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

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

Helix surfaces in the special linear group

We characterize helix surfaces (constant angle surfaces) in the special linear group $\mathrm{SL}(2,\r)$. In particular, we give an explicit local description of these surfaces in terms of a suitable curve and a 1-parameter family of isometries of $\mathrm{SL}(2,\r)$.Comment: Minor corrections. To appear in Annali di Matematica Pura e Applicata. arXiv admin note: substantial text overlap with arXiv:1206.127

On the Stability of the Equator Map for Higher Order Energy Functionals

Let Bn ⊂ ℝn and Sn ⊂ Rn+1 denote the Euclidean n-dimensional unit ball and sphere, respectively. The extrinsic k-energy functional is defined on the Sobolev space Wk,2 (Bn, Sn) as follows: Ekext(u) = ∫Bn |Δs u|2 dx when k = 2s, and Ekext(u) = ∫Bn|∇ Δs u|2 dx when k = 2s + 1. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map u∗: Bn → Sn, defined by u∗(x) = (x/|x|,0), is a critical point of Ekext(u) provided that n ≥ 2k + 1. The main aim of this paper is to establish necessary and sufficient conditions on k and n under which u∗: Bn → Sn is minimizing or unstable for the extrinsic k-energy