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First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature. (English) Zbl 1155.53030

For an \(n\)-dimensional compact hypersurface \(M\) with constant scalar curvature \(n(n-1)r\), \(r>1\), in a unit sphere \(S^{n+1}(r)\), the author studies the first eigenvalue \(\lambda _{1}\) of the Jacobi operator \(J_{S}\), following the definition given by H. Alencar, M. do Carmo and A. G. Colares [Math. Z. 213, No. 1, 117–131 (1993; Zbl 0792.53057)].
An inequality involving \(\lambda _{1}\), the mean curvature \(H\), the radius \(r \) and the dimension \(n\) of the hypersurface \(M\) is proved. Equality holds if and only if either \(M\) is totally umbilical and non-totally geodesic or \(M\) is a Riemannian product \(S^{m}(c)\times S^{n-m}(\sqrt{1-c^{2}})\), \(1\leq m\leq n-1\), \(r>1\).

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 0792.53057
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Full Text: DOI

References:

[1] H. Alencar, M. do Carmo, and A. G. Colares, Stable hypersurfaces with constant scalar curvature, Math. Z. 213 (1993), no. 1, 117 – 131. · Zbl 0792.53057
[2] Hilário Alencar, Manfredo do Carmo, and Walcy Santos, A gap theorem for hypersurfaces of the sphere with constant scalar curvature one, Comment. Math. Helv. 77 (2002), no. 3, 549 – 562. · Zbl 1032.53045
[3] Luis J. Alías, Abdênago Barros, and Aldir Brasil Jr., A spectral characterization of the \?(\?)-torus by the first stability eigenvalue, Proc. Amer. Math. Soc. 133 (2005), no. 3, 875 – 884. · Zbl 1065.53046
[4] Luis J. Alías, Aldir Brasil Jr., and Luiz A. M. Sousa Jr., A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue, Bull. Braz. Math. Soc. (N.S.) 35 (2004), no. 2, 165 – 175. · Zbl 1068.53042
[5] J. Lucas Barbosa, Manfredo do Carmo, and Jost Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), no. 1, 123 – 138. · Zbl 0653.53045
[6] Qing-Ming Cheng, The rigidity of Clifford torus \?\textonesuperior (\sqrt 1/\?)\times \?\(^{n}\)\(^{-}\)\textonesuperior (\sqrt (\?-1)/\?), Comment. Math. Helv. 71 (1996), no. 1, 60 – 69. · Zbl 0874.53046
[7] Qing-Ming Cheng, Hypersurfaces in a unit sphere \?\(^{n}\)\(^{+}\)\textonesuperior (1) with constant scalar curvature, J. London Math. Soc. (2) 64 (2001), no. 3, 755 – 768. · Zbl 1023.53044
[8] Qing-Ming Cheng, Compact hypersurfaces in a unit sphere with infinite fundamental group, Pacific J. Math. 212 (2003), no. 1, 49 – 56. · Zbl 1050.53039
[9] Qing Ming Cheng and Hisao Nakagawa, Totally umbilic hypersurfaces, Hiroshima Math. J. 20 (1990), no. 1, 1 – 10. · Zbl 0711.53045
[10] Qing-Ming Cheng, Shichang Shu, and Young Jin Suh, Compact hypersurfaces in a unit sphere, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 6, 1129 – 1137. · Zbl 1091.53037
[11] Shiu Yuen Cheng and Shing Tung Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195 – 204. · Zbl 0349.53041
[12] Jorge Hounie and Maria Luiza Leite, Two-ended hypersurfaces with zero scalar curvature, Indiana Univ. Math. J. 48 (1999), no. 3, 867 – 882. · Zbl 0929.53033
[13] H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187 – 197. · Zbl 0174.24901
[14] Haizhong Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann. 305 (1996), no. 4, 665 – 672. · Zbl 0864.53040
[15] Oscar Perdomo, First stability eigenvalue characterization of Clifford hypersurfaces, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3379 – 3384. · Zbl 1014.53036
[16] Robert C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geometry 8 (1973), 465 – 477. · Zbl 0277.53030
[17] Harold Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211 – 239. · Zbl 0787.53046
[18] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62 – 105. · Zbl 0181.49702
[19] Chuan Xi Wu, New characterizations of the Clifford tori and the Veronese surface, Arch. Math. (Basel) 61 (1993), no. 3, 277 – 284. · Zbl 0791.53056
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