In this work we will consider compact submanifold
Mn immersed in the Euclidean sphere
Sn+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 μ
1 of
L, by showing that either μ
1=0 or μ
1≤−
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.
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