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), 291307
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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 1.
 H. Alencar, M. do Carmo and A. G. Colares, Stable hypersurfaces with constant scalar curvature. Math. Z., 213 (1993), 117131. 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), 165175. 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), 123138. MR 917854 (88m:53109)
 4.
 L. Cao and H. Li, rMinimal submanifolds in space forms. Ann. Global Anal. Geom., 32 (2007), 311341. MR 2346221 (2008i:53076)
 5.
 Q.M. Cheng, Hypersurfaces in a unit sphere with consatant scalar curvature. J. London Math. Soc., 64 (2001), 755768. MR 1865560 (2002k:53116)
 6.
 Q.M. Cheng, Compact hypersurfaces in a unit sphere with infinite fundamental group. Pacific J. Math., 212 (2003), 4956. 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), 33093318. MR 2407097 (2009a:53099)
 8.
 S. Y. Cheng and S.T. Yau, Hypersurfaces with constant scalar curvature. Math. Ann., 225 (1977), 195204. 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, Twoended hypersurfaces with zero scalar curvature. Indiana Univ. Math. J., 48 (1999), 867882. MR 1736975 (2001b:53077)
 12.
 H. Li, Hypersurfaces with constant scalar curvature in space forms. Math. Ann., 305 (1996), 665672. 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), 269291. 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), 153166. 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), 484490. 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), 559565. 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), 465477. MR 0341351 (49:6102)
 18.
 H. Rosenberg, Hypersurfaces of constant curvatures in space forms. Bull. Sci. Math., 117 (1993), 211239. MR 1216008 (94b:53097)
 19.
 J. Simons, Minimal varieties in Riemannian manifolds. Ann. of Math. (2), 88 (1968), 62105. MR 0233295 (38:1617)
 20.
 A. El Soufi and S. Ilias, Second eigenvalue of Schrödinger operators and mean curvature. Commun. Math. Phys., 208 (2000), 761770. MR 1736334 (2001g:58050)
 21.
 F. Urbano, Minimal surfaces with low index in the threedimensional sphere. Proc. Amer. Math. Soc., 108 (1990), 989992. MR 1007516 (90h:53073)
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 C. Wu, New characterization of the Clifford tori and the Veronese surface. Arch. Math. (Basel), 61 (1993), 277284. MR 1231163 (94h:53084)
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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:
xfwang06@mails.tsinghua.edu.cn
DOI:
http://dx.doi.org/10.1090/S00029939201110892X
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:
ChuuLian Terng
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
