The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices (Q6619967)

\textit{J. Simons} [Ann. Math. (2) 88, 62--105 (1968; Zbl 0181.49702)] proved the very well-known rigidity theorem for an \(n\)-dimensional compact minimal submanifold in \(S^{n+m}\)\N\[\N\int_{M}|B|^2[(2-\frac{1}{m})|B|^2-n]dM\geq 0,\N\]\Nwith the consequence that the condition \(0 \leq |B|^2 \leq \frac{n}{2-\frac{1}{m}}\) implies \(|B|^2\equiv 0\) or \(|B|^2\equiv \frac{n}{2-\frac{1}{m}}\).\N\NBy \textit{S. S. Chern} et al. [in: Functional analysis and related fields. Proceedings of a conference in honor of Porfessor Marshall Stone, held at the University of Chicago, May 1968. Berlin-Heidelberg-New York: Springer-Verlag. 59--75 (1970; Zbl 0216.44001)] and \textit{H. B. Lawson jun.} [Ann. Math. (2) 89, 187--197 (1969; Zbl 0174.24901)], \(|B|^2\equiv \frac{n}{2-\frac{1}{m}}\) implies that \(M\) is a Clifford torus or a Veronese surface in \(S^4\). \N\NThe fundamental matrix is defined as follows. \N\[\N(S_{\alpha \beta}):=(\sum_{i,j}\langle B_{e_{i}e_{j}},\nu_{\alpha}\rangle \langle B_{e_{i}e_{j}},\nu_{\beta}\rangle)\N\]\Nwhere \(\{e_{1},\dots e_{n}\}\) and \(\{\nu_{1},\dots ,\nu_{m}\}\) are orthonormal basis for tangent space and normal space at a point \(p\) in \(M\) and \(B\) is the second fundamental form of \(M\). \N\NIn this paper, the authors provide another rigidity theorem for \(n\)-dimensional compact minimal Legendrian submanifolds \(M\) in \(S^{2n+1}\) using the second large eigenvalue \(\lambda_{2}\) of the fundamental matrix. They show that if \(|B|^2+\lambda_{2}\leq n+1\), then \(M\) is either a totally geodesic sphere or a Calabi torus. This gives another characterization of the Calabi torus.\N\NAs a byproduct, they prove another rigidity theorem for \(\lambda_{3}\equiv 0\) for \(n\geq 3\).