Dirac tori

We consider conformal immersions f:T2→R3 with the property that H2f⁎gR3 is a flat metric. These so called Dirac tori have the property that their Willmore energy is uniformly distributed over the surface and can be obtained using spin transformations of the plane by eigenvectors of the standard Dirac operator for a fixed eigenvalue. We classify Dirac tori and determine the conformal classes realized by them. Note that the spinors of Dirac tori satisfy the same system of PDE's as the differential of Hamiltonian stationary Lagrangian tori in R4. These were classified in [4].