Among the classes of Riemannian submanifolds, there is that of Willmore surfaces, named after Thomas Willmore, although the topic had already made its appearance early in the nineteenth century, through the works of Germain and Poisson, and again in the 1920s, through the works of Blaschke and Thomsen, whose findings were forgotten and only brought to light after the increased interest on the subject motivated by the work of T. Willmore, in part due to the celebrated Willmore conjecture, now affirmed by Marques–Neves. From the early 1960s, Willmore devoted particular attention to the quest for the optimal immersion of a given closed surface into Euclidean 3-space, regarding the minimization of some natural energy, motivated by questions on the elasticity of biological membranes and the energetic cost associated with membrane-bending deformations. The Willmore energy of a surface in Euclidean 3-space is given by its total squared mean curvature. We present a manifestly conformally invariant formulation of the Willmore energy of a surface in n-dimensional space-form. Willmore surfaces are the critical points of the Willmore functional, characterized by the harmonicity of the mean curvature sphere congruence. This characterization will enable us to apply to the class of Willmore surfaces the well-developed integrable systems theory of harmonic maps.

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.

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.

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.

In this chapter, we introduce constrained Willmore surfaces, the generalization of Willmore surfaces that arises when we consider critical points of the Willmore functional with respect to infinitesimally conformal variations, rather than with respect to all variations. Constrained Willmore surfaces in space-forms constitute a Möbius invariant class of surfaces with strong links to the theory of integrable systems, which we shall explore throughout this work. The introduction of a constraint in the variational problem equips surfaces with Lagrange multipliers, as first proven by Burstall–Pedit–Pinkall, in a manifestly conformally invariant characterization of constrained Willmore surfaces in space-forms in terms of the Hopf differential and the Schwarzian derivative, which we deduce in this chapter. The class of constrained Willmore surfaces was later given yet another manifestly conformally invariant characterization, free of holomorphic charts, by Burstall–Calderbank, which we derive in this chapter from the variational problem.