Classification of Lagrangian surfaces of curvature ɛ in non-flat Lorentzian complex space form $\tilde M_1^2 (4\varepsilon )$)

Abstract

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form $\tilde M_1^2 (4\varepsilon )$ of constant holomorphic sectional curvature 4ɛ is of constant curvature ɛ. A natural question is “Besides totally geodesic ones how many Lagrangian surfaces of constant curvature ɛ in $\tilde M_1^2 (4\varepsilon )$ are there?” In an earlier paper an answer to this question was obtained for the case ɛ = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ɛ ≠ 0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ɛ in $\tilde M_1^2 (4\varepsilon )$ with ɛ ≠ 0. Conversely, every Lagrangian surface of curvature ɛ ≠ 0 in $\tilde M_1^2 (4\varepsilon )$ is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.