Classification of Lagrangian surfaces of constant curvature in complex projective plane

Abstract

From Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classification of Lagrangian immersions of real space forms into complex space forms. The purpose of this article is thus to classify Lagrangian surfaces of constant curvature in complex projective plane C P 2 . Our main result states that there are 29 families of Lagrangian surfaces of constant curvature in C P 2 . Twenty-two of the 29 families are constructed via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in C P 2 are obtained from the 29 families. As an immediate by-product, many interesting new examples of Lagrangian surfaces of constant curvature in C P 2 are discovered.