On some pinchings of minimal submanifolds (Q2779198)

By analogy with [\textit{S. Montiel}, and \textit{A. Ros}, and \textit{F. Urbano}, Math. Z. 191, 537-548 (1986; Zbl 0563.53046)], if \(M\) is an \(n\)-dimensional compact totally real minimal submanfold of \(\mathbb{C} P^n(c)\) with \(|h(v,v)|^2\leq {c\over 8}\) \(\forall v\in UM\), then the following cases are possible:NEWLINENEWLINENEWLINE(a) \(h(v,v)=0\) and \(M\) is totally geodesicNEWLINENEWLINENEWLINE(b) \(|h(v,v)|^2= {c\over 8}\), \(n=2\) and \(M\) is a finite Riemannian covering of the unique flat torus minimally embedded in \(\mathbb{C} P^2(c)\) with parallel \(h\).NEWLINENEWLINENEWLINE(c) \(|h(v,v) |^2={c\over 8}\), \(n>2\) and \(M\) is congruent to \(SU(3)/SO (3)\) for \(n=5\), to \(SU(6)/Sp(3)\) for \(n=14\), to \(SU(3)\) for \(n=8\), and to \(E_6/F_4\) for \(n=26\), where \(h\) denotes the second fundamental form. In [\textit{H. Gauchman}, Trans. Am. Math. Soc. 298, 779-791 (1986; Zbl 0608.53056)] a pinching result is obtained for totally real minimal submanifold of \(CP^n(c)\) and a similar result for compact minimal submanifolds of the Euclidean spheres is obtained here.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00018].