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

  • Hôngvân Lê, Guofang Wang
  • Journal:
    Compositio Mathematica / Volume 129 / Issue 1 / October 2001
    Published online by Cambridge University Press:
    04 December 2007, pp. 87-93
    Print publication:
    October 2001
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  • 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.