2-harmonic isometric immersions between Riemannian manifolds. (Chinese. English summary) Zbl 0596.53046

Let M and N be compact Riemannian manifolds. A 2-harmonic map \(f: M\to N\) is the critical point of the 2-energy functional \(E_ 2(f)=\int_{M}\| \tau (f)\|^ 2*1\), where \(\tau\) (f) is the tension field of f. In this paper, a 2-harmonic isometric immersion \(f: M\to N\) is studied by means of moving frames. When the target manifold N is a sphere, the Laplacian of the square norm of the second fundamental form \(\Delta \| B(f)\|^ 2\) is computed and some pinching phenomena for \(\| B(f)\|^ 2\) are revealed. Besides, the relationship between isometric immersion and its relevant Gauss map is also studied from the viewpoint of 2-harmonicity. Some pinching theorems of \(\| B(f)\|^ 2\) are obtained when the Gauss map is 2-harmonic.
Reviewer: Y.B.Shen


53C40 Global submanifolds
58E20 Harmonic maps, etc.