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On the Principal Curvatures of Complete Minimal Hypersurfaces in Space Forms

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Abstract

In recent decades, there has been an increase in the number of publications related to the hypersurfaces of real space forms with two principal curvatures. The works focus mainly on the case when one of the two principal curvatures is simple. The purpose of this paper is to study a slightly more general class of complete minimal hypersurfaces in real space forms of constant curvature c, namely those with \(\mathrm{n}-1\) principal curvatures having the same sign everywhere. From assumptions on the scalar curvature R and the Gauss–Kronecker curvature K we characterize Clifford tori if \(c > 0\) and prove that K is identically zero if \(c \le 0\).

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Acknowledgements

The authors would like to thank the editor and two anonymous referees whose detailed comments greatly improved the presentation of the paper. The second author would also like to thank Professors Marcos M. Alexandrino and Paolo Piccione for inviting him to visit Institute of Mathematics and Statistics of the University of São Paulo in Brazil and for supplying him financial support.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, 05508-090, Brazil

    Rosa M. B. Chaves & B. C. Valério

  2. Departamento de Matemática, Universidade Federal do Estado do Rio de Janeiro, Avenida Pasteur, 458, Urca, Rio de Janeiro, RJ, 22290-240, Brazil

    L. A. M. Sousa Jr.

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  1. Rosa M. B. Chaves
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  2. L. A. M. Sousa Jr.
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  3. B. C. Valério
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Correspondence to L. A. M. Sousa Jr..

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Chaves, R.M.B., Sousa, L.A.M. & Valério, B.C. On the Principal Curvatures of Complete Minimal Hypersurfaces in Space Forms. Results Math 76, 5 (2021). https://doi.org/10.1007/s00025-020-01309-x

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