Compactness of conformal metrics with constant $Q$-curvature of higher order
Saikat Mazumdar, Bruno Premoselli
TL;DR
This work addresses the compactness of conformal metrics with constant $Q$-curvature of order $2k$ on a closed manifold, for all $1\le k< n/2$, under a positivity-preserving hypothesis for the GJMS operator. The authors develop a robust blow-up analysis that does not rely on explicit formulas for $P_g$ by exploiting Juhl's recursive expansions, obtain Weyl vanishing at concentration points, and leverage a Pohozaev identity together with positive mass assumptions to preclude blow-up. The result is the first general compactness theorem for arbitrary $k$ in the polyharmonic setting and suggests that the critical dimension threshold for compactness grows with $k$, a phenomenon that deepens our understanding of higher-order $Q$-curvature problems and informs potential full-compactness results.
Abstract
Let $k\ge1$ 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 $1 \le k < \frac{n}{2}$, under the following assumptions: $(M,g)$ is locally conformally flat and $P_g$ has positive mass in $M$, or $2k+1 \le n \le 2k+5$ and $P_g$ has positive mass in $M$, or $n \ge 2k+4$ and $|\text{W}_g|_g >0$ in $M$. For an arbitrary $1 \le k < \frac{n}{2}$, 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 $1 \le k < \frac{n}{2}$ 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 $k \to + \infty$.