Deformations of Q-curvature II

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.