On cohomogeneity one biharmonic hypersurfaces into the Euclidean space

TL;DR

This paper proves that no cohomogeneity one G-invariant proper biharmonic hypersurfaces exist in Euclidean space under certain symmetry conditions, supporting the Chen conjecture through a unified equivariant geometric approach.

Contribution

It establishes a non-existence result for a broad class of biharmonic hypersurfaces, extending the understanding of the Chen conjecture with a unified method.

Findings

No such hypersurfaces exist under the specified conditions.

The proof applies equivariant differential geometry techniques.

The result covers hypersurfaces with up to seven principal curvatures.

Abstract

The aim of this paper is to prove that there exists no cohomogeneity one $G-$invariant proper biharmonic hypersurface into the Euclidean space ${\mathbb R}^n$, where $G$ denotes a tranformation group which acts on ${\mathbb R}^n$ by isometries, with codimension two principal orbits. This result may be considered in the context of the Chen conjecture, since this family of hypersurfaces includes examples with up to seven distinct principal curvatures. The paper uses the methods of equivariant differential geometry. In particular, the technique of proof provides a unified treatment for all these $G-$actions.