## 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

### MSC:

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