Classification of a family of Hamiltonian-stationary Lagrangian submanifolds in C$^{n}$

Abstract

A Lagrangian submanifold in the complex Euclidean $n$-space ${\bf C}^n$ is called Hamiltonian-stationary if it is a critical point of the area functional restricted to (compactly supported) Hamiltonian variations. In this article, we classify the family of Hamiltonian-stationary Lagrangian submanifolds of ${\bf C}^n$ which are Lagrangian $H$-umbilical.