Obstructions to prescribed Q-curvature of complete conformal metrics on $\mathbb{R}^n$

TL;DR

The work addresses obstructions to realizing a prescribed Q-curvature by complete conformal metrics $g=e^{2u}|dx|^2$ on $\mathbb{R}^n$ with finite total Q-curvature, recasting the problem as $(-\Delta)^{\frac{n}{2}}u=f e^{nu}$. It develops a framework based on integral estimates, logarithmic potentials, and Pohozaev-type identities to derive barrier results, including a Bonnet-Mayer type obstruction when $f$ is bounded below by 1 at infinity and a Pohozaev/Kazdan–Warner obstruction for positive $f$ with $\frac{x\cdot \nabla f}{f}\ge -\frac{n}{2}$. The paper further analyzes decay conditions on $f$ (positive, nonpositive, and sign-changing) and their implications for normal solutions and the finiteness of total Q-curvature, establishing sharpness in the decay rate and linking to existing 2D/4D results via Li’s Q-curvature theory. Collectively, these results clarify when complete conformal metrics with prescribed Q-curvature can exist and how decay and sign of $f$ constrain such existence, guiding both sharpness questions and future variational constructions.

Abstract

We provide some obstructions to the prescribed Q-curvature problem for the complete conformal metrics on $\mathbb{R}^n$ with finite total Q-curvature. One of them is a Bonnet-Mayer type theorem with respect to Q-curvature. Others are related to the decay rate of the prescribed functions.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Lemma 2.1