Journal of Geometry

, 110:6 | Cite as

Algebraic CMC hypersurfaces of order 3 in Euclidean spaces

  • Oscar Perdomo
  • Vladimir G. TkachevEmail author
Open Access


Understanding and finding of general algebraic constant mean curvature surfaces in the Euclidean spaces is a hard open problem. The basic examples are the standard spheres and the round cylinders, all defined by a polynomial of degree 2. In this paper, we prove that there are no algebraic hypersurfaces of degree 3 in \(\mathbb {R}^n\), \(n\ge 3\), with nonzero constant mean curvature.


Constant mean curvature algebraic surfaces 

Mathematics Subject Classification

53C42 53A10 



This work was done while the first author visited Linköping University. He would like to thank the Mathematical Institution of Linköping University for hospitality. The second author acknowledges support from G D Magnusons Fond, MG2017-0101.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsCentral Connecticut State UniversityNew BritainUSA
  2. 2.Department of MathematicsLinköping UniversityLinköpingSweden

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