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.

A fundamental construction in conformal geometry of surfaces is the mean curvature sphere congruence, or central sphere congruence, the bundle of 2-spheres tangent to the surface and sharing with it mean curvature vector at each point (although the mean curvature vector is not conformally invariant, under a conformal change of the metric, it changes in the same way for the surface and the osculating 2-sphere). The concept has its origin in the nineteenth century, with the introduction of the mean curvature sphere of a surface at a point, by Germain. By the turn of the century, the family of the mean curvature spheres of a surface was known as the central sphere congruence, cf. Blaschke. Nowadays, after Bryant's paper, it goes as well by the name of conformal Gauss map. We introduce it in this chapter, starting by recalling some fundamental concepts in Riemannian Geometry.