A duality theorem for Willmore surfaces. (English) Zbl 0555.53002

A compact immersed surface in \({\mathbb{R}}^ 3\) is called a Willmore surface if it is a critical point for the functional \(\int H^ 2dA\). One special class of Willmore surfaces is obtained as follows: Start with a complete minimal surface M in \({\mathbb{R}}^ 3\) with finite total curvature having only planar ends. Then a suitable inversion takes M into a compact Willmore surface. The author proves that in fact all Willmore immersions \(f: S^ 2\to {\mathbb{R}}^ 3\) are obtained in this way. This reduces the classification of all Willmore spheres in \({\mathbb{R}}^ 3\) to a problem in algebraic geometry.
Reviewer: U.Pinkall


53A05 Surfaces in Euclidean and related spaces
53A30 Conformal differential geometry (MSC2010)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text: DOI