A Characterization of Minimal Legendrian Submanifolds in $\mathbb S$2n+1

Abstract

Let x: Lⁿ → $\mathbb S$²ⁿ⁺¹ ⊂ $\mathbb R$²ⁿ⁺² be a minimal submanifold in $\mathbb S$²ⁿ⁺¹. In this note, we show that L is Legendrian if and only if for any A ∈ su(n + 1) the restriction to L of 〈Ax, √(−1)x〉 satisfies Δf = 2(n + 1)f. In this case, 2(n + 1) is an eigenvalue of the Laplacian with multiplicity at least ½(n(n + 3)). Moreover if the multiplicity equals to ½(n(n + 3)), then Lⁿ is totally geodesic.