a conformal metric $\tilde g$ of $g$ such that its $Q$-curvature $Q_{\tilde g}^6$ equals $f$.