Minimal helix submanifolds and Minimal Riemannian foliations

TL;DR

This paper classifies minimal helix submanifolds in Euclidean space, showing that ruled minimal helix submanifolds are cylinders and complex helix submanifolds are extrinsic products, with applications to Riemannian foliations.

Contribution

It provides a classification of minimal helix submanifolds and Riemannian foliations, establishing new geometric characterizations and structures.

Findings

Ruled minimal helix submanifolds are cylinders

Complex helix submanifolds are extrinsic products with a complex line

Helix hypersurfaces with constant mean curvature are cylinders or hyperplanes

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 $\mathbb{C}^n$: 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.