Construction of Hamiltonian-stationary Lagrangian submanifolds of constant curvature $\varepsilon$ in complex space forms $\tilde M^n(4\varepsilon)

Abstract: Lagrangian submanifolds of a Kaehler manifold are called Hamiltonian-stationary (or $H$-stationary for short) if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In [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 $\tilde{M}^{n}(4c)$, Math. Proc. Cambridge Philo. Soc. 124 (1998), 107-125], an effective method to constructing Lagrangian submanifolds of constant curvature $\varepsilon$ in complex space form $M^n(4\varepsilon)$ was introduced. In this article we survey recent results on construction of Hamiltonian-stationary Lagrangian submanifolds in complex space forms using this method.