Classification of Lagrangian surfaces of curvature ɛ in non-flat Lorentzian complex space form % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8WrFz0dbbf9q8 % WrFfeuY-Hhbbf9v8qqaqFr0dd9qqFj0dXdbba91qpepGe9FjuP0-is % 0dXdbba9pGe9xq-Jbba9suk9fr-xfr-xfrpeWZqaaeaabiGaciaaca % qabeaadaabauaaaOqaaiqbd2eanzaaiaWaa0baaSqaaiabigdaXaqa % aiabikdaYaaakiabcIcaOiabisda0eXafv3ySLgzGmvETj2BSbacgi % Gae8xTduMaeiykaKcaaa!4CAA! $$ \tilde M_1^2 (4\varepsilon ) $$ )

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.