On scalar curvature for totally real minimal submanifolds in \(\mathbb{C} P^ n\) (Q1895523)

In this note, a conjecture proposed by the first author [cf. Chin. J. Contemp. Math. 12, 259-265 (1991); translation from Acta Math. Sin. 28, 85-90 (1985; Zbl 0567.53042)] is proved. Let \(M\) be an \(n\)-dimensional compact totally real minimal submanifold in \(\mathbb{C} P^n(c)\) of constant holomorphic sectional curvature \(c\). If the scalar curvature \(\rho\) of \(M\) satisfies \(\rho \geq 3(n - 2) nc/16\); or equivalently, \(|\sigma |^2 \leq n(n + 2)c/16\), where \(|\sigma |^2\) denotes the length square of the second fundamental form \(\sigma\) of \(M\), then \(M\) is an isotropic submanifold in \(\mathbb{C} P^n\) with parallel second fundamental form. In such a case, \(M\) is only one of the following: (i) \(\rho = n (n -1)c/4\) and \(M\) is totally geodesic; (ii) \(\rho = 0\), \(n =2\) and \(M\) is a finite Riemannian covering of the unique flat torus minimally imbedded in \(\mathbb{C} P^2\) with parallel second fundamental form; or (iii) \(\rho = 3n(n -2)c / 16\), \(n > 2\) and \(M\) is an imbedded submanifold congruent to the standard imbedding of one of the following submanifolds: SU(3)/SO(3) for \(n = 5\), SU(3) for \(n = 8\), SU(6)/Sp(3) for \(n =14\), or \(E_6/ F_4\) for \(n = 26\).