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


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


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