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Geometriae Dedicata

, Volume 200, Issue 1, pp 303–320 | Cite as

The one dimensional case of the singular minimal surfaces with density

  • Rafael LópezEmail author
Original Paper

Abstract

Given \(\alpha ,\lambda \in \mathbb {R}\), a singular minimal surface with density vector \(\mathbf {v}\) is a surface \(\Sigma \) in Euclidean space whose mean curvature H satisfies \(2H(p)= \alpha \langle N(p),\mathbf {v}\rangle /\langle p,\mathbf {v}\rangle +2\lambda \), \(p\in \Sigma \), being N the Gauss map of \(\Sigma \). In this paper we classify the class of these surfaces that are invariant by a one-parameter group of translations.

Keywords

Singular minimal surface Group of translations Manifold with density Phase plane 

Mathematics Subject Classification

53A10 53C44 53C21 53C42 

Notes

Acknowledgements

The author wishes to express his thanks to the referee for several helpful comments, specially concerning to the discussion of the portrait of phase plane.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Departamento de Geometría y Topología Instituto de Matemáticas (IEMath-GR)Universidad de GranadaGranadaSpain

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