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Constant mean curvature surfaces at the intersection of integrable geometries. (English) Zbl 1382.53005

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 12th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 4–9, 2010. Sofia: Bulgarian Academy of Sciences. Geometry, Integrability and Quantization, 305-319 (2011).
Summary: The constant mean curvature surfaces in three-dimensional space-forms are examples of isothermic constrained Willmore surfaces, characterized as the constrained Willmore surfaces in three-space admitting a conserved quantity. Both constrained Willmore spectral deformation and constrained Willmore Bäcklund transformation preserve the existence of a conserved quantity. The class of constant mean curvature surfaces in three-dimensional space-forms lies, in this way, at the intersection of several integrable geometries, with classical transformations of its own, as well as constrained Willmore transformations and transformations as a class of isothermic surfaces. Constrained Willmore transformation is expected to be unifying to this rich transformation theory.
For the entire collection see [Zbl 1245.00049].

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
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