22 research outputs found
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 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
invariant proper biharmonic hypersurface into the Euclidean space , where denotes a tranformation group which acts on 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 actions.Comment: 13 page
The energy density of biharmonic quadratic maps between spheres
In this paper, we first prove that a quadratic form from to
is non-harmonic biharmonic if and only if it has constant energy
density . 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