3 - Application: Isothermic and Willmore surfaces

Summary

As a second application of the classical model of Möbius geometry we will discuss some aspects of isothermic surfaces and Willmore surfaces. As unrelated as the two surface classes seem to be at first, they occur as two (the two nontrivial) solutions of the same problem that we will refer to) as “Blaschke's problem,” since it arises in [29]. The contents of this chapter are loosely arranged around the solution of this problem, collecting many of Blaschke's key results in surface theory in the conformal) 3-sphere. Isothermic as well as Willmore surfaces are a field of current interest after they have been neglected for quite some time — Willmore surfaces even seem to have been forgotten and had a different name in classical times. The study of both surface classes traces back to the first half of the nineteenth century; however, the origins remain obscure to the author. It is also true for both surface classes that physical considerations play a role — as indicated for isothermic surfaces by their name.

“Willmore surfaces” were apparently (re-) discovered at least three times.

First, in the early nineteenth century, Germain [126] (cf., [127]) studied “elastic surfaces” — analogous to the elastic curves discussed in the works of Bernoulli and Euler [185].