The classification of Hamiltonian stationary Lagrangian tori in the complex projective plane by their spectral data

TL;DR

This paper classifies Hamiltonian stationary Lagrangian tori in the complex projective plane using spectral data, extending integrable systems methods to encode these immersions and analyze their structure.

Contribution

It provides a detailed construction and classification framework for Hamiltonian stationary Lagrangian tori via spectral data, incorporating a quadratic loop of flat connections.

Findings

Classification of tori via spectral data

Extension of integrable systems techniques

Handling quadratic loop of flat connections

Abstract

It is known that all weakly conformal Hamiltonian stationary Lagrangian immersions of tori in the complex projective plane may be constructed by methods from integrable systems theory. This article describes the precise details of a construction which leads to a form of classification. The immersion is encoded as spectral data in a similar manner to the case of minimal Lagrangian tori in the complex projective plane, but the details require a careful treatment of both the "dressing construction" and the spectral data to deal with a loop of flat connexions which is quadratic in the loop parameter.