Global Convergence of the Gursky-Malchiodi $Q$-curvature Flow

In their seminal work, Gursky and Malchiodi introduced a non-local conformal flow in dimensions $n \geq 5$ to resolve the constant $Q$-curvature problem. They proved sequential convergence of the flow for initial metrics with positive scalar curvature and $Q$-curvature, provided the energy was sufficiently small. In this paper, we prove the global convergence of the flow for arbitrary initial energy under the same positivity assumptions by establishing a non-local version of the Łojasiewicz-Simon inequality for the Paneitz-Sobolev quotient along the flow. We construct test bubbles and estimate their Paneitz-Sobolev quotients, a strategy that was carried out in the celebrated work of Brendle in the context of the Yamabe flow. We develop a more geometric and systematic proof that addresses the algebraic and computational complexity inherent in the $Q$-curvature and the Paneitz operator. Along the way, we derive a stability inequality for the Paneitz-Sobolev quotient using a higher-order Koiso-Bochner formula established in recent work of Bahuaud, Guenther, Isenberg, and Mazzeo.

💡 Research Summary

In this paper the authors settle a long‑standing open problem concerning the global behavior of the non‑local conformal flow introduced by Gursky and Malchiodi for prescribing constant Q‑curvature in dimensions (n\ge5). The original work proved long‑time existence and sequential convergence only under a small‑energy hypothesis, assuming the initial metric has non‑negative scalar curvature and semi‑positive Q‑curvature. The present work removes the small‑energy restriction and establishes that the flow converges for arbitrary initial energy, provided the same positivity conditions hold.

The central analytic tool is a non‑local version of the Łojasiewicz–Simon inequality adapted to the Paneitz–Sobolev quotient \