Note on the spectrum of the Hodge-Laplacian for \(k\) -forms on minimal Legendre submanifolds in \(S^{2n+1}\) (Q1349265)

Let \(\mathbb S^{2n+1}\) be the unit sphere endowed with the standard Riemann metric and contact structure induced from \(\mathbb R^{2n+2}\equiv\mathbb C^{n+1}\). Let \(L\) a minimal Legendre submanifold of dimension \(n\) of \(\mathbb S^{2n+1}\). It is proved that for \(1\leq k\leq n\), \(n+1-k\) is an eigenvalue of the Hodge-Laplace operator acting on \(k\) and \((k-1)\)-forms on \(L\); and the dimensions of the eigenspaces \(\text{ Eig}_k(n+1-k)\) and \(\text{ Eig}_{k-1}(n+1-k)\) are at least \(n\choose k\).