Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The stability of Fubini-Study metric on $ \mathbb{CP}^n$


Authors: Xi Guo and Haizhong Li
Journal: Proc. Amer. Math. Soc. 146 (2018), 325-333
MSC (2010): Primary 53C20, 58E11
DOI: https://doi.org/10.1090/proc/13594
Published electronically: September 27, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we study the stability of a critical point of a conformally invariant functional $ \mathcal {F}$. For $ n\geq 3$, by use of the variational formulas, we prove that the Fubini-Study metric on $ \mathbb{CP}^n$ is a strictly stable critical point of $ \mathcal {F}$.


References [Enhancements On Off] (What's this?)

  • [1] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
  • [2] Xianzhe Dai, Xiaodong Wang, and Guofang Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161 (2005), no. 1, 151-176. MR 2178660, https://doi.org/10.1007/s00222-004-0424-x
  • [3] Matthew J. Gursky and Jeff A. Viaclovsky, A new variational characterization of three-dimensional space forms, Invent. Math. 145 (2001), no. 2, 251-278. MR 1872547, https://doi.org/10.1007/s002220100147
  • [4] Matthew J. Gursky and Jeff A. Viaclovsky, Rigidity and stability of Einstein metrics for quadratic curvature functionals, J. Reine Angew. Math. 700 (2015), 37-91. MR 3318510, https://doi.org/10.1515/crelle-2013-0024
  • [5] Xi Guo, Haizhong Li, and Guoxin Wei, On variational formulas of a conformally invariant functional, Results Math. 67 (2015), no. 1-2, 49-70. MR 3304030, https://doi.org/10.1007/s00025-014-0393-3
  • [6] Zhen Guo, Haizhong Li, and Changping Wang, The second variational formula for Willmore submanifolds in $ S^n$, Dedicated to Shiing-Shen Chern on his 90th birthday, Results Math. 40 (2001), no. 1-4, 205-225. MR 1860369, https://doi.org/10.1007/BF03322706
  • [7] Zejun Hu and Haizhong Li, A new variational characterization of $ n$-dimensional space forms, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3005-3023. MR 2052939, https://doi.org/10.1090/S0002-9947-03-03486-X
  • [8] Osamu Kobayashi, On a conformally invariant functional of the space of Riemannian metrics, J. Math. Soc. Japan 37 (1985), no. 3, 373-389. MR 792982, https://doi.org/10.2969/jmsj/03730373
  • [9] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
  • [10] Norihito Koiso, Nondeformability of Einstein metrics, Osaka J. Math. 15 (1978), no. 2, 419-433. MR 504300
  • [11] Norihito Koiso, On the second derivative of the total scalar curvature, Osaka J. Math. 16 (1979), no. 2, 413-421. MR 539596

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C20, 58E11

Retrieve articles in all journals with MSC (2010): 53C20, 58E11


Additional Information

Xi Guo
Affiliation: Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, People’s Republic of China
Email: guoxi@hubu.edu.cn

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

DOI: https://doi.org/10.1090/proc/13594
Keywords: Weyl curvature tensor, Riemannian functional, stability
Received by editor(s): June 18, 2016
Received by editor(s) in revised form: November 17, 2016
Published electronically: September 27, 2017
Additional Notes: The first author was supported by grant NSFC-11501184
The second author was supported by grant NSFC-11671224
Communicated by: Guofang Wei
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society