The Paneitz-Sobolev constant of a closed Riemannian manifold and an application to the nonlocal $\mathbf{Q}$-curvature flow

TL;DR

This paper proves that for certain high-dimensional, non-conformally flat closed manifolds, the Paneitz-Sobolev constant is strictly less than that of the sphere, and applies this to establish convergence of a nonlocal Q-curvature flow.

Contribution

It establishes a Paneitz-Sobolev constant inequality for high-dimensional non-conformally flat manifolds and applies this to prove convergence of the nonlocal Q-curvature flow.

Findings

Paneitz-Sobolev constant is less than that of the sphere for certain manifolds.

The result generalizes Aubin's 1976 analogy to higher dimensions.

The convergence of the nonlocal Q-curvature flow is recovered using this inequality.

Abstract

In this paper, we establish that: Suppose a closed Riemannian manifold $(M^n,g_0)$ of dimension $\geq 8$ is not locally conformally flat, then the Paneitz-Sobolev constant of $M^n$ has the property that $q(g_0)<q(S^n)$. The analogy of this result was obtained by T. Aubin in 1976 and had been used to solve the Yamabe problem on closed manifolds. As an application, the above result can be used to recover the sequential convergence of the nonlocal Q-curvature flow on closed manifolds recently introduced by Gursky-Malchiodi.