The Relationship Between K-harmonic Maps And Harmonic Maps

In this paper, we mainly study the k-harmonic map φ (k≥2) from complete Riemannian manifold (M, g) into Riemannian manifold (N, h) with non-positive curvature. The energy and j-energy of φ are denoted as E1(φ)= E(φ) and Ej(φ). We get the following result which can generalize the relevant conclusion for 2-harmonic maps.(i) Every k-harmonic map φ:(M, g)→(N, h) (k≥2) with finite j-energy until j= 2k-2, must be harmonic.(ii) In the case of Vol(M,g)=∞, every k-harmonic map φ:(M, g)→ (N,h) (k≥2) with finite j-energy Ej(φ) for all j= 2,4,…,2k-2 is harmonic.In addition, as the application of a k-harmonic map, a submanifold (M,g) is called a k-harmonic submanifold in (N, h) if the map φ:(M, g)→(N, h)(k≥ 2) is an isometric immersion. Therefore we consider the 3-harmonic submanifolds especially, and obtain sonic sufficient conditions of the 3-harmonic hypersurfaces in constant curvature spaces to be minimal or totally geodesic, which with parallel second fundamental form.