Eigenvalue inequalities and minimal submanifolds

Abstract

Let (S^m, 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 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, ∫_MHdv_M=0 and σ²{≤}λ₁, then H=0 or σ²=λ₁.
3.   If H is parallel and λ_k=n for some k, then H=0 or σ²(x){≥}λ_k+1−λ_k for some x∈Mⁿ.
4.   λ₁{≤}\frac{nV²}{V²−(∫_MHdv_M)²}. Especially, if ∫_MHdv_M=0, then λ₁{≤}n, and that λ₁=n implies that φ is minimal.