Abstract
We investigated minimal helix submanifolds of any dimension and codimension immersed in Euclidean space. Our main result proves that a ruled minimal helix submanifold is a cylinder. As an application we classify complex helix submanifolds of \({\mathbb {C}}^n\): They are extrinsic products with a complex line as a factor. The key tool is Corollary 1.3 which allows us to classify Riemannian foliations of open subsets of the Euclidean space with minimal leaves. Finally, we consider the case of a helix hypersurface with constant mean curvature and prove that it is either a cylinder or an open part of a hyperplane.
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Acknowledgments
We would like to thank Francisco Vittone for several useful comments. G. Ruiz-Hernández thanks the hospitality of DISMA at Politecnico di Torino where this work was started.
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This research was partially supported by Ministero degli Affari Esteri from Italy and CONACYT from Mexico. The second named author was partially supported by DGAPA-UNAM-PAPIIT, under Project IN100414.
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Di Scala, A.J., Ruiz-Hernández, G. Minimal helix submanifolds and minimal Riemannian foliations. Bol. Soc. Mat. Mex. 22, 229–250 (2016). https://doi.org/10.1007/s40590-015-0074-6
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DOI: https://doi.org/10.1007/s40590-015-0074-6