  In this paper, we give some simple conditions under which a Hamiltonian
stationary Lagrangian submanifold of a K\"ahler-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 $L^p$ space when the
ambient space is a simply connected complex space form.
