Minimal Darboux transformations

TL;DR

This paper establishes a permutability theorem for transformations of isothermic surfaces, linking Darboux pairs of minimal surfaces, curved flats, and flat fronts across different geometries.

Contribution

It provides a new permutability theorem for classical transformations of isothermic surfaces, simplifying the understanding of their relations across various geometric contexts.

Findings

Derived a permutability theorem for Christoffel, Goursat, and Darboux transformations.

Established a relation between Darboux pairs of minimal surfaces, curved flats, and flat fronts.

Provided a simplified proof connecting these geometric objects.

Abstract

We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats in the 2-sphere and flat fronts in hyperbolic space.