Special isothermic surfaces of type \(d\) (Q2880403)

From authors' abstract: The special isothermic surfaces, discovered by Darboux in connection with deformations of quadrics, admit a simple explanation via the gauge-theoretic approach to isothermic surfaces. We find that they fit into a hierarchy of special classes of isothermic surfaces and extend the theory to arbitrary codimension.NEWLINENEWLINE(From the author's Introduction) The purpose in this paper is to show that Darboux's special isothermic surfaces have a simple explanation in terms of the integrable systems approach to isothermic surfaces. As fruits of this analysis, we shall generalize the classical theory to arbitrary codimension and see that the special isothermic surfaces are a particular case of a hierarchy of natural classes of isothermic surfaces filtered by an interger \(d\). In codimension \(1\), the first three of these classes are the maps to \(S^2\) (\(d=0\)); the surfaces of constant mean curvature (\(d=1\)) and the special isothermic surfaces described about (\(d=2\)).NEWLINENEWLINEBeautiful examples and pictures of soliton versions of these surfaces are given in [\textit{E. Musso} and \textit{D. Nicolodi}, Contemp. Math. 288, 129--148 (2001; Zbl 1076.53004)].