Classification of Lagrangian Surfaces of Curvature ε in Non-flat Lorentzian Complex Space Form (M)over-tilde12 (4ε)

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form (M) over tilde (2)(1) (4 epsilon) of constant holomorphic sectional curvature 4 epsilon is of constant curvature epsilon. A natural question is "Besides totally geodesic ones how many Lagrangian surfaces of constant curvature epsilon in (M) over tilde (2)(1) (4 epsilon) are there?" In an earlier paper an answer to this question was obtained for the case epsilon = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case epsilon not equal 0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature epsilon in (M) over tilde (2)(1) (4 epsilon) with epsilon not equal 0. Conversely, every Lagrangian surface of curvature epsilon not equal 0 in (M) over tilde (2)(1) (4 epsilon) is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.