2025/02/20 by Liuwei Gong, Gong, Liuwei, Seunghyeok Kim +3 · 2 citations
Mathematics · Computer Science · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2502.14237
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 C4-compact in dimensions 5 ≤ n ≤ 24. For n ≥ 25, an example of an L∞-unbounded sequence of solutions has been known for over a decade (Wei and Zhao). Additionally, the compactness result for 5 ≤ n ≤ 9 was established by Li and Xiong. Secondly, we demonstrate that the solution set of the sixth-order constant Q-curvature problem is C6-compact in dimensions 7 ≤ n ≤ 26, whereas a blow-up example exists for n ≥ 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.