| 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 {\bf sequential convergence} of the flow for initial metrics with positive scalar curvature and $Q$-curvature, provided the initial energy is sufficiently {\bf small}. The question of global convergence for large initial energy has remained open.
In this talk, we resolve this problem by proving {\bf global convergence} of the flow for {\bf arbitrary} initial energy under the same positivity assumptions. Our approach centers on establishing a non-local version of the \L{}ojasiewicz-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). |
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