Abstract :

We study some minimization problems for Hamiltonian stationary Lagrangian surfaces in R^4. We show that the flat Lagrangian torus S^1 x S^1 minimizes the Willmore functional among Hamiltonian stationary tori of its isotopy class, which gives a new proof of the fact that it is area minimizing in the same class. Considering the Lagrangian flat cylinder as a torus in some quotient space of R^4, we show that it is also area minimizing in its isotopy class.