Let (
Sm,
g0) be the unit sphere, (
Mn,
g) its submanifold, λ
1 the first nonzero eigenvalue of (
Mn,
g),
H the mean curvature vector field of
Mn. By Takahashi theorem, if
Mn is minimal, then λ
1{≤}
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 λ
k{≥}
n or λ
k+1{≤}
n, then φ is minimal and λ
k=
n or λ
k+1=
n.
2. If
H is parallel, ∫
MHdvM=0 and σ
2{≤}λ
1, then
H=0 or σ
2=λ
1.
3. If
H is parallel and λ
k=
n for some
k, then
H=0 or σ
2(
x){≥}λ
k+1−λ
k for some
x∈
Mn.
4. λ
1{≤}\frac{
nV2}{
V2−(∫
MHdvM)
2}. Especially, if ∫
MHdvM=0, then λ
1{≤}
n, and that λ
1=
n implies that φ is minimal.
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