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

Abstract

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