Q curvature prescription; forbidden functions and the GJMS null space

TL;DR

This paper investigates the constraints on Q-curvature functions on even conformal manifolds with non-trivial GJMS kernel, identifying forbidden functions and conditions preventing constant Q-curvature.

Contribution

It introduces a finite-dimensional space related to the GJMS null space and characterizes functions that cannot be realized as Q-curvature.

Findings

Identifies a finite-dimensional space N(Q) linked to the GJMS kernel.

Describes an infinite class of functions that cannot be Q-curvature.

Shows certain functions in N(Q) prevent constant Q-curvature.

Abstract

On an even conformal manifold $(M,c)$, such that the critical GJMS operator has non-trivial kernel, we identify and discuss the role of a finite dimensional vector space $N(Q)$ of functions determined by the conformal structure. Using these we describe an infinite dimensional class of functions that cannot be the Q-curvature $Q^g$ for any $g$ in $c$. If certain functions arise in $N(Q)$ then $Q^g$ cannot be constant for any $g$ in $c$.