548 research outputs found
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 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 , 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
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