Abstract
We define a transformation on harmonic maps \({N:\,M \to S^2}\) from a Riemann surface M into the 2-sphere which depends on a parameter \({\mu \in \mathbb{C}_*}\), the so-called μ-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface \({f:\,M \to \mathbb{R}^3}\) and μ is real, the Darboux transformation of −N is the Gauss map of a classical Darboux transform of f. More generally, for all parameter \({\mu \in \mathbb{C}_*}\) 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 (Hertrich-Jeromin and Pedit Doc Math J DMV 2:313–333, 1997; Burstall Integrable systems, geometry, and topology, 2006; Inoguchi and Kobayashi Int J Math 16(2):101–110, 2005): every μ-Darboux transform is a simple factor dressing, and vice versa.
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Burstall, F.E., Dorfmeister, J.F., Leschke, K. et al. Darboux transforms and simple factor dressing of constant mean curvature surfaces. manuscripta math. 140, 213–236 (2013). https://doi.org/10.1007/s00229-012-0537-2
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DOI: https://doi.org/10.1007/s00229-012-0537-2