A Method Using Differential Equations for Characterizing Warped Product Lagrangian Submanifolds in Complex Space Forms

In this paper, we establish new inequalities for the length of the second fundamental form of an n-dimensional compact warped product Lagrangian submanifold Qn in a complex space form Qcn(4c). These estimates are expressed in terms of the Laplacian and the gradient of the warping function, together with a positive lower bound on the Ricci curvature. To the best of our knowledge, this provides the first nontrivial result of this type obtained without imposing higher-order curvature assumptions. The characterization results are derived using Obata-type and Tashiro-type differential equations on suitable Riemannian manifolds. By combining the Bochner formula with second-order differential equation techniques, we further characterize the warped product Qn via the first nonzero eigenvalue associated with the warping function. In particular, under appropriate conditions on the second fundamental form, we show that Qn is isometric to either the Euclidean space Rn or the standard sphere Sn.