A characterization of minimal Legendrian submanifolds in \({\mathbb{S}}^{2n+1}\) (Q2756167)

Assume \(x:L^n\to S^{2n+1} \subset\mathbb{R}^{2n+2}\) is a minimal submanifold of \(S^{2n+1}\). The author proves that \(L\) is Legendrian if and only if for any \(A\in su(n+1)\) the restriction to \(L\) of \(\langle Ax,\sqrt{-1}x \rangle\) satisfies the equation \(\Delta f=2(n+1)f\). It follows that \(2(n+1)\) is an eigenvalue of the Laplacian with multiplicity at least \({1\over 2}(n(n+3))\). Furthermore, if the multiplicity equals to \({1\over 2}(n(n+3))\), then \(L^n\) is totally geodesic. The proofs of the theorems are given in a gentle style.