Biharmonic submanifolds with parallel mean curvature vector field in spheres

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.Comment: 15 pages. Minor changes made and one section adde

Reduction methods for the bienergy

This paper, in which we develop ideas introduced in \cite{MR}, focuses on \emph{reduction methods} (basically, group actions or, more generally, simmetries) for the bienergy. This type of techniques enable us to produce examples of critical points of the bienergy by reducing the study of the relevant fourth order PDE's system to ODE's. In particular, we shall study rotationally symmetric biharmonic conformal diffeomorphisms between \emph{models}. Next, we will adapt the reduction method to study an ample class of $G-$invariant immersions into the Euclidean space. At present, the known instances in these contexts are far from reaching the depth and variety of their companions which have provided fundamental solutions to classical problems in the theories of harmonic maps and minimal immersions. However, we think that these examples represent an important starting point which can inspire further research on biharmonicity. In this order of ideas, we end this paper with a discussion of some open problems and possible directions for further developments.Comment: to appear in REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES. arXiv admin note: text overlap with arXiv:1507.03964, arXiv:1109.620

On cohomogeneity one biharmonic hypersurfaces into the Euclidean space

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.Comment: 13 page

The energy density of biharmonic quadratic maps between spheres

In this paper, we first prove that a quadratic form from $\mathbb{S}^m$ to $\mathbb{S}^n$ is non-harmonic biharmonic if and only if it has constant energy density $(m+1)/2$. Then, we give a positive answer to an open problem concerning the structure of non-harmonic biharmonic quadratic forms. As a direct application, using classification results for harmonic quadratic forms, we infer classification results for non-harmonic biharmonic quadratic forms