Extrinsic polyharmonic maps into the sphere

In the first part we shall prove that the inverse of the stereographic projection π^{−1} : R^n → S^n (n ≥2)is extrinsically k-harmonic if and only if n =2k. In the second part we shall study minimizing properties and stability of its restriction to the closed ball B^n(R). In this context we shall prove that there exists a small enough positive upper bound R^∗_k such that π^{−1} :B^n(R)→ S^n is a minimizer provided that 0 < R ≤R^∗_k . By contrast, we shall show that π^{−1} :B^n(R)→ S^n is not energy minimizing when R >1. Finally, in some cases we shall obtain stability with respect to rotationally symmetric variations (equivariant stability) for values of R which are greater than 1.