Minimal helix submanifolds and Minimal Riemannian foliations

2015/04/15 by Antonio J. Di Scala, Di Scala, Antonio J., Gabriel Ruiz-Hernandez +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1504.03748

23 pages

arxiv created 2015/04/15 · arxiv updated 2015/04/16

Abstract

We investigate 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 ℂⁿ: 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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