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)

 
 

 

Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature


Authors: Haizhong Li and Xianfeng Wang
Journal: Proc. Amer. Math. Soc. 140 (2012), 291-307
MSC (2010): Primary 53C42; Secondary 58J50
DOI: https://doi.org/10.1090/S0002-9939-2011-10892-X
Published electronically: May 6, 2011
MathSciNet review: 2833541
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ x:M\to\mathbb{S}^{n+1}(1)$ be an $ n$-dimensional compact hypersurface with constant scalar curvature $ n(n-1)r,~r\geq 1$, in a unit sphere $ \mathbb{S}^{n+1}(1),$ $ n\geq 5$, and let $ J_s$ be the Jacobi operator of $ M$. In 2004, L. J. Alías, A. Brasil and L. A. M. Sousa studied the first eigenvalue of $ J_s$ of the hypersurface with constant scalar curvature $ n(n-1)$ in $ \mathbb{S}^{n+1}(1),~n\geq 3$. In 2008, Q.-M. Cheng studied the first eigenvalue of the Jacobi operator $ J_s$ of the hypersurface with constant scalar curvature $ n(n-1)r, r>1$, in $ \mathbb{S}^{n+1}(1)$. In this paper, we study the second eigenvalue of the Jacobi operator $ J_s$ of $ M$ and give an optimal upper bound for the second eigenvalue of $ J_s$.


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

  • 1. H. Alencar, M. do Carmo and A. G. Colares, Stable hypersurfaces with constant scalar curvature. Math. Z., 213 (1993), 117-131. MR 1217674 (94d:53080)
  • 2. L. J. Alías, A. Brasil and L. A. M. Sousa, A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue. Bull. Braz. Math. Soc., 35 (2004), 165-175. MR 2081021 (2005e:53089)
  • 3. J. L. Barbosa, M. do Carmo and M. Eschenburg, Stability of hypersurfaces with constant mean curvature in Riemannian manifolds. Math. Z., 197 (1988), 123-138. MR 917854 (88m:53109)
  • 4. L. Cao and H. Li, r-Minimal submanifolds in space forms. Ann. Global Anal. Geom., 32 (2007), 311-341. MR 2346221 (2008i:53076)
  • 5. Q.-M. Cheng, Hypersurfaces in a unit sphere $ \mathbb{S}^{n+1}(1)$ with consatant scalar curvature. J. London Math. Soc., 64 (2001), 755-768. MR 1865560 (2002k:53116)
  • 6. Q.-M. Cheng, Compact hypersurfaces in a unit sphere with infinite fundamental group. Pacific J. Math., 212 (2003), 49-56. MR 2016567 (2004g:53059)
  • 7. Q.-M. Cheng, First eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature. Proc. Amer. Math. Soc., 136 (2008), 3309-3318. MR 2407097 (2009a:53099)
  • 8. S. Y. Cheng and S.-T. Yau, Hypersurfaces with constant scalar curvature. Math. Ann., 225 (1977), 195-204. MR 0431043 (55:4045)
  • 9. S. S. Chern, Minimal submanifolds in a Riemannian manifold. Dept. of Math. Tech. Report 19 (New Series), Univ. of Kansas, Lawrence, KS, 1968. MR 0248648 (40:1899)
  • 10. G. Hardy, J. E. Littlewood and G. Pólya, Inequalities. 2nd Edition, Cambridge Univ. Press, 1989. MR 0046395 (13:727e)
  • 11. J. Hounie and M. L. Leite, Two-ended hypersurfaces with zero scalar curvature. Indiana Univ. Math. J., 48 (1999), 867-882. MR 1736975 (2001b:53077)
  • 12. H. Li, Hypersurfaces with constant scalar curvature in space forms. Math. Ann., 305 (1996), 665-672. MR 1399710 (97i:53073)
  • 13. P. Li and S.-T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math., 69 (1982), 269-291. MR 674407 (84f:53049)
  • 14. S. Montiel and A. Ros, Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math., 83 (1986), 153-166. MR 813585 (87d:53109)
  • 15. K. Nomizu and B. Smyth, On the Gauss mapping for hypersurfaces of constant mean curvature in the sphere. Comment. Math. Helv., 44 (1969), 484-490. MR 0257939 (41:2588)
  • 16. O. Perdomo, On the average of the scalar curvature of minimal hypersurface of spheres with low stability index. Illinois J. Math., 48 (2004), 559-565. MR 2085426 (2006a:53076)
  • 17. R. C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Diff. Geom., 8 (1973), 465-477. MR 0341351 (49:6102)
  • 18. H. Rosenberg, Hypersurfaces of constant curvatures in space forms. Bull. Sci. Math., 117 (1993), 211-239. MR 1216008 (94b:53097)
  • 19. J. Simons, Minimal varieties in Riemannian manifolds. Ann. of Math. (2), 88 (1968), 62-105. MR 0233295 (38:1617)
  • 20. A. El Soufi and S. Ilias, Second eigenvalue of Schrödinger operators and mean curvature. Commun. Math. Phys., 208 (2000), 761-770. MR 1736334 (2001g:58050)
  • 21. F. Urbano, Minimal surfaces with low index in the three-dimensional sphere. Proc. Amer. Math. Soc., 108 (1990), 989-992. MR 1007516 (90h:53073)
  • 22. C. Wu, New characterization of the Clifford tori and the Veronese surface. Arch. Math. (Basel), 61 (1993), 277-284. MR 1231163 (94h:53084)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C42, 58J50

Retrieve articles in all journals with MSC (2010): 53C42, 58J50


Additional Information

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

Xianfeng Wang
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: xf-wang06@mails.tsinghua.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2011-10892-X
Keywords: Hypersurface with constant scalar curvature, second eigenvalue, Jacobi operator, mean curvature, principal curvature
Received by editor(s): August 23, 2010
Received by editor(s) in revised form: October 31, 2010
Published electronically: May 6, 2011
Additional Notes: The first author was supported in part by NSFC Grant #10971110 and Tsinghua University–K.U. Leuven Bilateral Scientific Cooperation Fund.
The second author was supported in part by NSFC Grant #10701007 and Tsinghua University–K.U. Leuven Bilateral Scientific Cooperation Fund.
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society