In this paper,we study the minimal norm tensor for the totally symmetric fourth-order co-variant tensors.Firstly,we obtain the general expression of the minimal norm tensor.Then,we prove the equivalence relation between the minimal norm tensor and the trace-free decomposition.Finally,by use of the non-negativity of minimal norm,we prove that on a minimal hypersurface of the unit sphere,the inequality|▽2h|2≥3/2[S×trh4+n·S-(trh3)2-2·S2]+3S(S-n)2/2(n+4)is always true,we also find that a great circle or a part of it or the Clifford torus satisfy the sharp condition in this in-equality.
维普
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