A generalisation of the Hopf Construction and harmonic morphisms into \s^2

In this paper we construct a new family of harmonic morphisms \varphi:V^5\to\s^2, where $V^5$ is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of \c^4=\r^8. These harmonic morphisms admit a continuous extension to the completion ${V^{\ast}}^5$, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.Comment: 10 page

Biminimal immersions

We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i) biminimal curves in a Riemannian manifold, with particular care to the case of curves in a space form ii) isometric immersions of codimension one in a Riemannian manifold, in particular for surfaces of a three-dimensional manifold. We describe two methods to construct families of biminimal surfaces using both Riemannian and horizontally homothetic submersions.Comment: Dedicated to Professor Renzo Caddeo on his 60th birthday. 2 figure

Classification results for biharmonic submanifolds in spheres

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.Comment: Dedicated to Professor Vasile Oproiu on his 65th birthday, 14 page

Properties of biharmonic submanifolds in spheres

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.Comment: 10 pages; contribution to the Proceedings of the 11-th International Conference on Geometry, Integrability and Quantization, Varna 2009, Bulgari