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On regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces

Part of: Classical differential geometry Global differential geometry

Published online by Cambridge University Press:  06 September 2021

Alexandre Paiva Barreto ,
Francisco Fontenele  and
Luiz Hartmann Open the ORCID record for Luiz Hartmann [Opens in a new window]
Alexandre Paiva Barreto
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, Brazil (alexandre@dm.ufscar.br, alexandre.paiva@ufscar.br)
Francisco Fontenele
Affiliation:
Departamento de Geometria, Universidade Federal Fluminense, Niterói, RJ, Brazil (fontenele@mat.uff.br)
Luiz Hartmann
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, Brazil (luizhartmann@ufscar.br, hartmann@dm.ufscar.br)
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Abstract

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

Keywords

Constant mean curvaturealgebraic hypersurface

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 1081 - 1088
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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