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

TL;DR

This paper introduces a new symmetric 2-tensor called J-tensor associated with Q-curvature on Riemannian manifolds, providing insights into higher-order curvature analogues and applications to Q-singular metrics and curvature estimates.

Contribution

It defines the J-tensor as a higher-order analogue of the Ricci tensor and explores its relation to Q-curvature, extending understanding of curvature properties and inequalities.

Findings

J-tensor is canonically associated with Q-curvature.

Established an Almost-Schur Lemma for Q-curvature.

Applied results to Q-singular metrics and curvature estimates.

Abstract

In this article, we define a symmetric 2-tensor canonically associated to Q-curvature 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.