Deformations of Q-curvature II

Abstract

This is the second article of a sequence of research on deformations of Q-curvature. In the previous one, we studied local stability and rigidity phenomena of Q-curvature. In this article, we mainly investigate the volume comparison with respect to Q-curvature. In particular, we show that volume comparison theorem holds for metrics close to strictly stable positive Einstein metrics. This result shows that Q-curvature can still control the volume of manifolds under certain conditions, which provides a fundamental geometric characterization of Q-curvature. Applying the same technique, we derive the local rigidity of strictly stable Ricci-flat manifolds with respect to Q-curvature, which shows the non-existence of metrics with positive Q-curvature near the reference metric.

26 pages, equality case in Corollary 6.3 discussed, new references added

References in corpus (3)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• Sign of Green's function of Paneitz operators and the Q curvature