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Simple factor dressing and the López-Ros deformation of minimal surfaces in Euclidean 3-space. (English) Zbl 1428.58014

Authors’ abstract: The paper investigates a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. The authors study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known López-Ros deformation of minimal surfaces is a special case of this transformation. They express the simple factor dressing and the López-Ros deformation explicitly in terms of the minimal surface and its conjugate surface. In particular, the periods and end behaviour of the simple factor dressing can be controlled. This allows to construct new examples of doubly-periodic minimal surfaces arising as simple factor dressings of Scherk’s first surface.

MSC:

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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