4 - Willmore Surfaces

Summary

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.