Willmore surfaces in the four-sphere
The main subject of the paper under review is the classification of the Willmore immersions f: \(S^ 2\to S^ 4\). As in the three-dimensional case the Willmore immersions are defined as the extremals of the conformal area. In the first three sections of the article the structure formulas for m-dimensional oriented submanifolds of the Möbius space \(S^ m\) are derived. Section 4 contains the general setting for Willmore surfaces f: \(M^ 2\to S^ n\); especially it is proved that the set of umbilic points of a non totally umbilic Willmore surface is a closed set without interior points. In the remaining sections 5-7 the author considers Willmore immersions of \(S^ 2\) into \(S^ 4\) which are not contained in an \(S^ 3\). He defines a conformal transform \(\hat f:\) \(S^ 2\to S^ 4\) for such immersions in an invariant way; a Willmore immersion f is said to be of type I, if its conformal transform \(\hat f\) is constant, and of type II, if its conformal Gauss map is holomorphic or anti-holomorphic. The author proves that a linearly full Willmore immersion f: \(S^ 2\to S^ 4\) which is not of type II must be of type I. With any Willmore immersion of type I the author associates a complete minimal immersion with finite total curvature by stereographic projection of \(S^ 4\) from the image point \(\hat f(S^ 2)\). This is used to reduce the classification of these immersions to the classification of meromorphic immersions \(h: S^ 2\to {\mathbb{C}}^ 4\) (Weierstrass representation formula). The value \(W(f)\) of the Willmore functional of f is given by \(W(f)=4\pi(d+1),\) where d denotes the number of the ends of the associated minimal surface. Naturally, the Willmore immersions of type II lead to the classification of certain holomorphic maps. For the value \(W(f)\) of these immersions the author proves \(W(f)=4\pi m,\) where m is a positive integer.
scientific article; zbMATH DE number 4157424