Compactness of singular solutions to the sixth order GJMS equation

TL;DR

The paper studies the compactness of the moduli space of complete, conformally flat metrics on a finitely punctured sphere with constant sixth-order $Q^6$-curvature. It recasts the sixth-order PDE via a cylindrical (Emden–Fowler) transform, analyzes canonical Delaunay and spherical models, and introduces necksize and Pohozaev invariants to control degenerations. The main result shows that, provided punctures stay separated and necksizes stay away from zero, the corresponding subset is sequentially compact in the Gromov–Hausdorff topology, with a detailed blow-up/removable-singularity analysis underpinning the argument. The work also develops a homological invariant and highlights implications for compactifications of higher-order conformal moduli spaces.

Abstract

We study compactness properties of the set of conformally flat singular metrics with constant, positive sixth order Q-curvature on a finitely punctured sphere. Based on a recent classification of the local asymptotic behavior near isolated singularities, we introduce a notion of necksize for these metrics in our moduli space, which we use to characterize compactness. More precisely, we prove that if the punctures remain separated and the necksize at each puncture is bounded away from zero along a sequence of metrics, then a subsequence converges with respect to the Gromov--Hausdorff metric. Our proof relies on an upper bound estimate which is proved using moving planes and a blow-up argument. This is combined with a lower bound estimate which is a consequence of a removable singularity theorem. We also introduce a homological invariant which may be of independent interest for upcoming research.

Figures (1)

• Figure 1: The two possible degenerations in the moduli space $\mathcal{M}_4^6$.

Theorems & Definitions (38)

• Remark 1.1
• Theorem A
• Theorem B
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Definition 2.1
• Remark 2.2