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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The second variational formula for the functional $ \int v^{(6)}(g)dV_g$


Authors: Bin Guo and Haizhong Li
Journal: Proc. Amer. Math. Soc. 139 (2011), 2911-2925
MSC (2010): Primary 53A30; Secondary 35J20, 35J60
Published electronically: December 30, 2010
MathSciNet review: 2801632
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Abstract: In this paper, we compute the second variational formula for the functional $ \int_M v^{(6)}(g)dv_g$, which was introduced by Graham-Juhl; the first variational formula was obtained by Chang-Fang. We also prove that an Einstein manifold (with dimension $ \ge 7$) is a strict local maximum within its conformal class unless the manifold is isometric to a round sphere with the standard metric up to a multiple of a constant.


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Additional Information

Bin Guo
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Address at time of publication: Department of Mathematics, Rutgers University, 23995 BPO WAY, Piscataway, New Jersey 08854-8139
Email: guob07@mails.tsinghua.edu.cn

Haizhong Li
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: hli@math.tsinghua.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10703-7
PII: S 0002-9939(2010)10703-7
Keywords: Second variation, renormalized volume coefficients, Bach tensor, Einstein metric.
Received by editor(s): July 14, 2010
Published electronically: December 30, 2010
Additional Notes: This work is supported by grant NSFC-10971110.
Communicated by: Jianguo Cao
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.