Darboux transforms and simple factor dressing of constant mean curvature surfaces

Reference:

Burstall, F. E., Dorfmeister, J. F., Leschke, K. and Quintino, A. C., 2013. Darboux transforms and simple factor dressing of constant mean curvature surfaces. Manuscripta Mathematica, 140 (1-2), pp. 213-236.

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Official URL:

http://dx.doi.org/10.1007/s00229-012-0537-2

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.

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