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Hopf tori in $$S^ 3$$. (English) Zbl 0585.53051
Let $$\pi$$ : $$S^ 3\to S^ 2$$ be the Hopf fibration. The author proves that the inverse image of any closed curve on $$S^ 2$$ is an immersed torus in $$S^ 3$$. This torus is called a Hopf torus. Using Hopf tori, he obtains: Every compact Riemann surface of genus one can be conformally embedded in the unit sphere $$S^ 3\subset R^ 4$$ as a flat torus. The embedding can be chosen as the intersection of $$S^ 3$$ with a quartic hypersurface in $$R^ 4$$. As a corollary, he gets: Every compact Riemann surface of genus one can be conformally embedded in $$R^ 3$$ as an algebraic surface of degree eight.
A surface in $$R^ 3$$ is called a Willmore surface if it is an extremal surface for the variational functional $$\int H^ 2dA$$ (H the mean curvature). The only known examples of compact Willmore surfaces are the stereographic projections of compact minimal surfaces in $$S^ 3$$. By showing that Hopf tori of some kind are Willmore surfaces, the author obtains an infinite series of compact Willmore surfaces that do not stem from minimal surfaces in $$S^ 3$$.
Reviewer: T.Ishihara

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A05 Surfaces in Euclidean and related spaces 57R40 Embeddings in differential topology
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##### References:
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