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

Let k≥1 be a positive integer and let Pg be the GJMS operator Pg 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 Pg we establish compactness, for an arbitrary 1 ≤ k < n/2, under the following assumptions: $(M,g)$ is locally conformally flat and Pg has positive mass in M, or 2k+1 ≤ n ≤ 2k+5 and Pg has positive mass in M, or n ≥ 2k+4 and |Wg|g >0 in M. For an arbitrary 1 ≤ k < n/2, the expression of Pg is not explicit, which is an obstacle to proving compactness. We overcome this by relying on Juhl's celebrated recursive formulae for Pg to perform a refined blow-up analysis for solutions of the Q-curvature equation and to prove a Weyl vanishing result for Pg. 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 → + ∞.