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 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 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 CP 2 (4).

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

“…For Q 2 we have τ = 8, inf K r = 0 and δ r 4 = 8. Thus Q 2 is a non-totally geodesic complex surface satisfying the equality case of (59). Thus Theorem 52 implies that Q 2 is a strongly minimal Kaehler surface in CP 3 (4).…”

“…Lagrangian submanifolds in complex space forms satisfying the improved inequalities (33) for δ(2) have been studied extensively in [7,[52][53][54][57][58][59][60][61][62][63] among others. For δ(2, 2)-ideal Lagrangian submanifolds, we refer to [16,18,64].…”

Lagrangian submanifolds in complex space forms satisfying equality in the optimal inequality involving $$\delta (2,\ldots ,2)$$

“…Theorem 1.3 Bolton and Vrancken (2007) Let M be a 3-dimensional non-minimal δ(2)-ideal Lagrangian submanifold of CP 3 (4). Then there is a minimal Lagrangian surfaceW in CP 2 (4) such that M can be locally written as [E 0 ], where…”

Classification of δ(2,n− 2)-ideal Lagrangian submanifolds in n-dimensional complex space forms

“…Case (b): ω 3 1 (E 1 ) = 0. In this case, the choice of E 3 we made before becomes arbitrary and thus equation (5.21) gives ω 3 1 (E 1 ) = µω 3 2 (E 1 ) = 0 for arbitrary E 3 . Thus, we have…”