Minimal helix surfaces in Nn×ℝ
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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.
KeywordsMinimal and helix surfaces Constant angle surfaces Eikonal functions
Mathematics Subject Classification53C40 53C42
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