抄録
Let (Sm, g0) be the unit sphere, (Mn, g) its submanifold, λ1 the first nonzero eigenvalue of (Mn, g), H the mean curvature vector field of Mn. By Takahashi theorem, if Mn is minimal, then λ1{≤}n. In this paper, we establish some eigenvalue inequalities and use them to prove:
1. If x is mass symmetric and of order {k, k+1} for some k such that λk{≥}n or λk+1{≤}n, then φ is minimal and λk=n or λk+1=n.
2. If H is parallel, ∫MHdvM=0 and σ2{≤}λ1, then H=0 or σ2=λ1.
3. If H is parallel and λk=n for some k, then H=0 or σ2(x){≥}λk+1−λk for some x∈Mn.
4. λ1{≤}\frac{nV2}{V2−(∫MHdvM)2}. Especially, if ∫MHdvM=0, then λ1{≤}n, and that λ1=n implies that φ is minimal.
