A Symmetric 2-Tensor canonically associated to Q-curvature and its applications

We define a symmetric 2-tensor, called the J-tensor, canonically associated to the Q-curvature on any Riemannian manifold with dimension at least three. The relation between the J-tensor and the Q-curvature is like that between the Ricci tensor and the scalar curvature. Thus the J-tensor can be interpreted as a higher-order analogue of the Ricci tensor. This tensor can be used to understand the Chang-Gursky-Yang theorem on 4-dimensional Q-singular metrics. We show that an almost-Schur lemma holds for the Qcurvature, yielding an estimate of the Q-curvature on closed manifolds.