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

Liuwei Gong, Sanghoon Lee, Juncheng Wei

TL;DR

The paper tackles the global convergence of the non-local fourth-order $Q$-curvature flow on closed manifolds with $n \ge 5$, under positivity assumptions on $Q$-curvature and nonnegative scalar curvature. By proving a new non-local Łojasiewicz–Simon inequality for the Paneitz–Sobolev quotient and constructing geometry-informed test bubbles, the authors obtain uniform energy control and a detailed blow-up analysis. A higher-order Koiso–Bochner formula is employed to manage Weyl-curvature terms, and a conformal geometric framework with mass-like quantities ensures sharp energy estimates. The main result is global convergence of the flow to a conformal metric with constant positive $Q$-curvature, resolving an open problem in the non-local, higher-order conformal setting and enriching the toolkit for non-local geometric flows.

Abstract

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.

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

TL;DR

The paper tackles the global convergence of the non-local fourth-order -curvature flow on closed manifolds with , under positivity assumptions on -curvature and nonnegative scalar curvature. By proving a new non-local Łojasiewicz–Simon inequality for the Paneitz–Sobolev quotient and constructing geometry-informed test bubbles, the authors obtain uniform energy control and a detailed blow-up analysis. A higher-order Koiso–Bochner formula is employed to manage Weyl-curvature terms, and a conformal geometric framework with mass-like quantities ensures sharp energy estimates. The main result is global convergence of the flow to a conformal metric with constant positive -curvature, resolving an open problem in the non-local, higher-order conformal setting and enriching the toolkit for non-local geometric flows.

Abstract

In their seminal work, Gursky and Malchiodi introduced a non-local conformal flow in dimensions to resolve the constant -curvature problem. They proved sequential convergence of the flow for initial metrics with positive scalar curvature and -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 -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.
Paper Structure (25 sections, 76 theorems, 406 equations)

This paper contains 25 sections, 76 theorems, 406 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Novelties of the method
  4. Organization of this paper
  5. Notations
  6. Preliminaries
  7. Elementary estimates
  8. Blow-up profile
  9. Variational theories for $Q$-curvature and Weyl tensor
  10. Deformations of $Q$-curvature
  11. Diffeomorphism invariance of curvatures
  12. Higher-order Koiso-Bochner formula
  13. Construction of test bubbles: $n \ge 8$ case
  14. Splitting of $S_2$ on standard sphere
  15. The second variation of total $Q$-curvature
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(M,g_0)$ be a closed manifold of dimension $n \ge 5$ that is not conformally equivalent to the standard sphere $S^n$. Assume that $(M,g_0)$ has semi-positive $Q$-curvature and nonnegative scalar curvature. Then the non-local flow where $u$ satisfies with exists for all time and converges to a metric with constant $Q$-curvature.

Theorems & Definitions (140)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: MR3420504
  • Lemma 2.2: MR3420504
  • Lemma 2.3: MR3420504
  • Corollary 2.4: MR3420504
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 130 more