6 - Transformations of Generalized Harmonic Bundles and Constrained Willmore Surfaces

Summary

Willmore surfaces in space-forms are characterized by the harmonicity of the mean curvature sphere congruence. In this chapter, we introduce the concept of perturbed harmonicity of a bundle, which will apply to the mean curvature sphere congruence to provide a characterization of constrained Willmore surfaces in space-forms. A generalization of the well-developed integrable systems theory of harmonic maps emerges. The starting point is a zero-curvature characterization of constrained Willmore surfaces, due to Burstall–Calderbank, which we derive in this chapter. Constrained Willmore surfaces come equipped with a family of flat metric connections. We then define a spectral deformation of perturbed harmonic bundles, by the action of a loop of flat metric connections, and Bäcklund transformations, defined by the application of a version of the Terng–Uhlenbeck dressing action by simple factors. Transformations on the level of perturbed harmonic bundles prove to give rise to transformations on the level of constrained Willmore surfaces, via the mean curvature sphere congruence. We establish a permutability between spectral deformation and Bäcklund transformation and show that all these transformations corresponding to the zero Lagrange multiplier preserve the class of Willmore surfaces. We define, more generally, transformations of complexified surfaces and prove that, for special choices of parameters, both spectral deformation and Bäcklund transformation preserve reality conditions.