Lagrangian submanifolds attaining equality in the improved Chen's inequality

Abstract

Recently Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of $\mathbb{C}{P}^n(4)$. For minimal submanifolds this inequality coincides with the original previously proved version. We consider here those non minimal 3-dimensional Lagrangian submanifolds in $\mathbb{C}{P}^3(4)$ 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 $\mathbb{C}{P}^2(4)$ and a suitable horizontal curve in $S^3(1)$.