Minimal submanifolds with bounded second fundamental form (Q2276779)

Let \(\sigma\) be the second fundamental form of an n-dimensional compact minimal totally real submanifold M in a complex projective space \({\mathbb{C}}P^ n(c)\) of holomorphic curvature c. For any \(u\in TM\), set \(\delta (u)=| \sigma (u,u)|^ 2.\) We prove that if \(\delta\) (u)\(\leq c/8\) for any unit vector \(u\in TM\), then either \(\delta\) (u)\(\equiv 0\) (i.e., M is totally geodesic) or \(\delta\) (u)\(\equiv c/8\). All n-dimensional compact minimal totally real submanifolds of \({\mathbb{C}}P^ n(c)\) satisfying \(\delta\) (u)\(\equiv c/8\) are determined. A pinching constant for the square norm of the second fundamental form of a minimal totally real submanifold in \({\mathbb{C}}P^ n(c)\) obtained by Chen and Ogiue is improved and a simple proof of Gauchman's pinching theorem for minimal submanifolds in a sphere is also given.