Eric Loubeau

On the biharmonic and harmonic indices of the Hopf map

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalize harmonic maps. We consider the Hopf map and modify it into a nonharmonic biharmonic map . We show to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawas determination of its harmonic index and nullity.

The index of biharmonic maps in spheres

Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese map and the Clifford torus.

Biharmonic surfaces of constant mean curvature

We compute a Simons type formula for the stress-energy tensor of biharmonic maps from surfaces. Specializing to Riemannian immersions, we prove several rigidity results for biharmonic CMC surfaces, putting in evidence the inuence of the Gaussian curvature on pseudo-umbilicity. Finally, the condition of biharmonicity is shown to enable an extension of the classical Hopf theorem to CMC surfaces in any ambient Riemannian manifold.

The harmonicity of nearly cosymplectic structures

Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Rie- mannian manifold and prove curvature identities which imply the harmonicity of their parametrizing section, thus complementing earlier results on nearly- Kahler almost complex structures.

A NOTE ON EXPONENTIALLY HARMONIC MORPHISMS

We prove that exponentially harmonic morphisms are precisely the Riemannian submersions with minimal fibres.