Compactness of conformal metrics with constant Q-curvature. I

We study compactness for nonnegative solutions of the fourth order constant Q-curvature equations on smooth compact Riemannian manifolds of dimension ≥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 ≤9. Moreover, assuming in addition that a positive mass type theorem holds for the Paneitz operator, we prove compactness in C⁴. Positive mass type theorems have been verified recently on LCF manifolds or manifolds of dimension ≤7, when the Yamabe invariant is positive. We also prove that, for dimension ≥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 ≥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.