The Yamabe problem for Q-curvature

TL;DR

This paper proves the existence of conformal metrics with constant Q-curvature on smooth closed Riemannian manifolds of dimension greater than four, using variational methods, Sobolev inequalities, and blow-up analysis.

Contribution

It establishes the existence of constant Q-curvature metrics under general conditions, combining variational techniques with new positivity arguments rooted in conformal covariance.

Findings

Existence of conformal metrics with constant Q-curvature on manifolds of dimension >4.

Development of a novel positivity argument for solutions.

Application of Sobolev inequalities and blow-up theory to solve the problem.

Abstract

In this paper we demonstrate that under general conditions there exists a metric in the conformal class of an arbitrary metric on a smooth, closed Riemannian manifold of dimension greater than four such that the $Q$-curvature of the metric is a constant. Existence of solutions is obtained through the combination of variational methods, second order Sobolev inequalities, and the $W^{2,2}$ blow-up theory developed by Hebey and Robert. Positivity of the solutions is obtained from a novel argument proven here for the first time that is rooted in the conformal covariance property of the Paneitz-Branson operator and the positive semidefiniteness of the second derivative of a $C^2$ function at a local minimum.