Deformations of Metrics and Biharmonic Maps

We construct biharmonic non-harmonic maps between Riemannian manifolds (M, g) and (N, h) by first making the ansatz that φ : (M, g) → (N, h) be a harmonic map and then deforming the metric on N by{\tilde h_\alpha } = \alpha h + \left( {1 - \alpha } \right){\rm{d}}f \otimes {\rm{d}}fto render φ biharmonic, where f is a smooth function with gradient of constant norm on (N, h) and α ∈ (0, 1). We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

Index and nullity of proper biharmonic maps in spheres

In recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional has been thoroughly studied. We shall focus on this problem and, in particular, we shall compute the exact index and nullity of some known examples of proper biharmonic maps. Moreover, we shall analyze a case where the domain is not compact. More precisely, we shall prove that a large family of proper biharmonic maps [Formula: see text] is strictly stable with respect to compactly supported variations. In general, the computations involved in this type of problems are very long. For this reason, we shall also define and apply to specific examples a suitable notion of index and nullity with respect to equivariant variations.