Complete Lagrangian ideal δ(2) submanifolds in the complex projective space

摘要

In the 1990s B.Y. Chen introduced a sequence of curvature invariants, constructed using appropriate sectional curvatures of the manifold. Each of these invariants is used to obtain an optimal lower bound for the length of the mean curvature vector for an immersion in a real space form. This upper bound remains valid for Lagrangian immersions in complex space forms. However in that setting it is no longer optimal. Recently B. Y. Chen, F. Dillen, J. Van der Veken and the author obtained an improved upper bound for Lagrangian immersions which is optimal. A Lagrangian submanifold is called δ-ideal if at every point it realises equality in this inequality. Besides the scalar curvature, the easiest and most studied of these invariants is the so called δ(2) invariant. In this paper we study complete Lagrangian δ(2)-ideal submanifolds in the complex projective space. We obtain that such submanifolds can only occur in dimension 3.