Darboux Transforms and Spectral Curves of Hamiltonian Stationary Lagrangian Tori

The multiplier spectral curve of a conformal torus f : T 2 → S in the 4– sphere is essentially [3] given by all Darboux transforms of f . In the particular case when the conformal immersion is a Hamiltonian stationary torus f : T 2 → R in Euclidean 4–space, the left normal N : M → S of f is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of N , as defined in [16]. We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces of genus zero. Moreover, we prove that all Darboux transforms, which arise from generic points on the spectral curve, are Hamiltonian stationary whereas we also provide examples of Darboux transforms which are not even Lagrangian.