Proper r-harmonic submanifolds into ellipsoids and rotation hypersurfaces

The study of r-harmonic maps was proposed by Eells–Sampson in 1965 and by Eells–Lemaire in 1983. These maps are a natural generalization of harmonic maps and are defined as the critical points of the r-energy functional Er(φ)=(1∕2)∫M|(d∗+d)r(φ)|2dvM, where φ:M→N denotes a smooth map between two Riemannian manifolds. If an r-harmonic map φ:M→N is an isometric immersion and it is not minimal, then we say that φ(M) is a proper r-harmonic submanifold of N. In this paper we prove the existence of several new, proper r-harmonic submanifolds into ellipsoids and rotation hypersurfaces.