A new characterization of submanifolds with parallel mean curvature vector in S^n+p

抄録

In this work we will consider compact submanifold Mⁿ immersed in the Euclidean sphere S^n+p with parallel mean curvature vector and we introduce a Schrödinger operator L=−Δ+V, where Δ stands for the Laplacian whereas V is some potential on Mⁿ which depends on n, p and h that are respectively, the dimension, codimension and mean curvature vector of Mⁿ. We will present a gap estimate for the first eigenvalue μ₁ of L, by showing that either μ₁=0 or μ₁≤−n(1+H²). As a consequence we obtain new characterizations of spheres, Clifford tori and Veronese surfaces that extend a work due to Wu [W] for minimal submanifolds.