Darboux transforms and simple factor dressing of constant mean curvature surfaces

Details Darboux transforms and simple factor dressing of constant mean curvature surfaces Darboux transforms and simple factor dressing of constant mean curvature surfaces

2010-09-27

arxiv.org/abs/1009.5274v1

Mathematics

Differential Geometry

20 pages

Scientific paper

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.

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