New examples of r-harmonic immersions into the sphere

Polyharmonic, or $r$-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper $r$-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion $i: S^{n-1}(R)\to S^n$ is a proper $r$-harmonic submanifold of $S^n$ if and only if the radius $R$ is equal to $1/ \sqrt{r}$. We shall also prove the existence of proper $r$-harmonic generalized Clifford's tori into the sphere.