Hamiltonian Stationary Lagrangian Surfaces in Euclidean Four-Space and Related Minimization Problems. Part II: Some Minimization Results

Abstract

We study some minimization problems for Hamiltonian stationaryLagrangian surfaces in R4. We show that the flat Lagrangian torusS1 × S1 minimizes the Willmore functional among Hamiltonianstationary tori of its isotopy class, which gives a new proof of thefact that it is area minimizing in the same class. Considering theLagrangian flat cylinder as a torus in some quotient space R4/v Z, we show that it is also area minimizing in its isotopy class.