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Hypersurfaces with constant higher order mean curvature in Euclidean space

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Abstract

In this paper we derive sharp estimates for the infimum and for the supremum of the squared norm of the second fundamental form of complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. Besides, we characterize those hypersurfaces for which any of these bounds is attained. Our results will be an application of a purely geometric result on the principal curvatures of the hypersurface, the so called principal curvature theorem, given by Smyth and Xavier (Invent Math 90:443–450, 1987).

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Acknowledgments

The authors would like to thank the anonymous referee for his/her valuable suggestions and corrections which contributed to improve this paper. The second author is very grateful to Luis J. Alías and the Department of Mathematics of the University of Murcia for the hospitality and support.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Murcia, 30100, Espinardo, Murcia, Spain

    Luis J. Alías & Josué Meléndez

Authors
  1. Luis J. Alías
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  2. Josué Meléndez
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Corresponding author

Correspondence to Luis J. Alías.

Additional information

Luis J. Alías was partially supported by MINECO/FEDER project MTM2012-34037, Spain. Josué Meléndez was supported by CONACYT (México) Grant 234865.

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Alías, L.J., Meléndez, J. Hypersurfaces with constant higher order mean curvature in Euclidean space. Geom Dedicata 182, 117–131 (2016). https://doi.org/10.1007/s10711-015-0131-3

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  • DOI: https://doi.org/10.1007/s10711-015-0131-3

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