Minimal helix surfaces in Nn×ℝ
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- Ruiz-Hernández, G. Abh. Math. Semin. Univ. Hambg. (2011) 81: 55. doi:10.1007/s12188-011-0052-5
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An immersed surface M in Nn×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 Nn×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.