Hamiltonian stationary Lagrangian tori contained in a hypersphere

TL;DR

This paper explicitly constructs all Hamiltonian stationary Lagrangian tori contained in a hypersphere in complex Euclidean space, showing they are homogeneous tori, using spectral curve methods and quaternionic connections.

Contribution

It provides a complete explicit construction of Hamiltonian stationary Lagrangian tori in a hypersphere, linking them to homogeneous tori via spectral curve analysis.

Findings

All such tori are homogeneous.

Construction uses spectral curves and quaternionic connections.

Explicit parametrizations are obtained.

Abstract

The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori which is contained in a hypersphere in the complex Euclidean plane are constructed explicitly. Then it is shown that they are homogeneous tori. For the construction, flat quaternionic connections of Hamiltonian stationary Lagrangian tori are considered and a spectral curve of an associated family of them is used.