Hamiltonian stationary Lagrangian surfaces of constant Gaussian curvature in complex space forms

Let $M$ be a connected Hamiltonian stationary Lagrangian surface of constant Gaussian curvature $K$ in a two-dimensional complex space form ${\text{Mtilde}}^2(4\epsilon )$ of constant holomorphic sectional curvature $4\epsilon$, and let $H$ denote its mean curvature vector field. We prove that either $K=\epsilon$ or $K=0\mathrm{\ne }\epsilon$ and $\mathrm{\mid }H\mathrm{\mid }$ is constant. In particular, if $\mathrm{\mid }H\mathrm{\mid }$ is non-constant, then $K=\epsilon$.