8 - Constrained Willmore Surfaces and the Isothermic Surface Condition

Summary

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.