Hamiltonian-stationary Lagrangian surfaces of constant curvature εε in complex space form M̃2(4ε)

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 εε in a complex space form M̃2(4ε). Then, by applying the traveling wave solutions, we construct families of type II Hamiltonian-stationary Lagrangian surfaces of constant curvature εε in complex space forms M̃2(4ε) via an effective method of [B.Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Lagrangian isometric immersions of a real-space-form Mn(c)Mn(c) into a complex-space-form M̃n(4c), 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=0ε=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.