TY - JOUR

T1 - Compactness of conformal metrics with constant Q-curvature. I

AU - Li, Yan Yan

AU - Xiong, Jingang

N1 - Funding Information:
Supported in part by NSF grants DMS-1203961 and DMS-1501004.Supported in part by NSFC 11501034, a key project of NSFC 11631002 and NSFC 11571019.

PY - 2019/3/17

Y1 - 2019/3/17

N2 - 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.

AB - 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.

KW - Compactness

KW - Fourth order equation

KW - Q-curvature

UR - http://www.scopus.com/inward/record.url?scp=85059856618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059856618&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.01.020

DO - 10.1016/j.aim.2019.01.020

M3 - Article

AN - SCOPUS:85059856618

VL - 345

SP - 116

EP - 160

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -