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Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature. (English) Zbl 1252.53071
This article deals with compact orientable hypersurfaces of the unit sphere with constant scalar curvature \(n(n- 1)r\). It is shown that the Jacobi operator is elliptic. Estimates for its first eigenvalue are known by work of L. J. Alías, A. Brasil and L. A. M. Sousa, [Bull. Braz. Math. Soc. (N.S.) 35, No. 2, 165–175 (2004; Zbl 1068.53042)] and Q.-M. Cheng [Proc. Am. Math. Soc. 136, No. 9, 3309–3318 (2008; Zbl 1155.53030)]. The first eigenvalue satisfies \(\lambda_1\leq-n(n-1)r\sqrt{r- 1}\). Here, an estimate for the second eigenvalue is established. If \(n\geq 5\) and \(H_3\neq 0\) then it is shown that the inequality \(\lambda_2\leq -{1\over 2}n(n-1)(n-2)\min|H_3|\) holds. The case of equality is also discussed. It is attained for certain sphere products.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C24 Rigidity results
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