A symmetric 2-tensor canonically associated to Q-curvature and its applications

Abstract

In this article, we define a symmetric 2-tensor canonically associated to Qcurvature called J-tensor on any Riemannian manifold with dimension at least three. The relation between J-tensor and Q-curvature is precisely like Ricci tensor and scalar curvature. Thus it can be interpreted as a higher-order analogue of Ricci tensor. This tensor can also be used to understand Chang-Gursky-Yang's theorem on 4-dimensional Q-singular metrics. Moreover, we show an Almost-Schur Lemma holds for Q-curvature, which gives an estimate of Q-curvature on closed manifolds.

Conformally variational Riemannian invariants

“…More recently, De Lellis and Topping [49] proved an L 2 -version of this result under the name "almost Schur lemma" for manifolds with nonnegative Ricci curvature; see [28,29] and [47] for analogues for manifolds with nonnegative scalar curvature and Ricci curvature bounded below, respectively. The almost Schur lemma was generalized to the fourth-order Qcurvature and its associated tensor S by the second-and third-named authors [52]. In this section we show that their result generalizes to any CVI; note that the observations leading to the analogue of Schur's lemma have already been pointed out by Gover and Ørsted [33].…”

“…Indeed, we conclude that (M, g) is not L-critical. Generalizing the approach to R-critical [53] and Q 4 -critical metrics [52], we prove Theorem 1.3 by using the weak unique continuation property. Proof.…”

“…This work is heavily influenced by previous work of Gover and Ørsted [33] which explored obstructions to prescribing the first such class of scalar invariants, conformally variational invariants, within a conformal class. It is also strongly influenced by recent generalizations of some of the above results to the Q-curvature [16,51,52] and the renormalized volume coefficients [14,15,37].…”

The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications

“…From this general result, we will extract as corollaries an almost-Schur inequality for the scalar curvature, analogous to the one proved by De Lellis-Topping in [13]; for Q-curvature, analogous to [22], and for the mean curvature of AE-hypersurfaces in Ricci-flat spaces, analogous to [6]. Some of these results, in the compact case, where summarized and put together in [7].…”

“…Finally, we will present an almost-Schur-type inequality for Q-curvature on AEmanifolds. Following [13]- [2], it has been shown in [22] that such an inequality holds in the context of closed manifolds. It should be stress that problems related with Q-curvature have gained plenty of attention, see [17] for a survey.…”