First Eigenvalue Characterisation of Clifford Hypersurfaces and Veronese Surfaces

We give a sharp estimate for the first eigenvalue of the Schrödinger operator L:=-Δ -σ which is defined on the closed minimal submanifold Mⁿ in the unit sphere Sⁿ⁺ᵐ , where σ is the square norm of the second fundamental form.