Deformations of surfaces preserving conformal or similarity invariants

Constant mean curvature surfaces (abbriviated as CMC surfaces) in the space forms are typical examples of isothermic surfaces. Bonnet showed that every constant mean curvature surface admits a one-parameter family of isometric deformations preserving the mean curvature. A surface which admits such a family of deformations is called a Bonnet surface. Both the isothermic surfaces and Bonnet surfaces are regarded as geometric generalizations of constant mean curvature surfaces. On the other hand, from the viewpoint of integrable system theory, Bobenko introduced the notion of surface with harmonic inverse mean curvature (HIMC surface, in short) in Euclidean 3-space R3. The first named author extended the notion of HIMC surface in R3 to that of 3-dimensional space forms [19]. HIMC surfaces have deformation families (associated family) which preserve the conformal structure of the surface and the harmonicity of the reciprocal mean curvature. There exist local bijective conformal correspondences between HIMC surfaces in different space forms. It should be remarked that while Bonnet surfaces are isothermic, HIMC surfaces are not necessarily isothermic. In fact, the associated family of a Bonnet surface or a HIMC surface preserves the Möbius metric, while