## 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

### Citations:

Zbl 1068.53042; Zbl 1155.53030
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### References:

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