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Characterization of hypersurfaces via the second eigenvalue of the Jacobi operator


Author: Abraão Mendes
Journal: Proc. Amer. Math. Soc. 147 (2019), 3515-3521
MSC (2010): Primary 53C24
DOI: https://doi.org/10.1090/proc/14477
Published electronically: March 21, 2019
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Abstract: In this work we characterize certain immersed closed hypersurfaces of some ambient manifolds via the second eigenvalue of the Jacobi operator. First, we characterize the Clifford torus as the surface which maximizes the second eigenvalue of the Jacobi operator among all closed immersed orientable surfaces of $ \mathbb{S}^3$ with genus bigger than zero. After, we characterize the slices of the warped product $ I\times _h\mathbb{S}^n$, under a suitable hypothesis on the warping function $ h:I\subset \mathbb{R}\to \mathbb{R}$, as the only hypersurfaces which saturate a certain integral inequality involving the second eigenvalue of the Jacobi operator. As a consequence, we obtain that if $ \Sigma $ is a closed immersed hypersurface of $ \mathbb{R}\times \mathbb{S}^n$, then the second eigenvalue of the Jacobi operator of $ \Sigma $ satisfies $ \lambda _2\le n$ and the slices are the only hypersurfaces which satisfy $ \lambda _2=n$.


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Abraão Mendes
Affiliation: Instituto de Matemática, Universidade Federal de Alagoas, Maceió, Alagoas, 57072-970, Brazil
Email: abraao.mendes@im.ufal.br

DOI: https://doi.org/10.1090/proc/14477
Received by editor(s): August 2, 2018
Received by editor(s) in revised form: November 16, 2018
Published electronically: March 21, 2019
Communicated by: Jiaping Wang
Article copyright: © Copyright 2019 American Mathematical Society