Hamiltonian-stationary Lagrangian surfaces of constant curvature [formula omitted] in complex space form [formula omitted

Abstract

A Lagrangian surface in a Kaehler manifold is called Hamiltonian-stationary if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In the first part of this article, we study the traveling wave solutions of an over-determined PDE system arising from a study of Hamiltonian-stationary Lagrangian surfaces of constant curvature ε in a complex space form M ̃ 2 ( 4 ε ) . Then, by applying the traveling wave solutions, we construct families of type II Hamiltonian-stationary Lagrangian surfaces of constant curvature ε in complex space forms M ̃ 2 ( 4 ε ) via an effective method of [B.Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Lagrangian isometric immersions of a real-space-form M n ( c ) into a complex-space-form M ̃ n ( 4 c ) , Math. Proc. Cambridge Philo. Soc. 124 (1998) 107–125]. In the second part, we are able to completely solve the over-determined PDE system for the case ε = 0 and to determine their corresponding Hamiltonian-stationary Lagrangian surfaces. As an immediate by-product, some interesting new families of Hamiltonian-stationary Lagrangian surfaces of constant curvature are obtained.