Lagrangian submanifolds in complex space forms satisfying an improved equality involving δ(2,2)

Abstract

It was proved in [8, 9] that every Lagrangian submanifold M of a complex space form M˜ 5 (4c) of constant holomorphic sectional curvature 4c satisfies the following optimal inequality: δ(2, 2) ≤ 25 4 H 2 + 8c, (A) where H 2 is the squared mean curvature and δ(2, 2) is a δ-invariant on M introduced by the first author. This optimal inequality improves a special case of an earlier inequality obtained in [B.-Y. Chen, Japan. J. Math. 26 (2000), 105-127]. The main purpose of this paper is to classify Lagrangian submanifolds of M˜ 5 (4c) satisfying the equality case of the improved inequality (A).Fundación Cámara (Universidad de Sevilla)National Natural Science Foundation of Chin