Compactness Properties for Geometric Fourth order Elliptic equations with Application to the Q-curvature Flow

TL;DR

This paper establishes the compactness of solutions to fourth order elliptic equations related to Q-curvature on compact 4-manifolds, leading to results on the global behavior of the Q-curvature flow and addressing an open problem in the field.

Contribution

It proves the compactness of solutions to perturbed Q-curvature equations and demonstrates the global existence and convergence of the Q-curvature flow on generic 4-manifolds.

Findings

Solutions are compact under L^1-perturbations

Global existence and convergence of Q-curvature flow

Positive answer to Malchiodi's open question

Abstract

We prove the compactness of solutions to general fourth order elliptic equations which are L^1-perturbations of the Q-curvature equation on compact Riemannian 4-maniods. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by product, we give a positive answer to an open question by A. Malchiodi on the existence of bounded Palais-Smale sequences for the Q-curvature problem when the Paneitz operator is positive with trivial kernel.