In this chapter, we introduce the concept of conserved quantity of a constrained Willmore surface in a space-form. In codimension 1, the existence of a conserved quantity for a constrained Willmore surface characterizes the constancy of the mean curvature of the surface in some space-form. In codimension 2, surfaces with holomorphic mean curvature vector in some space-form are examples of constrained Willmore surfaces admitting a conserved quantity. Both constrained Willmore spectral deformation and Bäcklund transformation preserve the existence of a conserved quantity of a constrained Willmore surface, 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, as we shall prove.

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.