Compactness of singular solutions to the sixth order GJMS equation

TL;DR

This paper investigates the compactness of conformally flat singular metrics with positive sixth order Q-curvature on punctured spheres, introducing a necksize concept and proving convergence under certain conditions.

Contribution

It introduces a new notion of necksize for these metrics and establishes conditions for compactness and convergence in the moduli space.

Findings

Bounded necksize ensures subsequence convergence in the Gromov--Hausdorff sense.

A classification of local asymptotic behavior near singularities is utilized.

A homological invariant is introduced for further research applications.

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.