Intrinsic and extrinsic structures of Lagrangian surfaces in complex space forms

Abstract

Lagrangian $H$-umbilical submanifolds introduced in $[1, 2]$ can be regarded as the simplest Lagrangian submanifolds in Kaehler manifolds next to totally geodesic ones. It was proved in [1] that Lagrangian $H$-umbilical submanifolds of dimension $\geq 3$ in complex Euclidean spaces are complex extensors, Lagrangian pseudo-spheres, and flat Lagrangian $H$-umbilical submanifolds. Lagrangian $H$-umbilical submanifolds of dimension $\geq 3$ in non-flat complex space forms are classified in [2]. In this paper we deal with the remaining case; namely, non-totally geodesic Lagrangian $H$-umbilical surfaces in complex space forms. Such Lagrangian surfaces are characterized by a very simple property; namely, $JH$ is an eigenvector of the shape operator $A_{H}, where $H$ is the mean curvature vector field. The main operator of this paper is to determine both the intrinsic and the extrinsic structures of Lagrangian $H$-umbilical surfaces.