New examples of r-harmonic immersions into the sphere

Abstract

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:Sn−1(R)↪Sn is a proper r-harmonic submanifold of Sn if and only if the radius R is equal to 1/r. We shall also prove the existence of proper r-harmonic generalized Clifford's tori into the sphere.