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

TL;DR

This paper surveys recent methods for constructing Hamiltonian-stationary Lagrangian submanifolds of constant curvature in complex space forms, highlighting advances in geometric analysis and variational techniques.

Contribution

It reviews recent developments in constructing Hamiltonian-stationary Lagrangian submanifolds using a method introduced in prior work, expanding understanding of their geometric properties.

Findings

Effective construction methods for Hamiltonian-stationary Lagrangian submanifolds.

New examples of such submanifolds in complex space forms.

Enhanced understanding of their geometric and variational properties.

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.