Infinitesimal Darboux transformations of the spectral curves of tori in the four-space

TL;DR

This paper investigates how conformal transformations affect the Dirac operator associated with tori in four-dimensional space, revealing flows described by nonlinear integrable systems that mostly preserve spectral properties but can alter the spectral curve.

Contribution

It introduces a new analysis of conformal transformation flows on the Dirac operator's potential, showing their integrable structure and impact on spectral curves of tori in four-space.

Findings

Flows preserve Floquet multipliers but not the spectral curve in general.

The flows are governed by nonlinear Melnikov-type systems.

In certain cases, the flows are described by integrable systems on tori.

Abstract

We study the action of conformal transformations of the ambient space on the Dirac operator coming into the Weierstrass (or spinor) representation of a torus in the Euclidean four-space. It is showed that such an action generates a flow acting on the potential of the operator, that this flow is described by a nonlinear system of the Melnikov type and that it preserves the Floquet multipliers of the Dirac operator with double-periodic potential. However this flow is only almost isospectral since it does not preserve the spectral curve in general and its action may result in adding or removing multiple points corresponding to the same multipliers. We demonstrate that in some important geometrical examples after reparameterization of the temporary variable such flows are governed by integrable systems on whiskered tori.