Compactness of conformal metrics with constant Q-curvature of higher order

2025/10/01 by Saikat Mazumdar, Mazumdar, Saikat, Bruno Premoselli +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2510.00888

openalex publication_date 2025/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k≥1 be a positive integer and let P_g be the GJMS operator P_g of order 2k on a closed Riemannian manifold (M,g) of dimension n>2k. We investigate the compactness of the set of conformal metrics to g with prescribed constant positive Q-curvature of order 2k- or, equivalently, of the set of positive solutions for the 2k-th order Q-curvature equation. Under a natural positivity-preserving condition on P_g we establish compactness, for an arbitrary 1 ≤ k < (n)/(2), under the following assumptions: (M,g) is locally conformally flat and P_g has positive mass in M, or 2k+1 ≤ n ≤ 2k+5 and P_g has positive mass in M, or n ≥ 2k+4 and |W_g|_g >0 in M. For an arbitrary 1 ≤ k < (n)/(2), the expression of P_g is not explicit, which is an obstacle to proving compactness. We overcome this by relying on Juhl's celebrated recursive formulae for P_g to perform a refined blow-up analysis for solutions of the Q-curvature equation and to prove a Weyl vanishing result for P_g. This is the first compactness result for an arbitrary 1 ≤ k < (n)/(2) and the first successful instance where Juhl's formulae are used to yield compactness. Our result also hints that the threshold dimension for compactness for the 2k-th order Q-curvature equation diverges as k → + ∞.

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Cited by

• Multiple Blow-Up Phenomena for Q-Curvature in High Dimensions

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• Local estimates for conformal Q-curvature equations