9 - The Special Case of Surfaces in 4-Space

Summary

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.