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Proceedings of the American Mathematical Society
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
Published electronically: May 6, 2011
MathSciNet review: 2833541
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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$.


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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: http://dx.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.



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