Compactness within the space of complete, constant Q-curvature metrics on the sphere with isolated singularities

TL;DR

The paper proves a sequential compactness result for the moduli space of complete, conformally flat metrics on the sphere minus k points with constant positive Q-curvature, by showing that if mutual puncture distances and Delaunay necksizes stay uniformly away from zero, the set is compact in the Gromov-Hausdorff sense. It achieves this via a detailed blow-up analysis, leveraging refined asymptotics near singularities, the Delaunay family of solutions, and new radial Pohozaev invariants that link necksize to integral monotonicity. The authors establish sharp a priori bounds, isolate potential blow-up scenarios, and exclude them using invariants and Liouville-type classifications, thereby obtaining a robust compactness theorem analogous to Pollack’s singular Yamabe results. The work advances understanding of the structure of singular Q-curvature metrics and provides tools potentially adaptable to related higher-order curvature problems.

Abstract

In this paper we consider the moduli space of complete, conformally flat metrics on a sphere with k punctures having constant positive Q-curvature and positive scalar curvature. Previous work has shown that such metrics admit an asymptotic expansion near each puncture, allowing one to define an asymptotic necksize of each singular point. We prove that any set in the moduli space such that the distances between distinct punctures and the asymptotic necksizes all remain bounded away from zero is sequentially compact, mirroring a theorem of D. Pollack about singular Yamabe metrics. Along the way we define a radial Pohozaev invariant at each puncture and refine some a priori bounds of the conformal factor, which may be of independent interest.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2: Chang, Han and Yang
• Theorem 3: Caristi and Mitidieri
• Theorem 4: Gursky and Malchiodi
• Lemma 5
• proof
• Remark 1
• Lemma 6
• Theorem 7
• Remark 2