Compactness of conformal metrics with constant $Q$-curvature. I

Abstract

We study compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, we prove that all solutions are universally bounded. If the $Q$-curvature is $1$, assuming that Paneitz operator's kernel is trivial and its Green function is positive, we establish universal energy bounds on manifolds which are either locally conformally flat (LCF) or of dimension $\le 9$. Moreover, assuming in addition that a positive mass type theorem holds for the Paneitz operator, we prove compactness in $C^4$. Positive mass type theorems have been verified recently on LCF manifolds or manifolds of dimension $\le 7$, when the Yamabe invariant is positive. We also prove that, for dimension $\ge 8$, the Weyl tensor has to vanish at possible blow up points of a sequence of blowing up solutions. This implies the compactness result in dimension $\ge 8$ when the Weyl tensor does not vanish anywhere. To overcome difficulties stemming from fourth order elliptic equations, we develop a blow up analysis procedure via integral equations.

40 pages, published version

References in corpus (1)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)

Cited by in corpus (8)

• Lectures on the fourth order Q curvature equation
• A non-compactness result on the fractional Yamabe problem in large dimensions
• Constant Q-curvature metrics with Delaunay ends: the nondegenerate case
• Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary
• Improved Beckner's inequality for axially symmetric functions on ${\mathbb{S}}^n$
• Classification theorems for solutions of higher order boundary conformally invariant problems, I
• Compactness of solutions to nonlocal elliptic equations
• Compactness of solutions to higher order elliptic equations