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Minimal helix surfaces in N n×ℝ

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Abstract

An immersed surface M in N n×ℝ is a helix if its tangent planes make constant angle with ∂ t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N n×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.

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Authors and Affiliations

  1. Mathematics Department, MIT, Cambridge, MA, 02139, USA

    Gabriel Ruiz-Hernández

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  1. Gabriel Ruiz-Hernández
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Correspondence to Gabriel Ruiz-Hernández.

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Communicated by V. Cortés.

The author was supported by Conacyt.

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Ruiz-Hernández, G. Minimal helix surfaces in N n×ℝ. Abh. Math. Semin. Univ. Hambg. 81, 55–67 (2011). https://doi.org/10.1007/s12188-011-0052-5

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  • DOI: https://doi.org/10.1007/s12188-011-0052-5

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