Darboux transforms and simple factor dressing of constant mean curvature surfaces

TL;DR

This paper introduces a new transformation called mu-Darboux for harmonic maps into the 2-sphere, linking it to classical Darboux transforms and simple factor dressing of constant mean curvature surfaces, thus unifying these concepts.

Contribution

It establishes that mu-Darboux transformations are equivalent to simple factor dressing for constant mean curvature surfaces, generalizing classical Darboux transform results.

Findings

mu-Darboux transformation coincides with simple factor dressing

Every mu-Darboux transform is a simple factor dressing

Generalizes classical Darboux transforms for CMC surfaces

Abstract

We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N is the Gauss map of a classical Darboux transform f. More generally, for all complex parameter the transformation on the harmonic Gauss map of f is induced by a (generalized) Darboux transformation on f. We show that this operation on harmonic maps coincides with simple factor dressing, and thus generalize results on classical Darboux transforms of constant mean curvature surfaces: every mu-Darboux transform is a simple factor dressing, and vice versa.