| Abstract: |
| In their seminal work [JEMS 2015], 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 initial energy is sufficiently small. The question of global convergence for large initial energy has remained open.
In this talk, we resolve this problem by proving global convergence of the flow for arbitrary initial energy under the same positivity assumptions. Our approach centers on establishing a non-local version of the \Lojasiewicz-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 (Invent 2006) 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 (2025). |
|