The Moutard transformation of two-dimensional Dirac operators and the Mobius geometry

TL;DR

This paper explores how the Moutard transformation relates to the Mobius inversion of surfaces in three-dimensional space, revealing a geometric interpretation of the transformation in terms of Dirac operators.

Contribution

It establishes a connection between the Moutard transformation of Dirac operators and Mobius geometry, providing a geometric understanding of surface inversion effects.

Findings

The Moutard transformation corresponds to Mobius inversion of surfaces.

It maps the potential of a surface to the potential of its inverted image.

The work links Dirac operator transformations with classical geometric transformations.

Abstract

We describe the action of the (Mobius) inversion on the data of the Weierstrass representation of surfaces in the three-space and show that the Moutard transformation of two-dimensional Dirac operators has a geometrical meaning: it maps the potential $U$ of a surface $S$ into the potential of its inversion.