Hamiltonian-minimal Lagrangian submanifold

A Lagrangian submanifold in a Kahler manifold is called Hamiltonian-minimal ($H$-minimal) if the variations of its volume along the Hamiltonian fields are zero. The notion of Hamiltonian minimality was introducted by Oh in connection with the Arnold conjecture on the number of fixed points of a Hamiltonian symplectomorphism.
In the talk I will discuss the properties of $H$-minimal submanifolds, and describe the known examples.