Compactness and non-compactness theorems of the fourth- and sixth-order constant $Q$-curvature problems

We provide a complete resolution to the question of compactness for the full solution sets of the fourth-order and sixth-order constant $Q$-curvature problems on smooth closed Riemannian manifolds not conformally diffeomorphic to the standard unit $n$-sphere, provided the associated conformally covariant differential operator has a positive Green's function. Firstly, we prove that the solution set of the fourth-order constant $Q$-curvature problem is $C^4$-compact in dimensions $5 \le n \le 24$. For $n \ge 25$, an example of an $L^{\infty}$-unbounded sequence of solutions has been known for over a decade (Wei and Zhao). Additionally, the compactness result for $5 \le n \le 9$ was established by Li and Xiong. Secondly, we demonstrate that the solution set of the sixth-order constant $Q$-curvature problem is $C^6$-compact in dimensions $7 \le n \le 26$, whereas a blow-up example exists for $n \ge 27$. Our main observation is that the linearized equations associated with both $Q$-curvature problems can be transformed into overdetermined linear systems, which admit nontrivial solutions due to unexpected algebraic structures of the Paneitz operator and the sixth-order GJMS operator. This key insight not only plays a crucial role in deducing the compactness result for high-dimensional manifolds, but also reveals an elegant hierarchical pattern with respect to the order of the conformally covariant operators, suggesting the possibility of a unified theory of the compactness of the constant $Q$-curvature problems of all admissible even integer orders.