Hamiltonian Stationary Tori in the Complex Projective Plane

Hamiltonian stationary Lagrangian surfaces are Lagrangian surfaces of a given four-dimensional manifold endowed with a symplectic and a Riemannian structure, which are critical points of the area functional with respect to a particular class of infinitesimal variations preserving the Lagrangian constraint: the compactly supported Hamiltonian vector fields. The Euler–Lagrange equations of this variational problem are highly simplified when we assume that the ambient manifold N is Kähler. In that case we can make sense of a Lagrangian angle function β along any simply-connected Lagrangian submanifold Σ ⊂ N (uniquely defined up to the addition of a constant). And as shown in [19] the mean curvature vector of the submanifold is then ~ H = J ∇β, where J is the complex structure on N and ∇β is the gradient of β along Σ. It turns out that Σ is Hamiltonian stationary if and only if β is a harmonic function on Σ.