Hamiltonian stationary Lagrangian surfaces with non-negative Gaussian curvature in Kähler-Einstein surfaces

In this paper, we give some simple conditions under which a Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold must have a Euclidean factor or be a fiber bundle over a circle. We also characterize the Hamiltonian stationary Lagrangian surfaces whose Gaussian curvature is non-negative and whose mean curvature vector is in some L p L^p space when the ambient space is a simply connected complex space form.