This chapter is dedicated to the very special class of constant mean curvature surfaces. A classical result by Thomsen characterizes isothermic Willmore surfaces in 3-space as minimal surfaces in some 3-dimensional space-form. Constant mean curvature surfaces in 3-dimensional space-forms are examples of constrained Willmore surfaces, characterized by the existence of some conserved quantity. Both constrained Willmore spectral deformation and Bäcklund transformation prove to preserve the existence of such a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, in the latter case. The class of constant mean curvature surfaces in 3-dimensional space-forms lies, in this way, at the intersection of several integrable geometries, with classical transformations of its own, as well as transformations as a class of constrained Willmore surfaces, together with transformations as a subclass of the class of isothermic surfaces, as we explore in this chapter. Constrained Willmore transformation proves to be unifying to this rich transformation theory, as we shall conclude.

This chapter is dedicated to the special case of surfaces in 4-space. Our approach is quaternionic, based on the model of the conformal 4-sphere on the quaternionic projective space. We extend the Darboux transformation of Willmore surfaces in 4-space presented by Burstall–Ferus–Leschke–Pedit–Pinkall, based on the solution of a Riccati equation, to a transformation of constrained Willmore surfaces in 4-space into new ones. We prove that non-trivial Darboux transformation of constrained Willmore surfaces in 4-space can be obtained as a particular case of Bäcklund transformation. This Darboux transformation of constrained Willmore surfaces displays a striking similarity with the description of isothermic Darboux transformation of constant mean curvature surfaces in Euclidean 3-space presented by Hertrich-Jeromin−Pedit, which, in fact, proves to be obtainable as a particular case of constrained Willmore Bäcklund transformation.