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 be an -dimensional compact hypersurface with constant scalar curvature , in a unit sphere , and let be the Jacobi operator of . In 2004, L. J. Alías, A. Brasil and L. A. M. Sousa studied the first eigenvalue of of the hypersurface with constant scalar curvature in . In 2008, Q.-M. Cheng studied the first eigenvalue of the Jacobi operator of the hypersurface with constant scalar curvature , in . In this paper, we study the second eigenvalue of the Jacobi operator of and give an optimal upper bound for the second eigenvalue of .

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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:
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.