Hamiltonian-stationary Lagrangian surfaces of constant curvature ε in complex space form (M)over-tilde2(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 epsilon in a complex space form (M) over tilde (2)(4 epsilon). Then, by applying the traveling wave solutions, we construct families of type II Hamiltonian-stationary Lagrangian surfaces of constant curvature epsilon in complex space forms (M) over tilde (2)(4 epsilon) 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) over tilde (n)(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 epsilon = 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. (c) 2009 Elsevier Ltd. All rights reserved.