Bianchi-Bäcklund transforms and dressing actions, revisited. (English) Zbl 1201.53004
The Bianchi-Bäcklund transform is a complex surface assigned to a surface of constant Gauss curvature \(1\). In order to obtain a new CGC surface from the given one, one has to do two successive Bianchi-Bäcklund transformations matched in a particular way. Such a Bianchi-Bäcklund procedure amounts to dressing the extended framing associated to the (harmonic) Gauss map by a certain dressing matrix, as was shown in [A. Mahler, Bianchi-Bäcklund and dressing transformations on constant mean curvature surfaces. PhD thesis, University of Toledo (2002)].
The current paper follows a different approach to this result. For certain “simple factors”, the dressing action can be computed explicitly; and each single Bianchi-Bäcklund transform corresponds to a dressing action for a certain simple factor.
Moreover, some relation to the sinh-Gordon equation can be exploited to produce all Bianchi-Bäcklund transformations from more simple ones.
The current paper follows a different approach to this result. For certain “simple factors”, the dressing action can be computed explicitly; and each single Bianchi-Bäcklund transform corresponds to a dressing action for a certain simple factor.
Moreover, some relation to the sinh-Gordon equation can be exploited to produce all Bianchi-Bäcklund transformations from more simple ones.
Reviewer: Andreas Gastel (Duisburg)
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
| 58E20 | Harmonic maps, etc. |
| 53C43 | Differential geometric aspects of harmonic maps |
Keywords:
constant Gauss curvature surfaces; harmonic maps; dressing actions; Bianchi-Bäcklund transformsReferences:
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