Darboux transforms and simple factor dressing of constant mean curvature surfaces

Francis E Burstall, J F Dorfmeister, K Leschke, Aurea C Quintino

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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.
LanguageEnglish
Pages213-236
Number of pages14
JournalManuscripta Mathematica
Volume140
Issue number1-2
Early online date22 Feb 2012
DOIs
StatusPublished - Jan 2013

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Constant Mean Curvature
Harmonic Maps
Gauss Map
Darboux Transformation
Transform
Riemann Surface
Generalise

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Darboux transforms and simple factor dressing of constant mean curvature surfaces. / Burstall, Francis E; Dorfmeister, J F; Leschke, K; Quintino, Aurea C.

In: Manuscripta Mathematica, Vol. 140, No. 1-2, 01.2013, p. 213-236.

Research output: Contribution to journalArticle

Burstall FE, Dorfmeister JF, Leschke K, Quintino AC. Darboux transforms and simple factor dressing of constant mean curvature surfaces. Manuscripta Mathematica. 2013 Jan;140(1-2):213-236. Available from, DOI: 10.1007/s00229-012-0537-2
Burstall, Francis E ; Dorfmeister, J F ; Leschke, K ; Quintino, Aurea C. / Darboux transforms and simple factor dressing of constant mean curvature surfaces. In: Manuscripta Mathematica. 2013 ; Vol. 140, No. 1-2. pp. 213-236
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