Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori

TL;DR

This paper explores the relationship between Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori, revealing their biholomorphic nature and the conditions under which Darboux transforms preserve Hamiltonian stationarity.

Contribution

It establishes a biholomorphic correspondence between the multiplier spectral curve and the eigenline spectral curve for Hamiltonian stationary tori, and characterizes Darboux transforms that maintain Hamiltonian stationarity.

Findings

Multiplier spectral curve and eigenline spectral curve are biholomorphic of genus zero.

All Darboux transforms from generic spectral curve points are Hamiltonian stationary.

Some Darboux transforms are not Lagrangian, showing diversity in transform properties.

Abstract

The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of the left normal, as defined by Hitchin. 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.