A characterization of minimal Legendrian submanifolds in S2n+1

Let x: L-n --> S2n+1 subset of R2n+2 be a minimal submanifold in bb S2n+1. In this note, we show that L is Legendrian if and only if for any A is an e lement of su(n + 1) the restriction to L of < Ax, root -1x > satisfies Delt af = 2(n + 1)f. In this case, 2(n + 1) is an eigenvalue of the Laplacian wi th multiplicity at least 1/2(n(n + 3)). Moreover if the multiplicity equals to 1/2(n(n + 3)), then L-n is totally geodesic.