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

Yueh-Ju Lin and Wei Yuan

Vol. 291 (2017), No. 2, 425–438

DOI: 10.2140/pjm.2017.291.425

Abstract

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 $Q$-curvature, yielding an estimate of the $Q$-curvature on closed manifolds.

Keywords

$J$-tensor, $Q$-curvature, $Q$-singular metric

Mathematical Subject Classification 2010

Primary: 53C20, 53C25

Milestones

Received: 13 April 2016

Accepted: 2 June 2017

Published: 14 September 2017