Complete stable minimal submanifolds with a parallel unit normal section in a Euclidean space

Abstract

Let Σ be an n-dimensional complete stable minimal submanifold in Rn+p admitting a parallel unit normal section ν∈Γ(NΣ). We prove that if Σ has finite L2-norm of the second fundamental form Aν with respect to ν under an additional assumption on Aη for any unit normal section η∈Γ(NΣ) such that 〈ν,η〉=0, then Σ is an affine n-plane. We also obtain an upper bound for the fundamental tone of Σ under a certain condition on the second fundamental form.