Lagrangian submanifolds attaining equality in the improved Chen's inequality

Abstract

In [7] Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of $\mathbb CP^n(4)$. For minimal submanifolds this inequality coincides with a previous version proved in [5]. We consider here those non-minimal $3$-dimensional Lagrangian submanifolds in $\mathbb CP^3 (4)$ attaining at all points equality in the improved Chen inequality. We show how all such submanifolds may be obtained starting from a minimal Lagrangian surface in $\mathbb CP^2(4)$.