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.
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Acknowledgements
The authors would like to express their appreciations to Professors Jeffrey S. Case and Yen-Chang Huang for many inspiring discussions. We would like to thank Professor Yoshihiko Matsumoto for introducing his remarkable work [16] and Professor Mijia Lai for a valuable comment on Corollary 6.3. We also would like to express our appreciations to an anonymous referee for a very careful reading and many precious suggestions and comments. Yueh-Ju Lin would also like to thank Princeton University for the support, as part of the work was done when she was in Princeton.
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Communicated by S.A. Chang.
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Wei Yuan was supported by NSFC (Grant No. 12071489, No. 12025109, No. 11521101)
