On cohomogeneity one biharmonic hypersurfaces into the Euclidean space

The aim of this paper is to prove that there exists no cohomogeneity one GG-invariant proper biharmonic hypersurface into the Euclidean space RnRn, where GG denotes a transformation group which acts on RnRn 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 GG-actions.